Abstract
Having a distance measure between quantum states satisfying the right properties is of fundamental importance in all areas of quantum information. In this work, we present a systematic study of the geometric Rényi divergence (GRD), also known as the maximal Rényi divergence, from the point of view of quantum information theory. We show that this divergence, together with its extension to channels, has many appealing structural properties, which are not satisfied by other quantum Rényi divergences. For example we prove a chain rule inequality that immediately implies the “amortization collapse” for the geometric Rényi divergence, addressing an open question by Berta et al. [Letters in Mathematical Physics 110:2277–2336, 2020, Equation (55)] in the area of quantum channel discrimination. As applications, we explore various channel capacity problems and construct new channel information measures based on the geometric Rényi divergence, sharpening the previously bestknown bounds based on the maxrelative entropy while still keeping the new bounds singleletter and efficiently computable. A plethora of examples are investigated and the improvements are evident for almost all cases.
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1 Introduction
In information theory, an imperfect communication link between a sender and a receiver is modeled as a noisy channel. The capacity of such a channel is defined as the maximum rate at which information can be transmitted through the channel reliably. This quantity establishes the ultimate boundary between communication rates that are achievable in principle by a channel coding scheme and those that are not. A remarkable result by Shannon [1] states that the capacity of a classical channel is equal to the mutual information of this channel, thus completely settling this capacity problem by a singleletter formula. Quantum information theory generalizes the classical theory, incorporating quantum phenomena like entanglement that have the potential to enhance communication capabilities. Notably, the theory of quantum channels is much richer but less wellunderstood than that of its classical counterpart. For example, quantum channels have several distinct capacities, depending on what one is trying to use them for, and what additional resources are brought into play. These mainly include the classical capacity, private capacity and quantum capacity, with or without the resource assistance such as classical communication and prior shared entanglement. The only solved case for general quantum channels is the entanglementassisted classical capacity, which is given by the quantum mutual information of the channel [2] and is believed as the most natural analog to Shannon’s formula. The capacities in other communication scenarios are still under investigation. Some recent works (e.g [3, 4]) also extend the use of quantum channels to generate quantum resources such as magic state, a key ingredient for faulttolerant quantum computation. The capability of a channel to generate such resource is thus characterized by its corresponding generation capacity.
In general, the difficulty in finding exact expressions for the channel capacities has led to a wide body of works to construct achievable (lower) and converse (upper) bounds. We will defer the detailed discussion of these bounds to the following individual sections. There are several important and highly desirable criteria that one would like from any bound on channel capacities. Specifically, one is generally interested in bounds that are:

singleletter; i.e., the bound depends only on a single use of the channel. Several wellestablished channel coding theorems state that the quantum channel capacity is equal to its corresponding regularized information measure (e.g. the quantum capacity of a channel is equal to its regularized coherent information [5,6,7]). However, these regularized formulas are simply impossible to evaluate in general using finite computational resources, thus not informative enough in spite of being able to write down as formal mathematical expressions. A singleletter formula could be more mathematically tractable and provides a possibility of its evaluation in practice.

computable; i.e., the formula can be explicitly computed for a given quantum channel. This is essentially required by the nature of capacity that quantifies the “capability” of a channel to transmit information or generate resource. An (efficiently) computable converse bound can help to assess the performance of a channel coding scheme in practice and can also be used as a benchmark for the succeeding research. Note that a singleletter formula is not sufficient to guarantee its computability. An example can be given by the quantum squashed entanglement, which admits a singleletter formula but whose computational complexity is proved to be NPhard [8].

general; i.e., the bounds holds for arbitrary quantum channels without requiring any additional assumption on their structure, such as degradability or covariance. There are bounds working well for specific quantum channels with a certain structure or sufficient symmetry. However, the noise in practice can be much more versatile than expected and more importantly does not necessarily possess the symmetry we need. A general bound is definitely preferable for the sake of practical interest.

strong converse; i.e., if the communication rate exceeds this bound, then the success probability or the fidelity of transmission of any channel coding scheme converges to zero as the number of channel uses increases. In contrast, the (weak) converse bound only requires the convergence to a scalar not equal to one. Thus a strong converse bound is conceptually more informative than a weak converse bound, leaving no room for the tradeoff between the communication rate and its success probability or fidelity. We call strong converse capacity of a channel the best (smallest) possible strong converse bound. If the strong converse capacity coincides with the capacity of a channel, then we say this channel admits the strong converse property. This property is known to hold for all memoryless channels in the classical information theory [9] while it remains open in the quantum regime in general (except for the entanglementassisted classical capacity [10]). A strong converse bound may witness the strong converse property of certain quantum channels (e.g. [11, 12]), further sharpening our understanding of the quantum theory.
1.1 Main contributions
In this paper we propose new bounds on quantum channel capacities that satisfy all criteria mentioned above and that improve on previously known bounds. The main novelty of this work is that our bounds all rely on the socalled geometric Rényi divergence. We establish several remarkable properties for this Rényi divergence that are particularly useful in quantum information theory and show how they can be used to provide bounds on quantum channel capacities. A key ingredient used throughout the paper is the semidefinite programming formulation of the weighted matrix geometric mean [13, 14].
Geometric Rényi divergence The geometric Rényi divergence (GRD), is defined as [15]
The quantity \({{\widehat{D}}}_\alpha \) is also known as the maximalRényi divergence [15] as it can be shown to be the maximal divergence among all quantum Rényi divergences satisfying the dataprocessing inequality. Different from the widely studied Petz Rényi divergence [16] or sandwiched Rényi divergence [17, 18], the GRD converges to the Belavkin–Staszewski relative entropy [19] when \(\alpha \rightarrow 1\). The geometric Rényi divergence of two channels \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\) is defined in the usual way as:
where \({{{\mathcal {S}}}}(A)\) is the set of quantum states and \(\phi _{AA'}\) is a purification of \(\rho _A\). We establish the following key properties of GRD which hold for any \(\alpha \in (1,2]\):

1.
It lies between the Umegaki relative entropy and the maxrelative entropy,
$$\begin{aligned} D(\rho \Vert \sigma ) \le {{\widehat{D}}}_{\alpha }(\rho \Vert \sigma ) \le D_{\max }(\rho \Vert \sigma ). \end{aligned}$$ 
2.
Its channel divergence admits a closedform expression,
$$\begin{aligned} {{\widehat{D}}}_\alpha ({{{\mathcal {N}}}}_{A\rightarrow B}\Vert {{{\mathcal {M}}}}_{A\rightarrow B}) = \frac{1}{\alpha 1} \log \left\ {\text {Tr}}_B \left[ J_{{{{\mathcal {M}}}}}^{\frac{1}{2}} \left( J_{{{{\mathcal {M}}}}}^{\frac{1}{2}}J_{{{{\mathcal {N}}}}} J_{{{{\mathcal {M}}}}}^{\frac{1}{2}}\right) ^{\alpha }J_{{{{\mathcal {M}}}}}^{\frac{1}{2}}\right] \right\ _{\infty }, \end{aligned}$$where \(J_{{{{\mathcal {N}}}}}\) and \(J_{{{{\mathcal {M}}}}}\) are the corresponding Choi matrices of \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\) respectively.

3.
Its channel divergence is additive under tensor product of channels,
$$\begin{aligned} {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}_1\otimes {{{\mathcal {N}}}}_2\Vert {{{\mathcal {M}}}}_1\otimes {{{\mathcal {M}}}}_2) = {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}_1\Vert {{{\mathcal {M}}}}_1) + {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}_2\Vert {{{\mathcal {M}}}}_2). \end{aligned}$$ 
4.
Its channel divergence is subadditive under channel composition,
$$\begin{aligned} {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}_2\circ {{{\mathcal {N}}}}_1\Vert {{{\mathcal {M}}}}_2\circ {{{\mathcal {M}}}}_1) \le {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}_1\Vert {{{\mathcal {M}}}}_1) + {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}_2\Vert {{{\mathcal {M}}}}_2). \end{aligned}$$ 
5.
It satisfies the chain rule for any quantum states \(\rho _{RA}\), \(\sigma _{RA}\) and quantum channels \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\),
$$\begin{aligned} {\widehat{D}}_{\alpha }({{{\mathcal {N}}}}_{A\rightarrow B}(\rho _{RA}) \Vert {{{\mathcal {M}}}}_{A \rightarrow B}(\sigma _{RA}))&\le {\widehat{D}}_{\alpha }( \rho _{RA} \Vert \sigma _{RA}) + {\widehat{D}}_{\alpha }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}}). \end{aligned}$$
These properties set a clear difference of GRD with other Rényi divergences. Of particular importance is the chain rule property, which immediately implies that the “amortization collapse” for the geometric Rényi divergence, addressing an open question from [20, Eq. (55)] in the area of quantum channel discrimination. Moreover, due to the closedform expression of the channel divergence and the semidefinite representation of the matrix geometric means [14], any optimization \(\min _{{{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}} {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\) can be computed as a semidefinite program if \(\varvec{{{{\mathcal {V}}}}}\) is a set of channels characterized by semidefinite conditions.
Applications in quantum channel capacities We utilize the geometric Rényi divergence to study several different channel capacity problems, including (1) unassisted quantum capacity, (2) twoway assisted quantum capacity, (3) twoway assisted quantum capacity of bidirectional quantum channels, (4) unassisted private capacity, (5) twoway assisted private capacity, (6) unassisted classical capacity, (7) magic state generation capacity, as listed in Table 1. Most existing capacity bounds are based on the maxrelative entropy due to its nice properties, such as triangle inequality or semidefinite representations. However, these bounds are expected to be loose as the maxrelative entropy stands at the top among the family of quantum divergences. For the bounds based on the Umegaki’s relative entropy, they are unavoidably difficult to compute in general due to their minimax optimization formula. In this work, we construct new channel information measures based on the geometric Rényi divergence, sharpening the previous bounds based on the maxrelative entropy in general while still keeping the new bounds singleletter and efficiently computable. ^{Footnote 1} A plethora of examples are analyzed in each individual sections and the improvements are evident for almost all cases.
The significance of this work is at least twofold. First, from the technical side, we showcase that the geometric Rényi divergence, which has not been exploited so far in the quantum information literature, is actually quite useful for channel capacity problems. We regard our work as an initial step towards other interesting applications and expect that the technical tools established in this work can also be used in, for example, quantum network theory, quantum cryptography, as the maxrelative entropy also appears as the key entropy in these topics. We include another explicit application in the task of quantum channel discrimination in Appendix D. Second, our new capacity bounds meet all the aforementioned desirable criteria and improve the previously bestknown results in general, making them suitable as new benchmarks for computing the capacities of quantum channels.
2 Preliminaries
A quantum system, denoted by capital letters (e.g., A, B), is usually modeled by finitedimensional Hilbert spaces (e.g., \({{{\mathcal {H}}}}_A\), \({{{\mathcal {H}}}}_B\)). The set of linear operators and the set of positive semidefinite operators on system A are denoted as \({{{\mathcal {L}}}}(A)\) and \({{{\mathcal {P}}}}(A)\), respectively. The identity operator on system A is denoted by \({\mathbb {1}}_A\). The set of quantum state on system A is denoted as \({{{\mathcal {S}}}}(A)\equiv \{\rho _A\,\, \rho _A \ge 0,\,{\text {Tr}}\rho _A = 1\}\). A subnormalized state is a positive semidefinite operator with trace no greater than one. For any two Hermitian operators X, Y, we denote \(X \ll Y\) if their supports has the inclusion \({{\text {supp}}}(X) \subseteq {{\text {supp}}}(Y)\). The trace norm of X is given by \(\Vert X\Vert _1 \equiv {\text {Tr}}\sqrt{X^\dagger X}\). The operator norm \(\Vert X\Vert _\infty \) is defined as the maximum eigenvalue of \(\sqrt{X^\dagger X}\). The set of completely positive (CP) maps from A to B is denoted as \({\mathrm{CP}}(A:B)\). A quantum channel or quantum operation \({{{\mathcal {N}}}}_{A\rightarrow B}\) is a completely positive and tracepreserving linear map from \({{{\mathcal {L}}}}(A)\) to \({{{\mathcal {L}}}}(B)\). A subchannel or suboperation \({{{\mathcal {M}}}}_{A\rightarrow B}\) is a completely positive and trace nonincreasing linear map from \({{{\mathcal {L}}}}(A)\) to \({{{\mathcal {L}}}}(B)\). Let \(\Phi \rangle _{A'A} = \sum _{i} i\rangle _{A'}i\rangle _A\) be the unnormalized maximally entangled state. Then the Choi matrix of a linear map \({{{\mathcal {E}}}}_{A'\rightarrow B}\) is defined as \(J_{AB}^{{{{\mathcal {E}}}}} = ({{{\mathcal {I}}}}_{A}\otimes {{{\mathcal {E}}}}_{A'\rightarrow B})(\Phi \rangle \langle \Phi _{A'A})\). We will drop the identity map \({{{\mathcal {I}}}}\) and identity operator \({\mathbb {1}}\) if they are clear from the context. The logarithms in this work are taken in the base two.
2.1 Notation for semidefinite representation
For the simplicity of presenting a semidefinite program, we will introduce some new notations to denote semidefinite conditions. Denote the positive semidefinite condition \(X \ge 0\) as \(\llbracket X \rrbracket _{{\mathsf {P}}}\), the equality condition \(X = 0\) as \(\llbracket X \rrbracket _{{\mathsf {E}}}\), the Hermitian condition \(X = X^\dagger \) as \(\llbracket X \rrbracket _{{\mathsf {H}}}\) and the linear condition \(\llbracket X \rrbracket _{{\mathsf {L}}}\) if X is certain linear operator. We also denote the Hermitian part of X as \(X^{{\mathsf {H}}} \equiv X + X^\dagger \).
2.2 Quantum divergences
A functional \({\varvec{D}}: {{{\mathcal {S}}}}\times {{{\mathcal {P}}}}\rightarrow {\mathbb {R}}\) is a generalized divergence if it satisfies the dataprocessing inequality
For any \(\alpha \in [1/2,1) \cup (1,\infty )\), the sandwiched Rényi divergence is defined as [17, 18]
which is the smallest quantum Rényi divergence that satisfies a dataprocessing inequality, and has been widely used to prove the strong converse property (e.g. [11, 18]). In particular, the sandwiched Rényi divergence is nondecreasing in terms of \(\alpha \), interpolating the Umegaki relative entropy \(D(\rho \Vert \sigma ) \equiv {\text {Tr}}[\rho \, (\log \rho  \log \sigma )]\) [27] and the maxrelative entropy \(D_{\max }(\rho \Vert \sigma ) \equiv \min \{\log t\,\,\rho \le t \sigma \}\) [28, 29] as its two extreme cases,
Another commonly used quantum variant is the Petz Rényi divergence [16] defined as
which attains operational significance in the quantum generalization of Hoeffding’s and Chernoff’s bounds on the success probability in binary hypothesis testing [30, 31]. At the limit of \(\alpha \rightarrow 0\), the Petz Rényi divergence converges to the minrelative entropy [29],
where \(\Pi _\rho \) is the projector on the support of \(\rho \). Due to the the LiebThirring trace inequality [32], it holds for all \(\alpha \in (1,\infty )\) that
Both \({{\widetilde{D}}}_\alpha \) and \({\bar{D}}_\alpha \) recover the Umegaki relative entropy D at the limit of \(\alpha \rightarrow 1\). But they are not easy to optimize over in general.
For any generalized divergence \({\varvec{D}}\), the generalized channel divergence between quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) and subchannel \({{{\mathcal {M}}}}_{A'\rightarrow B}\) is defined as [33, 34]
where \(\phi _{AA'}\) is a purification of \(\rho _A\). In particular, the maxrelative channel divergence has a simple closed form [20, Lemma 12],
where \(J_{{{{\mathcal {N}}}}}\) and \(J_{{{{\mathcal {M}}}}}\) are the corresponding Choi matrices of \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\) respectively.
3 Geometric Rényi Divergence
In this section, we investigate the geometric Rényi divergence and its corresponding channel divergence. Our main contribution in this section is to prove several crucial properties of these divergences which are summarized in Theorem 3. These properties will be extensively used in the following sections.
3.1 Definitions and key properties
Definition 1
([15]). Let \(\rho \) be a quantum state and \(\sigma \) be a subnormalized state with \(\rho \ll \sigma \) and \(\alpha \in (1,2]\), their geometric Rényi divergence ^{Footnote 2} is defined as
where \(G_\alpha (X,Y)\) is the weighted matrix geometric mean defined as
Note that a useful fact of matrix geometric mean is that \(G_{\alpha }(X,Y) = G_{1\alpha }(Y,X)\) (see e.g. [36]).
Remark 1
The geometric Rényi divergence converges to the Belavkin–Staszewski relative entropy [19],
Note that \(D(\rho \Vert \sigma ) \le {{\widehat{D}}}(\rho \Vert \sigma )\) in general and they coincide for commuting \(\rho \) and \(\sigma \) [37]. Some basic properties such as jointconvexity, dataprocessing inequality and the continuity of the geometric Rényi divergence (or more generally, maximal fdivergence) of states can be found in [15]. Further studies on its reversibility under quantum operations are given in [38, 39]. Moreover, the weighted matrix geometric mean admits a semidefinite representation [14] (see also Lemma 46 in Appendix A).
Definition 2
For any quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\), subchannel \({{{\mathcal {M}}}}_{A'\rightarrow B}\), and \(\alpha \in (1,2]\), their geometric Rényi channel divergence is defined as
where \(\phi _{AA'}\) is a purification of \(\rho _A\).
The following Theorem summarizes several crucial properties of the geometric Rényi divergence and its channel divergence. We present their detailed proofs in the next section.
Theorem 3
(Main technical results). The following properties of the geometric Rényi divergence and its channel divergence hold.^{Footnote 3}

1.
(Comparison with D and \(D_{\max }\)): For any quantum state \(\rho \), subnormalized quantum state \(\sigma \) with \(\rho \ll \sigma \) and \(\alpha \in (1,2]\), it holds
$$\begin{aligned} D(\rho \Vert \sigma ) \le {{\widehat{D}}}_{\alpha }(\rho \Vert \sigma ) \le D_{\max }(\rho \Vert \sigma ). \end{aligned}$$(13) 
2.
(Closedform expression of the channel divergence): For any quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\), subchannel \({{{\mathcal {M}}}}_{A'\rightarrow B}\) and \(\alpha \in (1,2]\), the geometric Rényi channel divergence is given by
$$\begin{aligned} {{\widehat{D}}}_\alpha ({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}}) = \frac{1}{\alpha 1}\log \Big \Vert {\text {Tr}}_B G_{1\alpha }(J_{AB}^{{{{\mathcal {N}}}}},J_{AB}^{{{{\mathcal {M}}}}})\Big \Vert _\infty , \end{aligned}$$(14)where \(J_{AB}^{{{{\mathcal {N}}}}}\) and \(J_{AB}^{{{{\mathcal {M}}}}}\) are the corresponding Choi matrices of \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\) respectively. Moreover, for the Belavkin–Staszewski channel divergence, its has the closedform expression:
$$\begin{aligned} {{\widehat{D}}}({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}}) = \left\ {\text {Tr}}_B\, \left\{ (J^{{{{\mathcal {N}}}}}_{AB})^{\frac{1}{2}} \log \left[ (J^{{{{\mathcal {N}}}}}_{AB})^{\frac{1}{2}} (J^{{{{\mathcal {M}}}}}_{AB})^{1} (J^{{{{\mathcal {N}}}}}_{AB})^{\frac{1}{2}} \right] (J^{{{{\mathcal {N}}}}}_{AB})^{\frac{1}{2}} \right\} \right\ _\infty \, . \end{aligned}$$(15) 
3.
(Additivity under tensor product): Let \({{{\mathcal {N}}}}_1\) and \({{{\mathcal {N}}}}_2\) be two quantum channels and let \({{{\mathcal {M}}}}_1\) and \({{{\mathcal {M}}}}_2\) be two subchannels. Then for any \(\alpha \in (1,2]\), it holds that
$$\begin{aligned} {{\widehat{D}}}_\alpha ({{{\mathcal {N}}}}_1\otimes {{{\mathcal {N}}}}_2\Vert {{{\mathcal {M}}}}_1\otimes {{{\mathcal {M}}}}_2) = {{\widehat{D}}}_\alpha ({{{\mathcal {N}}}}_1\Vert {{{\mathcal {M}}}}_1) + {{\widehat{D}}}_\alpha ({{{\mathcal {N}}}}_2\Vert {{{\mathcal {M}}}}_2). \end{aligned}$$(16) 
4.
(Chain rule): Let \(\rho \) be a quantum state on \({{{\mathcal {H}}}}_{RA}\), \(\sigma \) be a subnormalized state on \({{{\mathcal {H}}}}_{RA}\) and \({{{\mathcal {N}}}}_{A\rightarrow B}\) be a quantum channel, \({{{\mathcal {M}}}}_{A\rightarrow B}\) be a subchannel and \(\alpha \in (1,2]\). Then
$$\begin{aligned} {\widehat{D}}_{\alpha }({{{\mathcal {N}}}}_{A\rightarrow B}(\rho _{RA}) \Vert {{{\mathcal {M}}}}_{A\rightarrow B}(\sigma _{RA}))&\le {\widehat{D}}_{\alpha }( \rho _{RA} \Vert \sigma _{RA}) + {\widehat{D}}_{\alpha }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}}) \, . \end{aligned}$$(17) 
5.
(Subadditivity under channel composition): For any quantum channels \({{{\mathcal {N}}}}^1_{A\rightarrow B}\), \({{{\mathcal {N}}}}^2_{B\rightarrow C}\), any subchannels \({{{\mathcal {M}}}}^1_{A\rightarrow B}\), \({{{\mathcal {M}}}}^2_{B\rightarrow C}\) and \(\alpha \in (1,2]\), it holds
$$\begin{aligned} {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}_2\circ {{{\mathcal {N}}}}_1\Vert {{{\mathcal {M}}}}_2\circ {{{\mathcal {M}}}}_1) \le {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}_1\Vert {{{\mathcal {M}}}}_1) + {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}_2\Vert {{{\mathcal {M}}}}_2). \end{aligned}$$(18) 
6.
(Semidefinite representation): Let \(\varvec{{{{\mathcal {V}}}}}\) be a set of subchannels from A to B characterized by certain semidefinite conditions. For any quantum channel \({{{\mathcal {N}}}}_{A\rightarrow B}\) and \(\alpha (\ell ) = 1+2^{\ell }\) with \(\ell \in {{{\mathbb {N}}}}\), the optimization \(\min _{{{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}} {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\) can be computed by a semidefinite program:
(19)where \(J_{{{{\mathcal {N}}}}}\) and \(J_{{{{\mathcal {M}}}}}\) are the corresponding Choi matrices of \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\) respectively. Here the short notation that \(\llbracket X \rrbracket _{{\mathsf {P}}}\), \(\llbracket X \rrbracket _{{\mathsf {E}}}\) and \(\llbracket X \rrbracket _{{\mathsf {H}}}\) represent the positive semidefinite condition \(X \ge 0\), the equality condition \(X = 0\) and the Hermitian condition \(X = X^\dagger \), respectively.
Remark 2
Inequality (13) acts as a starting point of our improvement on the previous capacity bounds built on the maxrelative entropy. The closedform expression of the channel divergence directly leads to the additivity property in Item 3 and the semidefinite representation in Item 6. These properties should be contrasted with the situation for the Petz or sandwiched Rényi divergence for channels, for which it is unclear how they can be calculated efficiently. The chain rule is another fundamental property that sets a difference of the geometric Rényi divergence with other variants. Using the notion of amortized channel divergence [20]
the chain rule is equivalent to
That is, the “amortization collapse” for the geometric Rényi divergence. This solves an open question from [20, Eq. (55)] in the area of quantum channel discrimination.
Remark 3
Note that the properties in Item 3,4,5 do not hold for the Umegaki relative entropy in general unless we consider the regularized channel divergence [40]. This implies that these properties are not satisfied by the Petz or sandwiched channel Rényi divergences for \(\alpha \) in the neighbourhood of 1. As such, defining a measure with such desirable properties requires going away from the Umegaki relative entropy. We also note that it is unclear whether the Petz or sandwiched channel Rényi divergences are efficiently computable, let alone having a simple closedform expression. These indicate that the results we obtained in this work based on the geometric Rényi divergence cannot be easily extended to the Umegaki relative entropy.
Remark 4
Except for the condition \({{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}\), the semidefinite representation in the above Item 6 with \(\alpha (\ell ) = 1 + 2^{\ell }\) is described by \(\ell + 3\) linear matrix inequalities, each of size no larger than \(2d \times 2d\) with \(d = AB\). Thus the computational complexity (timeusage) for computing \(\min _{{{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}} {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\) is the same as computing \(\min _{{{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}} D_{\max }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\). In practice, taking \(\ell = 0\) (\(\alpha = 2\)) already gives an improved result and choosing \(\ell \) around \(8  10\) will make the separation more significant. Moreover, a slight modification can be done [14] to compute the optimization for any rational \(\alpha \in (1,2]\). But we will restrict our attention, without loss of generality, to the discrete values \(\alpha (\ell ) = 1 + 2^{\ell }\) with \(\ell \in {{{\mathbb {N}}}}\).
3.2 Detailed proofs
In the following, we give a detailed proof of each property listed in Theorem 3.
Lemma 4
(Comparison with D and \(D_{\max }\)). For any quantum state \(\rho \), subnormalized quantum state \(\sigma \) with \(\rho \ll \sigma \) and \(\alpha \in (1,2]\), the following relation holds
Proof
The first two inequalities follow from Eqs. (3) and (6). The third inequality follows since the geometric Rényi divergence is the largest Rényi divergence satisfying the dataprocessing inequality (see [15] or [35, Eq. (4.34)]). It remains to prove the last one. Since the geometric Rényi divergence is monotonically nondecreasing with respect to \(\alpha \),^{Footnote 4} it suffices to show that \({{\widehat{D}}}_2(\rho \Vert \sigma ) \le D_{\max }(\rho \Vert \sigma )\). This has been proved in [28, Remark 5.3.2]. We provide here a different proof by using their semidefinite representations. Recall that \(D_{\max }(\rho \Vert \sigma ) = \min \{\log t\,\, \rho \le t \sigma \}\). Denote the optimal solution as t, and we have \(D_{\max }(\rho \Vert \sigma ) = \log t\) with \(0 \le \rho \le t \sigma \). Note that
where the last equality follows from the Schur complement characterization of the block positive semidefinite matrix. Take \(M = t \rho \), and we have
Thus \(M = t \rho \) is a feasible solution of optimization (23) which implies \({{\widehat{D}}}_2(\rho \Vert \sigma ) \le \log {\text {Tr}}[t \rho ] = \log t = D_{\max }(\rho \Vert \sigma )\). This completes the proof. \(\quad \square \)
Compared with \(D_{\max }\), it is clear that \({{\widehat{D}}}_\alpha \) gives a tighter approximation of the Umegaki relative entropy D from above. We provide a concrete example in Fig. 1 to give an intuitive understanding of the relations between different divergences.
Lemma 5
(Closedform expression). For any quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\), subchannel \({{{\mathcal {M}}}}_{A'\rightarrow B}\) and \(\alpha \in (1,2]\), the geometric Rényi channel divergence is given by
where \(J_{AB}^{{{{\mathcal {N}}}}}\) and \(J_{AB}^{{{{\mathcal {M}}}}}\) are the corresponding Choi matrices of \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\) respectively. Moreover, for the Belavkin–Staszewski channel divergence, its has the closedform expression:
Proof
Note that for any quantum state \(\rho _{A}\) and its purification \(\phi _{AA'}\), we have the relation
By definition of the geometric Rényi divergence we have
where the third step follows from the transformer inequality given in Lemma 47 in Appendix A and the fact that we can assume by continuity that \(\rho _A\) has full rank.^{Footnote 5} The last line follows from the semidefinite representation of the infinity norm \(\Vert X\Vert _{\infty } = \max _{\rho \in {{{\mathcal {S}}}}} {\text {Tr}}X\rho \).
The expression for \({{\widehat{D}}}\) follows exactly the same steps by using Corollary 48 in Appendix A and replacing the weighted matrix geometric mean with the operator relative entropy. \(\quad \square \)
Lemma 6
(Additivity). Let \({{{\mathcal {N}}}}_1\) and \({{{\mathcal {N}}}}_2\) be two quantum channels and let \({{{\mathcal {M}}}}_1\) and \({{{\mathcal {M}}}}_2\) be two subchannels. Then for any \(\alpha \in (1,2]\) it holds that
Proof
Due to the closedform expression in Lemma 5, we have
The first and last lines follow from Lemma 5. The second and third lines follow since the weighted matrix geometric mean and the infinity norm are multiplicative under tensor product. \(\quad \square \)
Lemma 7
(Chain rule). Let \(\rho \) be a quantum state on \({{{\mathcal {H}}}}_{RA}\), \(\sigma \) be a subnormalized state on \({{{\mathcal {H}}}}_{RA}\) and \({{{\mathcal {N}}}}_{A\rightarrow B}\) be a quantum channel, \({{{\mathcal {M}}}}_{A\rightarrow B}\) be a subchannel and \(\alpha \in (1,2]\). Then
Proof
Let \(\Phi \rangle _{SA} = \sum _i i\rangle _Si\rangle _A\) be the unnormalized maximally entangled state. Denote \(J^{{{{\mathcal {N}}}}}_{SB}\) and \(J^{{{{\mathcal {M}}}}}_{SB}\) as the Choi matrices corresponding to \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\), respectively. Then we have the identities (see e.g. [22, Eq.(11)])
For \(y = \Vert {\text {Tr}}_B\, G_{1\alpha }(J_{SB}^{{{{\mathcal {N}}}}},J_{SB}^{{{{\mathcal {M}}}}})\Vert _\infty \), Lemma 5 ensures that
and by definition of the infinity norm we find
By definition of the geometric Rényi divergence and by using (40) we can write
where the first inequality follows from the transformer inequality given in Lemma 47 in Appendix A. The third line follows from the multiplicativity of weighted matrix geometric mean under tensor product. The second inequality uses (42) and the fact that \(X \mapsto {\text {Tr}}\, K X\) is monotone for positive operator K. Equation (48) follows from the identity \(\langle \Phi _{SA} Y_{RA}\otimes {\mathbb {1}}_S\Phi _{SA}\rangle = {\text {Tr}}_A\, Y_{RA}\). \(\quad \square \)
Lemma 8
(Subadditivity). For any quantum channels \({{{\mathcal {N}}}}^1_{A\rightarrow B}\), \({{{\mathcal {N}}}}^2_{B\rightarrow C}\), any subchannels \({{{\mathcal {M}}}}^1_{A\rightarrow B}\), \({{{\mathcal {M}}}}^2_{B\rightarrow C}\) and \(\alpha \in (1,2]\), it holds
Proof
This is a direct consequence of the chain rule in Lemma 7. For any pure state \(\phi _{AR}\), we have
Taking a maximization of \(\phi _{AR}\) on the left hand side, we have the desired result. \(\quad \square \)
Lemma 9
(Semidefinite representation). Let \(\varvec{{{{\mathcal {V}}}}}\) be a set of subchannels from A to B characterized by certain semidefinite conditions. For any quantum channel \({{{\mathcal {N}}}}_{A\rightarrow B}\) and \(\alpha (\ell ) = 1+2^{\ell }\) with \(\ell \in {{{\mathbb {N}}}}\), the optimization \(\min _{{{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}} {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\) can be computed by a semidefinite program:
where \(J_{{{{\mathcal {N}}}}}\) and \(J_{{{{\mathcal {M}}}}}\) are the corresponding Choi matrices of \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\) respectively.
Proof
This is a direct consequence of the closedform expression in Lemma 5 and the semidefinite representation of the weighted matrix geometric means in [14] (see also Lemma 46 in Appendix A), as well as the semidefinite representation of the infinity norm of an Hermitian operator \(\Vert X\Vert _\infty = \min \{y\,\, X \le y{\mathbb {1}}\}\). \(\quad \square \)
4 Quantum Communication
4.1 Background
The quantum capacity of a noisy quantum channel is the maximum rate at which it can reliably transmit quantum information over asymptotically many uses of the channel. There are two different quantum capacities of major concern, the (unassisted) quantum capacity Q and the twoway assisted quantum capacity \(Q^{\leftrightarrow }\) , depending on whether classical communication is allowed between each channel uses.
The wellestablished quantum capacity theorem shows that the quantum capacity is equal to the regularized channel coherent information [5,6,7, 42,43,44],
where \(I_c({{{\mathcal {N}}}})\equiv \max _{\rho \in {{{\mathcal {S}}}}} \left[ H({{{\mathcal {N}}}}(\rho ))  H({{{\mathcal {N}}}}^c(\rho ))\right] \) is the channel coherent information, H is the von Neumann entropy and \({{{\mathcal {N}}}}^c\) is the complementary channel of \({{{\mathcal {N}}}}\). The regularization in (56) is necessary in general since the channel coherent information is nonadditive [45, 46] and an unbounded number of channel uses may be required to detect capacity [47]. For this reason, the quantum capacity is notoriously difficult to evaluate, not to mention the scenario with twoway classical communication assistance.
Substantial efforts have been made in providing singleletter lower and upper bounds on Q and \(Q^{\leftrightarrow }\) (e.g. [48,49,50,51,52,53]). Most of them require certain symmetries of the channel to be computable or relatively tight. Of particular interest is a strong converse bound given by Tomamichel et al. [11]. Inspired by the Rains bound from entanglement theory [54], they introduced the Rains information (R) of a quantum channel and further proved that it is a strong converse rate for quantum communication through the channel. However, R is not known to be computable in general due to its minimax optimization of the Umegaki relative entropy. For the ease of computability, Wang et al. [21] relaxed the Umegaki relative entropy to the maxrelative entropy, obtaining a variant known as the maxRains information (\(R_{\max }\)). Leveraging the semidefinite representation of the maxrelative entropy, they showed that \(R_{\max }\) is efficiently computable via a semidefinite program. It was later strengthened by Berta and Wilde [22] that \(R_{\max }\) is also a strong converse rate for quantum communication under twoway classical communication assistance. Since then, the maxRains information \(R_{\max }\) is arguably^{Footnote 6} the bestknown computable strong converse bound on both assisted and unassisted quantum capacities in general.
4.2 Summary of results
In this part, we aim to improve the bound given by the maxRains information in both assisted and unassisted scenarios. The structure of this part is organized as follows (see also a schematic diagram in Fig. 2).
In Sect. 4.3 we discuss the unassisted quantum communication. Based on the notion of the generalized Rains information in [11], we exhibit that the generalized Rains information induced by the geometric Rényi divergence (\({{\widehat{R}}}_{\alpha }\)) can be computed as a semidefinite program, improving the previously known result of the maxRains information [21] in general. That is, we show that
where \(Q({{{\mathcal {N}}}})\) and \(Q^{\dagger }({{{\mathcal {N}}}})\) denote the unassisted quantum capacity of channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively.
In Sect. 4.4, we study the quantum communication with PPT assistance, an assistance stronger than the twoway classical communication. We introduce the generalized Thetainformation which is a new variant of channel information inspired by the channel resource theory (similar to the Upsiloninformation in [57]). More precisely, we define the generalized Thetainformation as a “channel distance” to the class of subchannels given by the zero set of Holevo–Werner bound (\(Q_{\Theta }\)) [48]. Interestingly, we show that the maxRains information \(R_{\max }\) coincides with the generalized Thetainformation induced by the maxrelative entropy \(R_{\max ,\Theta }\), i.e., \(R_{\max } = R_{\max ,\Theta }\), thus providing a completely new perspective of understanding the former quantity. Moreover, we prove that the generalized Thetainformation induced by the geometric Rényi divergence (\({{\widehat{R}}}_{\alpha ,\Theta }\)) is a strong converse bound on the PPTassisted quantum capacity by utilizing an “amortization argument”. Together with its SDP formula, we conclude that \({{\widehat{R}}}_{\alpha ,\Theta }\) improves the previous result of the maxRains information [22] in general. That is, we show that
where \(Q^{{{\mathrm{PPT}}},\leftrightarrow }({{{\mathcal {N}}}})\) and \(Q^{{{\mathrm{PPT}}},\leftrightarrow ,\dagger }({{{\mathcal {N}}}})\) denote the PPTassisted quantum capacity of channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively.
In Sect. 4.5, we consider the PPTassisted quantum communication via bidirectional quantum channels, a more general model than the usual pointtopoint channels. We extend the results in Sect. 4.4 to this general model and demonstrate an improvement to the previous result of the bidirectional maxRains information (\(R_{\max }^{{\mathrm{bi}}}\)) [23]. That is, we show that
where \(Q^{{\mathrm{bi}},{{\mathrm{PPT}}},\leftrightarrow }({{{\mathcal {N}}}})\) and \(Q^{{\mathrm{bi}},{{\mathrm{PPT}}},\leftrightarrow ,\dagger }({{{\mathcal {N}}}})\) denote the PPTassisted quantum capacity of a bidirectional channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively.
Finally in Sect. 4.6 we investigate several fundamental quantum channels, demonstrating the efficiency of our new strong converse bounds. It turns out that our new bounds work exceptionally well and exhibit a significant improvement on the maxRains information for almost all cases.
4.3 Unassisted quantum capacity
In this section, we discuss converse bounds on the unassisted quantum capacity.^{Footnote 7}
Definition 10
([11]). For any generalized divergence \({\varvec{D}}\), the generalized Rains bound of a quantum state \(\rho _{AB}\) is defined as
where the minimization is taken over the Rains set \({{\mathrm{PPT}}}'(A:B) \equiv \big \{\sigma _{AB}\,\big \, \sigma _{AB} \ge 0,\,\big \Vert \sigma _{AB}^{{\mathsf {T}}_B}\big \Vert _1 \le 1\big \}\).
Definition 11
([11]). For any generalized divergence \({\varvec{D}}\), the generalized Rains information of a quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) is defined as
where \(\phi _{AA'}\) is a purification of quantum state \(\rho _A\).
In particular, the Rains information is induced by the Umegaki relative entropy [11],
The maxRains information is induced by the maxrelative entropy [21],
Denote \({{\widehat{R}}}_{\alpha }\) as the generalized Rains information induced by the geometric Rényi divergence. We have the following result.
Theorem 12
(Application 1). For any quantum channel \({{{\mathcal {N}}}}\) and \(\alpha \in (1,2]\), it holds
where \(Q({{{\mathcal {N}}}})\) and \(Q^{\dagger }({{{\mathcal {N}}}})\) denote the unassisted quantum capacity of channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively.
Proof
The first two inequalities follow since the Rains information \(R({{{\mathcal {N}}}})\) has been proved to be a strong converse bound on the unassisted quantum capacity [11]. The last two inequalities are direct consequences of the inequalities in Lemma 4. \(\quad \square \)
Remark 5
Note that in the limit of \(\alpha \rightarrow 1\), the bound \({{\widehat{R}}}_{\alpha }\) will converge to the Rains information induced by the Belavkin–Staszewski relative entropy due to Eq. (11).
The following result shows how to compute the newly introduced bound \({{\widehat{R}}}_{\alpha }({{{\mathcal {N}}}})\) as an SDP.
Proposition 13
(SDP formula). For any quantum channel \({{{\mathcal {N}}}}\) and \(\alpha (\ell ) = 1+2^{\ell }\) with \(\ell \in {\mathbb {N}}\), it holds
with \(S_\alpha ({{{\mathcal {N}}}})\) given by the following SDP
where \(J_{{{{\mathcal {N}}}}}\) is the Choi matrix of \({{{\mathcal {N}}}}\) and \(X^{{\mathsf {H}}} \equiv X + X^\dagger \) denotes the Hermitian part of X.
Proof
The proof involves a nontrivial scaling technique for variables replacement, which is important for simplifying the minimax optimization of \({{\widehat{R}}}_{\alpha }\) to a single SDP. More formally, this proof contains two steps. First we derive a suitable SDP formula for \({{\widehat{R}}}_\alpha (\rho _{AB})\) in terms of a maximization problem. Second, we replace \(\rho _{AB}\) as the channel’s output state \({{{\mathcal {N}}}}_{A'\rightarrow B}(\phi _{AA'})\) and maximize over all the input state \(\rho _A\). Since the SDP maximization formula for \({{\widehat{R}}}_\alpha (\rho _{AB})\) is not necessarily unique, we need to find a suitable one which is able to give us an overall semidefinite optimization in the second step.
Step One: Combining the semidefinite representation of the geometric Rényi divergence in Lemma 46 and the semidefinite representation of the Rains set \({{\mathrm{PPT}}}'(A:B)=\big \{\sigma _{AB}\ge 0 \,\, \sigma _{AB}^{{\mathsf {T}}_{ B}} = X_{AB}  Y_{AB},\, {\text {Tr}}(X_{AB} + Y_{AB}) \le 1,\, X_{AB} \ge 0,\, Y_{AB} \ge 0\big \}\), we have the SDP formula for the geometric Rényi Rains bound as,
By the Lagrange multiplier method, the dual SDP is given by
Due to the Slater’s condition, we can easily check that the strong duality holds. Note that both (64) and (65) are already SDPs for any quantum state \(\rho _{AB}\). However, the last condition in (65) will introduce a nonlinear term if we perform the second step of proof at this stage. The following trick will help us get rid of the variable f which is essential to obtain the final result. Note that the last condition above implies \(f \ge 0\) and together with the rest conditions we necessarily have \(f > 0\). Replacing the variables as
we obtain an equivalent SDP of \({{\widehat{R}}}_{\alpha }(\rho _{AB})\) as
Denote the objective function \(f^{{1}/({2^\ell + 1})} \cdot a  f\) with \(a = {\text {Tr}}\big [\big ({{\widetilde{K}}}^{{\mathsf {H}}}  \sum _{i=1}^\ell {{\widetilde{W}}}_i \big )\varvec{\cdot }\rho \big ] \ge 0\). For any fixed value a, the optimal solution is taken at \(f = [{a}/({2^\ell +1})]^{1+1/2^\ell }\) with the maximal value \(2^\ell [{a}/({2^\ell + 1})]^{1+1/2^\ell }\). Without loss of generality, we can replace the objective function with \(2^\ell [{a}/({2^\ell + 1})]^{1+1/2^\ell }\) and get rid of the variable f. Direct calculation gives us
Step Two: Note that \({{{\mathcal {N}}}}_{A'\rightarrow B}(\phi _{AA'}) = \sqrt{\rho _A} J_{{{{\mathcal {N}}}}} \sqrt{\rho _A}\) holds for any quantum state \(\rho _A\) with purification \(\phi _{AA'}\). Thus the final result is straightforward from (68) by replacing the input state \(\rho _{AB}\) as \(\sqrt{\rho _A} J_{{{{\mathcal {N}}}}} \sqrt{\rho _A}\), replacing \(K,Z_i,W_i\) as \(\rho _A^{1/2} K \rho _A^{1/2},\rho _A^{1/2} Z_i \rho _A^{1/2}, \rho _A^{1/2} W_i\rho _A^{1/2}\) respectively and maximizing over all input state \(\rho _A\). \(\quad \square \)
4.4 Twoway assisted quantum capacity
In this section, we discuss converse bounds on twoway assisted quantum capacity.^{Footnote 8} Recall that the Rains bound in (57) is essentially established as the divergence between the given state and the Rains set—a set of subnormalized states given by the zero set^{Footnote 9} of the lognegativity \(E_N(\rho _{AB})\equiv \log \Vert \rho _{AB}^{{\mathsf {T}}_B}\Vert _1\) [58]. With this in mind, we introduce a new variant of the channel’s analog of Rains bound, compatible with the notion of channel resource theory. Specifically, consider the Holevo–Werner bound [48]—a channel’s analog of the lognegativity,
where \(\Theta \) is the transpose map and \(\Vert {{{\mathcal {F}}}}_{A'\rightarrow B}\Vert _\diamondsuit \equiv \sup _{X_{AA'}\in {{{\mathcal {L}}}}(AA')} \Vert {{{\mathcal {F}}}}_{A'\rightarrow B}(X_{AA'})\Vert _1/ \Vert X_{AA'}\Vert _1\) is the diamond norm [59]. In particular, this bound can be represented as the following SDP,
Inspired by the formulation of the Rains set, we define the set of subchannels given by the zero set of the Holevo–Werner bound \(Q_{\Theta }\) as
Definition 14
(Thetainfo.). For any generalized divergence \({\varvec{D}}\), the generalized Thetainformation^{Footnote 10} of a quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) is defined as
where \({\varvec{{{{\mathcal {V}}}}}_\Theta }\) is the Theta set in (71) and \(\phi _{AA'}\) is a purification of quantum state \(\rho _A\).
Remark 6
On the r.h.s. of Eq. (72), the objective function is concave in \(\rho _A\) and convex in \({{{\mathcal {M}}}}\) [57, Proposition 8]. Thus we can swap the min and max by using Sion’s minimax theorem [60].
The following result compares the generalized Thetainformation in (72) and the generalized Rains information in (58) presented in the previous section. Interestingly, these two quantities coincide for the maxrelative entropy in general.
Proposition 15
For any generalized divergence \({\varvec{D}}\) and any quantum channel \({{{\mathcal {N}}}}\), it holds
Moreover, for the maxrelative entropy the equality always holds, i.e,
Proof
We prove the relation (73) first. Note that for any pure state \(\phi _{AA'}\) and \({{{\mathcal {M}}}}_{A'\rightarrow B} \in \varvec{{{{\mathcal {V}}}}}_\Theta \), we have
This implies \({{{\mathcal {M}}}}_{A'\rightarrow B}(\phi _{AA'}) \in {{\mathrm{PPT}}}'(A:B)\). Then it holds
The first and last line follow by definition. The inequality holds since \({{{\mathcal {M}}}}_{A'\rightarrow B}(\phi _{AA'}) \in {{\mathrm{PPT}}}'(A:B)\) and thus the first line is minimizing over a larger set. In the third line, we swap the min and max by the argument in Remark 6.
We next prove the Eq. (74). Recall that the SDP formula of the maxRains information is given by ([21, Proposition 5] or [61, Eq. (11)])
Replace V and Y with \(\mu V\) and \(\mu Y\) respectively, and then denote \(N = (V Y)^{{\mathsf {T}}_B}\), we have
Notice that the second to the last conditions define a set of CP maps
Combining (81) and (82), we obtain \(R_{\max }({{{\mathcal {N}}}}) = \min _{{{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}} D_{\max }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\). Thus it suffices for us to show the equivalence \(\varvec{{{{\mathcal {V}}}}}= \varvec{{{{\mathcal {V}}}}}_\Theta \). For any \({{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}_\Theta \), take \(V = (R + J_{{{{\mathcal {M}}}}}^{{\mathsf {T}}_B})/2\) and \(Y = (R  J_{{{{\mathcal {M}}}}}^{{\mathsf {T}}_B})/2\). Then \(V \ge 0\), \(Y \ge 0\), \(J_{{{{\mathcal {M}}}}}^{{\mathsf {T}}_B} = VY\) and \({\text {Tr}}_B (V+Y) = {\text {Tr}}_B R \le {\mathbb {1}}_A\), which implies \({{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}\). On the other hand, for any \({{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}\), take \(R = V + Y\). We can check that \({\text {Tr}}_B R = {\text {Tr}}_B(V+Y) \le {\mathbb {1}}_A\), \(R + J_{{{{\mathcal {M}}}}}^{{\mathsf {T}}_B} = (V+Y) + (VY) = 2V \ge 0\) and \(R  J_{{{{\mathcal {M}}}}}^{{\mathsf {T}}_B} = (V+Y)  (VY) = 2Y \ge 0\), which implies \({{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}_\Theta \). Finally we have
which completes the proof. \(\quad \square \)
We proceed to consider the geometric Rényi divergence and show its amortization property, a key ingredient to proving the strong converse bound on the assisted quantum capacity in Theorem 17.
Suppose Alice and Bob share a quantum state \(\rho _{A'AB'}\) with the system cut \(A'A:B'\). Their shared entanglement with respect to the measure \({{\widehat{R}}}_\alpha \) is given by \({{\widehat{R}}}_\alpha (\rho _{A'A:B'})\). If Alice redistributes part of her system A through the channel \({{{\mathcal {N}}}}_{A\rightarrow B}\) and Bob receives the output system B, then their shared state becomes to \(\omega _{A':BB'} = {{{\mathcal {N}}}}_{A\rightarrow B}(\rho _{A'A:B'})\) with the shared entanglement evaluated as \({{\widehat{R}}}_\alpha (\omega _{A':BB'})\). The amortization inequality shows that the amount of entanglement change after the state redistribution is upper bounded by the channel’s information measure \({{\widehat{R}}}_{\alpha ,\Theta }({{{\mathcal {N}}}})\).
Proposition 16
(Amortization). For any quantum state \(\rho _{A'AB'}\), any quantum channel \({{{\mathcal {N}}}}_{A\rightarrow B}\) and the parameter \(\alpha \in (1,2]\), it holds
Proof
This is a direct consequence of the chain rule property of the geometric Rényi divergence in Lemma 7. Suppose the optimal solutions of \({{\widehat{R}}}_\alpha (\rho _{A'A:B'})\) and \({{\widehat{R}}}_{\alpha ,\Theta }({{{\mathcal {N}}}}_{A\rightarrow B})\) are taken at \(\sigma _{A'AB'}\in {{\mathrm{PPT}}}'(A'A:B')\) and \({{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}_{\Theta }\), respectively. Let \( \gamma _{A'BB'} = {{{\mathcal {M}}}}_{A\rightarrow B}(\sigma _{A'AB'})\). We have
where the first inequality follows from the definition of diamond norm and the second inequality follows from the choice of \(\sigma _{A'AB'}\) and \({{{\mathcal {M}}}}_{A\rightarrow B}\). Thus \(\gamma _{A'BB'} \in {{\mathrm{PPT}}}'(A':BB')\) and forms a feasible solution for \({{\widehat{R}}}_\alpha (\omega _{A':BB'})\). Then we have
The second inequality follows from the chain rule of the geometric Rényi divergence in Lemma 7, and the last line follows by the optimality assumption of \({{{\mathcal {M}}}}\) and \(\sigma \). \(\quad \square \)
Theorem 17
(Application 2). For any quantum channel \({{{\mathcal {N}}}}\) and \(\alpha \in (1,2]\), it holds
where \(Q^{{{\mathrm{PPT}}},\leftrightarrow }({{{\mathcal {N}}}})\) and \(Q^{{{\mathrm{PPT}}},\leftrightarrow ,\dagger }({{{\mathcal {N}}}})\) denote the PPTassisted quantum capacity of channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively.
Proof
The first inequality holds by definition. The last inequality holds since we have \( {{\widehat{R}}}_{\alpha ,\Theta }({{{\mathcal {N}}}}) \le R_{\max ,\Theta }({{{\mathcal {N}}}}) = R_{\max }({{{\mathcal {N}}}})\) by Lemma 4 and Proposition 15, respectively. It remains to prove the second one. Once we have the amortization inequality in Proposition 16, the proof of the second inequality will closely follow the one in [22, Theorem 3]. Consider n round PPTassisted quantum communication protocol illustrated in Fig. 3. For the ith round, denote the input state of \({{{\mathcal {N}}}}\) as \(\rho _{A'AB'}^{\scriptscriptstyle (i)}\) and the output state as \(\sigma _{A'BB'}^{\scriptscriptstyle (i)}\). The final state after n rounds communication is denoted as \(\omega _{M_AM_B}\). Then we have
The first and third lines follow from the monotonicity of the geometric Rényi Rains bound \({{\widehat{R}}}_\alpha \) with respect to the PPT operations [11, Eq. (22)]. The second line follows since \(\rho _{A'A:B'}^{\scriptscriptstyle (1)}\) is a PPT state and thus \({{\widehat{R}}}_{\alpha }(\rho ^{\scriptscriptstyle (1)}_{A'A:B'}) = 0\). The last line follows from Proposition 16.
Note that any communication protocol is characterized by a triplet \((n,r,\varepsilon )\) with the number of rounds n, the communication rate r, and the error tolerance \(\varepsilon \). Denote \(k \equiv 2^{nr}\) and we have \({\text {Tr}}\Phi _k \,\omega \ge 1\varepsilon \) with \(\Phi _k\) being the kdimensional maximally entangled state. Moreover, for any \(\sigma \in {{\mathrm{PPT}}}'\), it holds \({\text {Tr}}\Phi _k \sigma \le 1/k\) [54]. Without loss of generality, we can assume that \(\varepsilon \le 12^{nr}\). Otherwise, any rate above the capacity would satisfy the strong converse property since \(1\varepsilon < 2^{nr}\). Thus for any \(\sigma \in {{\mathrm{PPT}}}'\) we have
Let \({{{\mathcal {N}}}}(\gamma ) = ({\text {Tr}}\Phi _k \gamma ) 0\rangle \langle 0 + ({\text {Tr}}({\mathbb {1}}\Phi _k) \gamma ) 1\rangle \langle 1\). Due to the dataprocessing inequality, we have
where \(\delta _\alpha (p\Vert q)\equiv \frac{1}{\alpha 1} \log \big [p^\alpha q^{1\alpha } + (1p)^\alpha (1q)^{1\alpha }\big ]\) is the binary classical Rényi divergence. The last inequality in (97) follows from the monotonicity property that \(\delta _\alpha (p'\Vert q) \le \delta _\alpha (p\Vert q)\) if \(p \le p' \le q\) and \(\delta _\alpha (p\Vert q') \le \delta _\alpha (p\Vert q)\) if \(p \le q' \le q\) [62]. Then we have
Combining Eqs. (91) and (98), we have
which is equivalent to
This implies that if the communication rate r is strictly larger than \({{\widehat{R}}}_{\alpha ,\Theta }({{{\mathcal {N}}}})\), the fidelity of transmission \(1\varepsilon \) decays exponentially fast to zero as the number of channel use n increases. Or equivalently, we have the strong converse inequality \(Q^{{{\mathrm{PPT}}},\leftrightarrow ,\dagger }({{{\mathcal {N}}}}) \le {{\widehat{R}}}_{\alpha ,\Theta }({{{\mathcal {N}}}}) \) and completes the proof. \(\quad \square \)
Let \(Q^{\leftrightarrow }\) and \(Q^{\leftrightarrow ,\dagger }\) be the twoway assisted quantum capacity and its strong converse capacity respectively. We have the following as a direct consequence of Theorem 17, since PPT assistance is stronger.
Corollary 18
For any quantum channel \({{{\mathcal {N}}}}\) and \(\alpha \in (1,2]\), it holds
Finally, we present how to compute \({{\widehat{R}}}_{\alpha ,\Theta }({{{\mathcal {N}}}})\) as an SDP.
Proposition 19
(SDP formula). For any quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) and \(\alpha (\ell ) = 1+2^{\ell }\) with \(\ell \in {\mathbb {N}}\), the geometric Rényi Thetainformation can be computed as an SDP:
where \(J_{{{{\mathcal {N}}}}}\) is the Choi matrix of \({{{\mathcal {N}}}}\).
Proof
This directly follows from Lemma 9 and the definition of the Theta set \(\varvec{{{{\mathcal {V}}}}}_\Theta \) in (71). \(\quad \square \)
4.5 Extension to bidirectional channels
In this section we showcase that the above results for the PPT/twoway assisted quantum capacity can be extended to a more general scenario where Alice and Bob share a bidirectional quantum channel.
A bipartite quantum channel \({{{\mathcal {N}}}}_{A_1B_1\rightarrow A_2B_2}\) is a completely positive tracepreserving map that sends composite system \(A_1B_1\) to \(A_2B_2\). This channel is called bidirectional channel if \(A_1A_2\) are held by Alice and \(B_1B_2\) are held by Bob. That is, Alice and Bob each input a state to this channel and receive an output [63], as depicited in Fig. 4. This is the most general setting for twoparty communications and will reduce to the usual pointtopoint channel when the dimensions of Bob’s input and Alice’s output are trivial, i.e., \(\dim ({{{\mathcal {H}}}}_{B_1}) = \dim ({{{\mathcal {H}}}}_{A_2}) = 1\).
In [23], the authors introduced the bidirectional version of the maxRains information as
Let \(Q^{{\mathrm{bi}},{{\mathrm{PPT}}},\leftrightarrow }\) and \(Q^{{\mathrm{bi}},{{\mathrm{PPT}}},\leftrightarrow ,\dagger }\) be the PPTassisted quantum capacity of a bidirectional channel and its strong converse capacity respectively.^{Footnote 11} It was proved in [23] that
Following a similar approach in Sect. 4.4, we can further strengthen this bound by exploiting the geometric Rényi divergence.
We start with a bidirectional version of the Werner–Holevo bound^{Footnote 12}
and define its zero set \(\varvec{{{{\mathcal {V}}}}}^{\mathrm{bi}}_{\Theta }\) which admits a semidefinite representation as
Using the same idea as the pointtopoint scenario, we defined the generalized Thetainformation of a bidirectional channel \({{{\mathcal {N}}}}_{A_1B_1\rightarrow A_2 B_2}\) as the “channel distance”^{Footnote 13}
where \({\varvec{D}}\) is a generalized divergence and the channel divergence follows from the usual definition
by maximizing over all the pure states \(\phi _{A_1B_1A_3B_3}\).
Following a similar proof of Proposition 15, we can show that the bidirectional maxRains information defined in (106) coincides with the bidirectional Thetainformation induced by the maxrelative entropy. That is,
Denote the bidirectional Rains bound as \({{\widehat{R}}}_{\alpha }^{{\mathrm{bi}}}(\rho ) \equiv \min _{\sigma \ge 0, \Vert \sigma ^{{\mathsf {T}}_{B_1B_2}}\Vert _1 \le 1} {{\widehat{D}}}_{\alpha }(\rho \Vert \sigma )\). A similar proof as Proposition 16 gives us the following amortization inequality.
Proposition 20
(Amortization). For any quantum state \(\rho _{A_1A_3:B_1B_3}\), any bidirectional quantum channel \({{{\mathcal {N}}}}_{A_1B_1\rightarrow A_2B_2}\) and \(\alpha \in (1,2]\), it holds
with the output state \(\omega _{A_2A_3:B_2B_3} = {{{\mathcal {N}}}}_{A_1B_1\rightarrow A_2B_2}(\rho _{A_1A_3:B_1B_3})\).
Using the amortization inequality in Proposition 20 and a standard argument as Theorem 17, we have the analog results of Theorem 17 and Corollary 18 for bidirectional channels as follows:
Theorem 21
(Main result 3). For any bidirectional channel \({{{\mathcal {N}}}}_{A_1B_1\rightarrow A_2B_2}\) and \(\alpha \in (1,2]\), it holds
where \(Q^{{\mathrm{bi}},{{\mathrm{PPT}}},\leftrightarrow }({{{\mathcal {N}}}})\) and \(Q^{{\mathrm{bi}},{{\mathrm{PPT}}},\leftrightarrow ,\dagger }({{{\mathcal {N}}}})\) denote the PPTassisted quantum capacity of a bidirectional channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively. As a consequence, it holds
where \(Q^{{\mathrm{bi}},\leftrightarrow }({{{\mathcal {N}}}})\) and \(Q^{{\mathrm{bi}},\leftrightarrow ,\dagger }({{{\mathcal {N}}}})\) denote the twoway assisted quantum capacity of a bidirectional channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively.
Proposition 22
(SDP formula). For any bidirectional channel \({{{\mathcal {N}}}}_{A_1B_1\rightarrow A_2B_2}\) and \(\alpha (\ell ) = 1+2^{\ell }\) with \(\ell \in {{{\mathbb {N}}}}\), the bidirectional geometric Rényi Thetainformation can be computed as an SDP:
where \(J_{{{{\mathcal {N}}}}}\) is the Choi matrix of \({{{\mathcal {N}}}}\).
Proof
This directly follows from Lemma 9 and the definition of \(\varvec{{{{\mathcal {V}}}}}^{\mathrm{bi}}_{\Theta }\) in (109). \(\quad \square \)
4.6 Examples
In this section, we investigate several fundamental quantum channels as well as their compositions. We use these toy models to test the performance of our new strong converse bounds, demonstrating the improvement on the previous results. The semidefinite programs are implemented in MATLAB via the CVX package [66, 67], by the solver “Mosek” [68] with the best precision.^{Footnote 14}
4.6.1 Fundamental quantum channels
The quantum depolarizing channel with dimension d is defined as
The quantum erasure channel is defined as
where \(e\rangle \) is an erasure state orthogonal to the input Hilbert space. The quantum dephasing channel is defined as
where \(Z = 0\rangle \langle 0  1\rangle \langle 1\) is the Pauliz operator. These three classes of channels are covariant with respect to the whole unitary group. The generalized amplitude damping (GAD) channel is defined as
with the Kraus operators
The GAD channel is one of the realistic sources of noise in superconductingcircuitbased quantum computing [69], which can viewed as the qubit analogue of the bosonic thermal channel. When \(N = 0\), it reduces to the conventional amplitude damping channel with two Kraus operators \(A_1\), \(A_2\).
4.6.2 Comparison for the unassisted quantum capacity
For the unassisted quantum capacity, we compare the qubit depolarizing channel \({{{\mathcal {D}}}}_p\), the qubit erasure channel \({{{\mathcal {E}}}}_p\), the qubit dephasing channel \({{{\mathcal {Z}}}}_p\) and the generalized amplitude damping channels \({{{\mathcal {A}}}}_{p,N}\) with different choices of parameter N.
Since \({{{\mathcal {D}}}}_p\), \({{{\mathcal {E}}}}_p\) and \({{{\mathcal {Z}}}}_p\) are covariant with respect to the unitary group, the optimal input state \(\rho _A\) of their Rains information is taken at the maximally mixed state [11, Proposition 2]. Therefore, their Rains information can be computed via the algorithm in [55, 56]. Moreover, for any parameters \(\gamma ,N \in [0,1]\), the GAD channel \({{{\mathcal {A}}}}_{\gamma ,N}\) is covariant with respect to the Pauliz operator Z. That is, \({{{\mathcal {A}}}}_{\gamma ,N}(Z\rho Z) = Z {{{\mathcal {A}}}}_{\gamma ,N}(\rho )Z\) for all quantum state \(\rho \). To compute its Rains information, it suffices to perform the maximization over input states with respect to the oneparameter family of states \(\rho _A = (1p)0\rangle \langle 0 + p 1\rangle \langle 1\) [70]. This can be handled, for example, by MATLAB function “fminbnd”.
The comparison results are shown in Fig. 5. It is clear that the geometric Rényi Rains information \({{\widehat{R}}}_{\alpha {\scriptscriptstyle (10)}}\) coincide with the Rains information R for all these channels except for the particular case \({{{\mathcal {A}}}}_{p,0}\) in subfigure (d). For all cases, \({{\widehat{R}}}_{\alpha {\scriptscriptstyle (10)}}\) sets a big difference from the maxRains information \(R_{\max }\).
4.6.3 Comparison for the twoway assisted quantum capacity
For the twoway assisted quantum capacity, we consider the channels \({{{\mathcal {D}}}}_p\), \({{{\mathcal {E}}}}_p\) and \({{{\mathcal {Z}}}}_p\) composed with the amplitude damping channel \({{{\mathcal {A}}}}_{p,0}\), and the generalized amplitude damping channel \({{{\mathcal {A}}}}_{p,N}\) with different choices of parameter N. Because these channels are not covariant w.r.t. the whole unitary group, their Rains information are not known as valid converse bounds on the twoway assisted quantum capacity.
The comparison result^{Footnote 15} for the twoway assisted quantum capacity is given in Fig. 6. The geometric Rényi Thetainformation \({{\widehat{R}}}_{\alpha {\scriptscriptstyle (10)},\Theta }\) demonstrates a significant improvement over the maxRains information \(R_{\max }\) for all these channels except for one particular case \({{{\mathcal {A}}}}_{p,0}\) in subfigure (d).
4.6.4 Comparison for the twoway assisted quantum capacity of bidirectional channels
Consider a typical noise in a quantum computer which is modeled as [23]
where S is the swap operator and \(U_\phi = 00\rangle \langle 00 + e^{i\phi } 01\rangle \langle 01 + e^{i\phi } 10\rangle \langle 10 + e^{2i\phi } 11\rangle \langle 11\) is the collective dephasing noise. The comparison result of our new bound \({{\widehat{R}}}^{\mathrm{bi}}_{\alpha {\scriptscriptstyle (10)},\Theta }\) with the previous bound \(R^{\mathrm{bi}}_{\max }\) is given in Fig. 7.
5 Private Communication
5.1 Background
The private capacity of a quantum channel is defined as the maximum rate at which classical information can be transmitted privately from the sender (Alice) to the receiver (Bob). By “private”, it means a third party (Eve) who has access to the channel environment cannot learn anything about the information that Alice sends to Bob. There are also two different private capacities of major concern, the (unassisted) private capacity P and the twoway assisted private capacity \(P^{\leftrightarrow }\), depending on whether classical communication is allowed between each channel uses.
In the same spirit of the quantum capacity theorem, the private capacity theorem states that the private capacity of a quantum channel is given by its regularized private information [7, 71],
where \(I_p({{{\mathcal {N}}}}) \equiv \max _{{\mathscr {E}}} \left[ \chi ({\mathscr {E}}, {{{\mathcal {N}}}})  \chi ({\mathscr {E}}, {{{\mathcal {N}}}}^c)\right] \) is the private information with the maximization taken over all quantum state ensembles \({\mathscr {E}} = \{p_i, \rho _i\}\), \(\chi ({\mathscr {E}}, {{{\mathcal {N}}}}) \equiv H(\sum _i p_i {{{\mathcal {N}}}}(\rho _i))  \sum _i p_i H({{{\mathcal {N}}}}(\rho _i))\) is the Holevo information of the ensemble \({\mathscr {E}}\), H is the von Neumann entropy and \({{{\mathcal {N}}}}^c\) is the complementary channel of \({{{\mathcal {N}}}}\). The regularization in (124) is necessary in general since the private information is proved to be nonadditive [72] and an unbounded number of channel uses may be required to achieve its private capacity [73].
Despite their importance in understanding the fundamental limits of quantum key distributions [74], much less is known about the converse bounds on private capacities, mostly due to their inherently involved settings. The squashed entanglement of a channel was proposed in [75] and proved to be a converse bound on the twoway assisted private capacity. But it remains unknown to be a strong converse rate and the quantity itself is difficult to compute exactly [8]. The entanglement cost of a channel was introduced in [76] and shown to be a strong converse bound on the twoway assisted private capacity [25]. But it was not given by a singleletter formula. A closely related quantity to this part is the relative entropy of entanglement of a channel (\(E_R\)), which was proved as a (weak) converse bound on the twoway assisted private capacity for channels with “covariant symmetry” [24]. This was later strengthened in [12] as a strong converse bound on the unassisted private capacity for general quantum channels and a strong converse bound on the twoway assisted private capacity for channels with “covariant symmetry”. Moreover, the maxrelative entropy of entanglement of a channel (\(E_{\max }\)) was proved as a strong converse bound on the twoway assisted private capacity in general [25].
5.2 Summary of results
In this part, we extend the techniques used in the previous sections to the task of private communication and aim to improve the maxrelative entropy of entanglement of a channel in both assisted and unassisted scenarios. The structure is organized as follows (see also a schematic diagram in Fig. 8).
In Sect. 5.3 we discuss the unassisted private communication. While the relative entropy of entanglement \(E_R\) established the best known strong converse bound in this case, the difficulties of its evaluation are twofold: the optimization over the set of separable states and the minimax optimization of the Umegaki relative entropy. The first difficulty will be automatically removed for qubit channels since separability can be completely characterized by the positive partial transpose conditions [77]. The second can be handled by relaxing the Umegaki relative entropy to a semidefinite representable one, such as the maxrelative entropy. Based on a notion of the generalized relative entropy of entanglement of a channel, we exhibit that the entanglement of a channel induced by the geometric Rényi divergence (\({{\widehat{E}}}_{\alpha }\)) lies between \(E_R\) and \(E_{\max }\). That is, we show that
where \(P({{{\mathcal {N}}}})\) and \(P^{\dagger }({{{\mathcal {N}}}})\) denote the unassisted private capacity of channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively. Moreover, \({{\widehat{E}}}_{\alpha }({{{\mathcal {N}}}})\) is given by a conic program in general and reduces to a semidefinite program for all qubit channels.
In Sect. 5.4, we study the private communication with twoway classical communication assistance. We introduce the generalized Sigmainformation which is a new variant of channel information inspired by the channel resource theory (similar to the Upsiloninformation in [57]). More precisely, we define the generalized Sigmainformation as a “channel distance” to the class of entanglement breaking subchannels. We show that the maxrelative entropy of entanglement \(E_{\max }\) coincides with the generalized Sigmainformation induced by the maxrelative entropy \(E_{\max ,\Sigma }\), i.e., \(E_{\max } = E_{\max ,\Sigma }\), thus providing a completely new perspective of understanding the former quantity. Moreover, we prove that the generalized Sigmainformation induced by the geometric Rényi divergence (\({{\widehat{E}}}_{\alpha ,\Sigma }\)) is a strong converse bound on the twoway assisted private capacity by utilizing an “amortization argument”, improving the previously bestknown result of the maxrelative entropy of entanglement [25] in general. That is, we show that
where \(P^{\leftrightarrow }({{{\mathcal {N}}}})\) and \(P^{\leftrightarrow ,\dagger }({{{\mathcal {N}}}})\) denote the twoway assisted private capacity of channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively. Moreover, \({{\widehat{E}}}_{\alpha ,\Sigma }({{{\mathcal {N}}}})\) is given by a conic program in general and reduces to a semidefinite program for all qubit channels.
5.3 Unassisted private capacity
In this section we discuss converse bounds on the unassisted private capacity.^{Footnote 16}
Definition 23
([78]). For any generalized divergence \({\varvec{D}}\), the generalized relative entropy of entanglement of a quantum state \(\rho _{AB}\) is defined as
where \({{\mathrm{SEP}}}_{\bullet }(A:B)\) is the set of subnormalized separable states between A and B.
If the generalized divergence satisfies the dominance property, i.e., \({\varvec{D}}(\rho \Vert \sigma ) \ge {\varvec{D}}(\rho \Vert \sigma ')\) if \(\sigma \le \sigma '\), then the optimal solution of the above minimization problem can always be taken at a normalized separable states. Since the dominance property is generic for most divergences of concern, the above definition is consistent with the one defined over the set of normalized separable states (e.g. [78]).
Definition 24
([24]). For any generalized divergence \({\varvec{D}}\), the generalized relative entropy of entanglement of a quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) is defined as
where \(\phi _{AA'}\) is a purification of quantum state \(\rho _A\).
In particular, the relative entropy of entanglement for a channel is induced by the Umegaki relative entropy [24],
The maxrelative entropy of entanglement for a channel is induced by the maxrelative entropy [25],
These two quantities are known as strong converse bounds for private communication with and without classical communication assistance, respectively. That is,
The computability of \(E_R\) is usually restricted to qubit covariant channels where the input state \(\rho _A\) can be taken as the maximally mixed states and the set of separable states coincides with the set of PPT states [77]. The following result relaxes \(E_R\) to its geometric Rényi version \({{\widehat{E}}}_\alpha \), which is SDP computable for all qubit channels and is tighter than \(E_{\max }\) in general.
Theorem 25
(Application 3). For any quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) and \(\alpha \in (1,2]\), it holds
where \(P({{{\mathcal {N}}}})\) and \(P^{\dagger }({{{\mathcal {N}}}})\) denote the unassisted private capacity and its corresponding strong converse capacity, respectively. Moreover, the bound \({{\widehat{E}}}_{\alpha }({{{\mathcal {N}}}})\) with \(\alpha (\ell ) = 1+2^{\ell }\) and \(\ell \in {\mathbb {N}}\) can be given as
with \(T_\alpha ({{{\mathcal {N}}}})\) given by the following conic program
where \(J_{{{{\mathcal {N}}}}}\) is the Choi matrix of \({{{\mathcal {N}}}}\) and \(\mathcal {BP}(A:B)\) is the set of blockpositive operators which reduces to a semidefinite cone if \(AB \le 6\).
Proof
The first inequality in (130) follows by definition. The second inequality in (130) was proved in [12]. The last two inequalities in (130) are direct consequences of Lemma 4. The derivation of the conic program (132) follows the same steps as Proposition 13. The block positive cone \(\mathcal {BP}(A:B)\) is the dual cone of the set of separable operators. When the channel dimension satisfies \(AB \le 6\), this cone admits a semidefinite representation as \(\mathcal {BP}(A:B)=\{X+Y^{{\mathsf {T}}_B}\,\, X \ge 0, Y \ge 0\}\) [79, Table 2.2]. Thus the conic program (132) reduces to a semidefinite program. \(\quad \square \)
5.4 Twoway assisted private capacity
In this section we discuss converse bounds on the twoway assisted private capacity.^{Footnote 17}
A quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) is called entanglement breaking if its output state \({{{\mathcal {N}}}}_{A'\rightarrow B}(\rho _{AA'})\) is separable for any input \(\rho _{AA'}\) or equivalently if its Choi matrix is separable [80]. Since every entanglement breaking channel can be simulated by a measurementpreparation scheme [80, 81], any twoway assisted private communication protocol via entanglement breaking channel will end up with a separable state. This indicates that these channels are useless for private communication. With this in mind, we consider the set of entanglement breaking subchannels as
where \(\mathcal {SEP}(A:B)\) denotes the cone of separable operators.
Definition 26
(Sigmainfo.). For any generalized divergence \({\varvec{D}}\), the generalized Sigmainformation of a quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) is defined as
where \(\phi _{AA'}\) is a purification of quantum state \(\rho _A\).
As mentioned in Remark 6, the min and max in the above definition can be swapped.
Analogous to Proposition 15, the following result establishes the relation between the generalized Sigmainformation \({\varvec{E}}_{\Sigma }\) and the generalized relative entropy of entanglement of a channel \({\varvec{E}}\).
Proposition 27
For any generalized divergence \({\varvec{D}}\) and any quantum channel \({{{\mathcal {N}}}}\), it holds
Moreover, for the maxrelative entropy the equality always holds, i.e,
Proof
This first inequality can be proved in a similar manner as Proposition 15 by using the fact that \({{{\mathcal {M}}}}_{A'\rightarrow B}(\phi _{AA'}) = \sqrt{\rho _A} J_{{{{\mathcal {M}}}}}\sqrt{\rho _A} \in {{\mathrm{SEP}}}_{\bullet }(A:B)\) for any \(\rho _A \in {{{\mathcal {S}}}}(A)\) and \({{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}_{\Sigma }\). We now prove the Eq. (136). It has been shown in [22, Lemma 7] that
Using the semidefinite representation of infinity norm and replacing \(Y_{AB} = t J_{{{{\mathcal {M}}}}}\), we have
By the definition of \(D_{\max }\) and \(\varvec{{{{\mathcal {V}}}}}_{\Sigma }\), we have
where the second equality follows from Eq. (8). \(\quad \square \)
Remark 7
The idea of considering the set of entanglement breaking channels also appears in [25, Theorem V.2], where an upper bound of \(E_{\max }({{{\mathcal {N}}}})\) is given as \( E_{\max }({{{\mathcal {N}}}}) \le B_{\max }({{{\mathcal {N}}}})\) with
However, the key difference here is that \(E_{\max ,\Sigma }\) is minimizing over all the entanglement breaking subchannels which is a strictly superset of entanglement breaking channels. Such extension is essential to get the equality \(E_{\max }({{{\mathcal {N}}}}) = E_{\max ,\Sigma }({{{\mathcal {N}}}})\) instead of an upper bound.
We further consider the Sigmainformation induced by the geometric Rényi divergence. Following a similar argument as Proposition 16, we can have the amortization property.
Proposition 28
(Amortization). For any quantum state \(\rho _{A'AB'}\) and quantum channel \({{{\mathcal {N}}}}_{A\rightarrow B}\) and the parameter \(\alpha \in (1,2]\), it holds
Proof
The proof is similar to that of Proposition 16. We only need to show that for any subnormalized state \(\sigma _{A'AB'} \in {{\mathrm{SEP}}}_{\bullet }(A'A:B')\) and map \({{{\mathcal {M}}}}_{A\rightarrow B} \in \varvec{{{{\mathcal {V}}}}}_{\Sigma }\), it holds \(\gamma _{A'BB'}\equiv {{{\mathcal {M}}}}_{A\rightarrow B}(\sigma _{A'AB'}) \in {{\mathrm{SEP}}}_{\bullet }(A':BB')\). This can be checked as follows. First it is clear that \({\text {Tr}}\gamma _{A'BB'} \le 1\) since both \({{{\mathcal {M}}}}\) and \(\sigma \) are subnormalized. Denote the tensor product decomposition \(\sigma _{A'AB'} = \sum _{i,j} \sigma ^i_{A'A}\otimes \sigma ^j_{B'}\) and \(J^{{{{\mathcal {M}}}}}_{SB} = \sum _{k,\ell } J^k_{S}\otimes J^\ell _B\). Let \(\Phi \rangle _{SA}\) be the unnormalized maximally entangled state. Then we have
where the r.h.s. belongs to \(\mathcal {SEP}(A':BB')\). This completes the proof. \(\quad \square \)
Combining the amortization inequality and a similar argument in [25, Theorem IV.1.], we can obtain an improved strong converse bound as follows:
Theorem 29
(Application 4). For any quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) and \(\alpha \in (1,2]\), it holds
where \(P^{\leftrightarrow }({{{\mathcal {N}}}})\) and \(P^{\leftrightarrow ,\dagger }({{{\mathcal {N}}}})\) denote the twoway assisted private capacity of channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively. Moreover, the bound \({{\widehat{E}}}_{\alpha ,\Sigma }({{{\mathcal {N}}}})\) with \(\alpha (\ell ) = 1+2^{\ell }\) and \(\ell \in {\mathbb {N}}\) can be given by a conic program
which reduces to a semidefinite program if the dimension satisfies \(AB \le 6\).
6 Classical Communication
6.1 Background
The classical capacity of a quantum channel is the maximum rate at which it can reliably transmit classical information over asymptotically many uses of the channel. Since classical messages are of major concern here, the communication assistance is usually given by the shared entanglement instead of the twoway classical communication discussed in the quantum/private communication scenarios. The entanglementassisted classical capacity has been completely solved as the mutual information of the channel [2], which is believed to be a natural counterpart in the classical Shannon theory. In this sense, shared entanglement simplifies the quantum Shannon theory.
When it comes to the unassisted classical capacity, the bestknown characterization is given by the classical capacity theorem, which states that the classical capacity of a quantum channel is given by its regularized Holevo information [82, 83],
where \(\chi ({{{\mathcal {N}}}}) \equiv \max _{{\mathscr {E}}} \chi ({\mathscr {E}}, {{{\mathcal {N}}}}) \) is the Holevo information with the maximization taken over all quantum state ensembles \({\mathscr {E}} = \{p_i, \rho _i\}\), \(\chi ({\mathscr {E}}, {{{\mathcal {N}}}}) \equiv H(\sum _i p_i{{{\mathcal {N}}}}(\rho _i))  \sum _i p_i H({{{\mathcal {N}}}}(\rho _i))\) is the Holevo information of the ensemble \({\mathscr {E}}\), and H is the von Neumann entropy. An impressive work by Hastings [84] shows that the Holevo information is not additive in general, indicating the necessity of the regularization in (145). Moreover, as computing \(\chi \) itself is already NPcomplete [85], its regularized quantity for a general quantum channel is expected to be more difficult to evaluate. Even for the qubit amplitude damping channel, its unassisted classical capacity is still unknown [26].
Deriving a singleletter expression for the classical capacity of a quantum channel remains a major open problem in quantum information theory. Several general converse bounds are given in [34] by an “continuity argument”, extending the idea in [51] from quantum capacity to classical capacity. However, those bounds are not known to be strong converse and typically work well only if the channel possesses certain structures, such as close to entanglement breaking channel or sufficiently covariant.^{Footnote 18} Two bestknown strong converse bounds are given by \(C_\beta \) and \(C_\zeta \) in [26], and both bounds are SDP computable. An attempt to improve the bound \(C_\beta \) is discussed in [57] by a notion called Upsiloninformation (\(\Upsilon \)), similar to the Thetainformation and Sigmainformation in the previous parts. However, a (weak) subadditivity of the Upsiloninformation induced by the sandwiched Rényi divergence is required for showing \(\Upsilon \) as a strong converse bound for general quantum channels. This subadditivity was only proved in [57] for covariant channels while the general case was left open.
6.2 Summary of results
In this part, we aim to push forward the analysis in [57] by considering the geometric Rényi divergence and improve both of the two strong converse bounds \(C_\beta \) and \(C_\zeta \) in general. The structure of this part is organized as follows (see also a schematic diagram in Fig. 9).
In Sect. 6.3, we first study the generalized Upsiloninformation induced by the maxrelative entropy (\(\Upsilon _{\max }\)) and prove that it is no greater than \(C_\beta \) and \(C_\zeta \) in general. We then discuss the generalized Upsiloninformation induced by the geometric Rényi divergence (\({{\widehat{\Upsilon }}}_{\alpha }\)) and show that it is a strong converse bound on classical capacity by proving its subadditivity. Due to the relation that \({{\widehat{D}}}_{\alpha } \le D_{\max }\), we have \({{\widehat{\Upsilon }}}_\alpha \le \Upsilon _{\max }\). Then we have an improved strong converse bound \({{\widehat{\Upsilon }}}_\alpha \) satisfying
where \(C({{{\mathcal {N}}}})\) and \(C^{\dagger }({{{\mathcal {N}}}})\) denote the unassisted classical capacity of channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively.
In Sect. 6.4, we investigate several fundamental quantum channels, demonstrating the efficiency of our new strong converse bounds. It turns out that our new bounds work exceptionally well and exhibit a significant improvement on previous results for almost all cases.
6.3 Unassisted classical capacity
In this section we discuss converse bounds on the unassisted classical capacity of a quantum channel.^{Footnote 19}
A quantum channel is called constant channel or replacer channel if it always output a fixed quantum state, i.e., there exists \(\sigma _B \in {{{\mathcal {S}}}}(B)\) such that \({{{\mathcal {N}}}}_{A\rightarrow B}(\rho _A) = \sigma _B\) for all \(\rho _A \in {{{\mathcal {S}}}}(A)\). Unlike quantum or private communication where the sets of useless channels are not completely determined yet, the useless set for classical communication is fully characterized by the set of constant channels. That is, \(C({{{\mathcal {N}}}}) = 0\) if and only if \({{{\mathcal {N}}}}\) is a constant channel.^{Footnote 20} As a natural extension, the work [57] proposed to consider the set of constantbounded subchannels,
It seems not easy to find a semidefinite representation for the set \(\varvec{{{{\mathcal {V}}}}}_{cb}\) directly. Thus a restriction of \(\varvec{{{{\mathcal {V}}}}}_{cb}\) was given in [57] as
This subset can be seen as the zero set^{Footnote 21} of the strong converse bound \(C_\beta ({{{\mathcal {N}}}}) \equiv \log \beta (J_{{{{\mathcal {N}}}}})\) [26]. As discussed in Appendix B, we will see that \(\varvec{{{{\mathcal {V}}}}}_{cb}\) can be approximated by a complete semidefinite hierarchy, where the subset \(\varvec{{{{\mathcal {V}}}}}_\beta \) can be considered as a symmetrized version of its first level. In the following, we proceed our analysis, without loss of generality, over the set \(\varvec{{{{\mathcal {V}}}}}_\beta \). A more detailed discussion of \(\varvec{{{{\mathcal {V}}}}}_{cb}\) can be found in Appendix B.
Definition 30
(Upsiloninfo. [57]). For any generalized divergence \({\varvec{D}}\), the generalized Upsiloninformation of a quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) with respect to the set \(\varvec{{{{\mathcal {V}}}}}_\beta \) is defined as
where \(\phi _{AA'}\) is a purification of quantum state \(\rho _A\).
As mentioned in Remark 6, the min and max in the above definition can be swapped.
Let us first consider the generalized Upsiloninformation induced by the maxrelative entropy \(\Upsilon _{\max }\).
Proposition 31
For any quantum channel \({{{\mathcal {N}}}}\), the generalized Upsiloninformation induced by the maxrelative entropy \(\Upsilon _{\max }({{{\mathcal {N}}}})\) is given as an SDP,
Proof
By definition we have \(\Upsilon _{\max }({{{\mathcal {N}}}})\equiv \min _{{{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}_\beta }} D_{\max }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}}) = \min _{{{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}_\beta } D_{\max }(J_{{{{\mathcal {N}}}}}\Vert J_{{{{\mathcal {M}}}}})\), where the second equality follows from Eq. (8). Then it is clear that
Replacing \(K = t J_{{{{\mathcal {M}}}}}\), \(S = t G\) and \(R = tW\), we have the desired result. \(\quad \square \)
Besides the bound \(C_\beta \), there is another SDP strong converse bound given in [26] as
We can show that \(\Upsilon _{\max }\) is no greater than both of these quantities in general.
Proposition 32
For any quantum channel \({{{\mathcal {N}}}}\), it holds \(\Upsilon _{\max }({{{\mathcal {N}}}}) \le \min \big \{C_\beta ({{{\mathcal {N}}}}),C_\zeta ({{{\mathcal {N}}}})\big \}\).
Proof
The result is clear by comparing their SDP formulas. Specifically, by restricting \(K_{AB} = J_{{{{\mathcal {N}}}}}\) in \(\Upsilon _{\max }\), we can retrieve \(C_\beta \). By restricting \(R_{AB} = {\mathbb {1}}_A \otimes S_B\) in \(\Upsilon _{\max }\), we can retrieve \(C_\zeta \). \(\quad \square \)
We further consider the generalized Upsiloninformation induced by the geometric Rényi divergence \({{\widehat{\Upsilon }}}_\alpha \). The following subadditivity is a key ingredient to proving that \({{\widehat{\Upsilon }}}_\alpha \) is a strong converse bound in Theorem 34.
Proposition 33
(Subadditivity). For any quantum channels \({{{\mathcal {N}}}}_1\), \({{{\mathcal {N}}}}_2\) and \(\alpha \in (1,2]\), it holds
Proof
This is a direct consequence of the additivity of the geometric Rényi channel divergence in Lemma 6 and the subadditivity of the quantity \(\beta (\cdot )\) in (147). More specifically, suppose the optimal solution of \({{\widehat{\Upsilon }}}_\alpha ({{{\mathcal {N}}}}_1)\) and \({{\widehat{\Upsilon }}}_\alpha ({{{\mathcal {N}}}}_2)\) are taken at \({{{\mathcal {M}}}}_1 \in \varvec{{{{\mathcal {V}}}}}_\beta ^1\) and \({{{\mathcal {M}}}}_2 \in \varvec{{{{\mathcal {V}}}}}_\beta ^2\) respectively. Then we can check that \({{{\mathcal {M}}}}_1\otimes {{{\mathcal {M}}}}_2 \in \varvec{{{{\mathcal {V}}}}}_\beta ^{12}\) which is a feasible solution for \({{\widehat{\Upsilon }}}_\alpha ({{{\mathcal {N}}}}_1\otimes {{{\mathcal {N}}}}_2)\). Thus we have
where the last inequality follows by the optimality assumption of \({{{\mathcal {M}}}}_1\) and \({{{\mathcal {M}}}}_2\). \(\quad \square \)
Based on the subadditivity, we are ready to show our improved strong converse bound.
Theorem 34
(Application 5). For any quantum channel \({{{\mathcal {N}}}}\) and \(\alpha \in (1,2]\), it holds
where \(C({{{\mathcal {N}}}})\) and \(C^{\dagger }({{{\mathcal {N}}}})\) denote the unassisted classical capacity of channel \({{{\mathcal {N}}}}\) and its corresponding strong converse capacity, respectively.
Proof
The first inequality holds by definition. The third inequality follows from Lemma 4. The last inequality was proved in Proposition 32. It remains to prove the second inequality \(C^\dagger ({{{\mathcal {N}}}}) \le {{\widehat{\Upsilon }}}_\alpha ({{{\mathcal {N}}}})\). For any classical communication protocol with a triplet \((r,n,\varepsilon )\), it holds by a standard argument [57, Proposition 20] that
where \({{\widetilde{\Upsilon }}}_\alpha \) is the Upsilon information induced by the sandwiched Rényi divergence \({{\widetilde{D}}}_\alpha \). Due to the subadditivity of \({{\widehat{\Upsilon }}}_\alpha \) in Proposition 33 and the inequality in Lemma 4, we have
Combining (154) and (155), we have
This implies that if the communication rate r is strictly larger than \({{\widehat{\Upsilon }}}_\alpha ({{{\mathcal {N}}}})\), the success probability of the transmission \(1\varepsilon \) decays exponentially fast to zero as the number of channel use n increases. Or equivalently, we have the strong converse inequality \(C^\dagger ({{{\mathcal {N}}}}) \le {{\widehat{\Upsilon }}}_\alpha ({{{\mathcal {N}}}})\) and completes the proof. \(\quad \square \)
Finally, we present how to compute the geometric Rényi Upsilon information.
Proposition 35
(SDP formula) For any quantum channel \({{{\mathcal {N}}}}_{A'\rightarrow B}\) and \(\alpha (\ell ) = 1+2^{\ell }\) with \(\ell \in {\mathbb {N}}\), the geometric Rényi Upsilon information can be computed by the following SDP:
Proof
This directly follows from Lemma 9 and the definition of the set \(\varvec{{{{\mathcal {V}}}}}_\beta \) in (147). \(\quad \square \)
6.4 Examples
In this section, we study several fundamental quantum channels as well as their compositions. We use these toy models to test the performance of our new strong converse bounds, demonstrating the improvement on the previously known results.
Consider the depolarizing channel \({{{\mathcal {D}}}}_p\) defined in (117), the erasure channel \({{{\mathcal {E}}}}_p\) defined in (118) and the dephrasure channel \({{{\mathcal {N}}}}_{p,q}(\rho )\equiv (1q)[(1p)\rho +pZ\rho Z] + q {\text {Tr}}(\rho )e\rangle \langle e\), where \(e\rangle \) is an erasure flag orthogonal to the input Hilbert space. Since these channels are covariant with respect to the unitary group, their Upsilon informations are known as strong converse bounds [57, Proposition 20] and can be computed via the algorithm in [55, 56]. As for the generalized amplitude damping (GAD) channel \({{{\mathcal {A}}}}_{\gamma ,N}\) defined in (120), its Upsilon information is not known as a valid converse bound.
The comparison results^{Footnote 22} are shown in Fig. 10. It is clear that \({{\widehat{\Upsilon }}}_{\alpha {\scriptscriptstyle (10)}}\) demonstrates significant improvements over \(C_\beta \) and \(C_{\zeta }\) for all these channels except for one particular case \({{{\mathcal {A}}}}_{p,0}\) in subfigure (d) where all bounds coincide. It is interesting to note that an analytical expression of the bounds \(C_\beta ({{{\mathcal {A}}}}_{\gamma ,N}) = C_\zeta ({{{\mathcal {A}}}}_{\gamma ,N}) = \log (1+\sqrt{1\gamma })\) is given in [70, Proposition 6], which is independent on the parameter N. However, this is clearly not the case for our new bound \({{\widehat{\Upsilon }}}_\alpha \). For covariant channels \({{{\mathcal {D}}}}_p\), \({{{\mathcal {E}}}}_p\) and \({{{\mathcal {N}}}}_{p,p^2}\), the bound \({{\widehat{\Upsilon }}}_{\alpha {\scriptscriptstyle (10)}}\) also coincides with the Upsilon information \(\Upsilon \) in subfigures (a–c). In particular, \({{\widehat{\Upsilon }}}_{\alpha {\scriptscriptstyle (10)}}\) is given by \(1p\) in subfigure (b), witnessing again the strong converse property of the qubit erasure channel \(C({{{\mathcal {E}}}}_p) = C^\dagger ({{{\mathcal {E}}}}_p) = 1p\) [87]. Such tightness can also be observed here for the dephrasure channel \({{{\mathcal {N}}}}_{p,q}\) and it would be easy to show that \(\chi ({{{\mathcal {N}}}}_{p,q}) = C({{{\mathcal {N}}}}_{p,q}) = C^{\dagger }({{{\mathcal {N}}}}_{p,q}) = 1q\) which is independent of the dephasing noise parameter p.
From Fig. (10 e), \({{\widehat{\Upsilon }}}_\alpha \) does not give improvement for the amplitude damping channel \({{{\mathcal {A}}}}_{\gamma ,0}\). However, when considering the composition channel \({{{\mathcal {M}}}}_p \equiv {{{\mathcal {A}}}}_{p,0}\circ {{{\mathcal {Z}}}}_p\), which was studied by Aliferis et al. [88] in the context of faulttolerant quantum computation, the strong converse bound \({{\widehat{\Upsilon }}}_\alpha \) works unexpectedly well, as shown in Fig. 11.
Since \({{{\mathcal {M}}}}_p\) is an entanglementbreaking channel at \(p = 1/2\), it is expected that \({{{\mathcal {M}}}}_p\) is approximately entanglementbreaking around this point. Therefore, a converse bound \(C_{\mathrm{EB}}\) was established in [34, Corollary III.7] by using a continuity argument of the classical capacity. It has been shown in [34, Figure 5] that this continuity bound \(C_{\mathrm{EB}}\) gives certain improvement on \(C_\beta \) for the interval around \(p = 1/2\). However, Fig. 11 shows that the new strong converse bound \({{\widehat{\Upsilon }}}_{\alpha {\scriptscriptstyle (10)}}\) is much tighter than both \(C_{\mathrm{EB}}\) and \(C_\beta \) for all \({{{\mathcal {M}}}}_p\) with \(p\in [0,0.75]\). The Holevo information \(\chi \) is also numerically computed by utilizing the algorithm^{Footnote 23} in [89]. We observe that the upper bound \({{\widehat{\Upsilon }}}_{\alpha {\scriptscriptstyle (10)}}\) and the lower bound \(\chi \) are very close, leading to a good estimation to the classical capacity of \({{{\mathcal {M}}}}_p\).
7 Magic State Generation
7.1 Background
The idea of faulttolerant quantum computation proposes a reliable framework to implement practical quantum computation against noise and decoherence (e.g. [90,91,92,93]). Due to the GottesmanKnill theorem [94, 95], quantum circuits constructed by stabilizer operations can be efficiently simulated by a classical computer. Therefore, to fully power the universal quantum computation the stabilizer operations must be supplemented with some other faulttolerant nonstabilizer resource. A celebrated scheme for this is given by the state injection technique that allows us to implement nonstabilizer operations by mixing the stabilizer operations with a key ingredient called “magic states” [96, 97]. These are nonstabilizer states that must be prepared using the experimentally costly process of magic state distillation (e.g. [98, 99]). While extensive efforts have been devoted to construct efficient distillation codes (e.g. [94, 99,100,101]), recent study in [3] as well as [4] initiate the investigation of magic state generation via a general quantum channel, aiming to exploit the power and the limitations of a noisy quantum channel in the scenario of faulttolerant quantum computation.
Of particular interest is the work [3] which identifies a larger class of operations (completely positive Wigner preserving (CPWP) operations), that can be efficiently simulated via classical algorithms. Based on this notion of free operations, the authors established a complete resource theory framework and introduced two efficiently computable magic measures for quantum channels, named mana (\({\mathbb {M}}\)) and maxThauma (\(\theta _{\max }\)) respectively. They proved several desirable properties of these two measures, and further showcased that these channel measures provided strong converse bounds for the task of magic state generation as well as lower bounds for the task of quantum channel synthesis.
7.2 Summary of results
In this part, we aim to push forward the study in [3] by considering the Thauma measure induced by the geometric Rényi divergence. Our results can be summarized as follows.
In Sect. 7.4, we prove that the geometric Rényi Thauma of a channel (\({{\widehat{\theta }}}_{\alpha }\)) possesses all the nice properties that are held by the mana and maxThauma, including the reduction to states, monotonicity under CPWP superchannels, faithfulness, amortization inequality, subadditivity under channel composition, additivity under tensor product as well as a semidefinite representation.
In Sect. 7.5, we exhibit that the geometric Rényi Thauma of a channel is a strong converse bound on the magic state generation capacity (the maximum number of magic state that can be produced per channel use of \({{{\mathcal {N}}}}\) via adaptive protocols), improving the maxThauma in general. More precisely, we show that
where \(C_\psi ({{{\mathcal {N}}}})\) and \(C_\psi ^\dagger ({{{\mathcal {N}}}})\) denote the capacity of a channel \({{{\mathcal {N}}}}\) to generate magic state \(\psi \) and its corresponding strong converse capacity, respectively, and \(\theta _{\min }(\psi )\) is a constant coefficient for given \(\psi \).
In Sect. 7.6, we show that the geometric Rényi Thauma can also provide lower bounds for the task of quantum channel synthesis. That is, we prove that the number of uses of a channel \({{{\mathcal {N}}}}'\) required to implement another channel \({{{\mathcal {N}}}}\) is bounded from below by \({{{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}})}/{{{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}}')}\) for all \(\alpha \in (1,2]\). This new bound complements the previous results by mana and maxThauma.
7.3 Preliminaries of the resource theory of magic
We first review some basic formalism of the resource theory of magic. Throughout this part, a Hilbert space implicitly has an odd dimension, and if the dimension is not prime, it should be understood to be a tensor product of Hilbert spaces each having odd prime dimension. Let \(\{j\rangle \}_{j=0}^{d1}\) be the standard computational basis. For a prime number d, the generalized Pauli operator (or sometimes called the shift and boost operators) X, Z are respectively defined as
where \(\oplus \) denotes the addition modulo d. The Heisenberg–Weyl operators are defined as^{Footnote 24}
For a system with composite Hilbert space \({{{\mathcal {H}}}}_A \otimes {{{\mathcal {H}}}}_B\), the Heisenberg–Weyl operators are the tensor product of the Heisenberg–Weyl operators on subsystems \(T_{{\varvec{u}}_A \otimes {\varvec{u}}_B} = T_{{\varvec{u}}_A} \otimes T_{{\varvec{u}}_B}\). For each point \({\varvec{u}}\in {{{\mathbb {Z}}}}_d \times {{{\mathbb {Z}}}}_d\) in the discrete phase space, there is a corresponding operator
The value of the discrete Wigner representation of a quantum state \(\rho \) at \({\mathbb {A}}_{{\varvec{u}}}\) is given by
The Wigner trace and Wigner spectral norm of an Hermitian operator V are defined as
respectively. For any Hermiticitypreserving map \({{{\mathcal {N}}}}\), its discrete Wigner function is defined as
with \(J_{{{{\mathcal {N}}}}}\) being the Choi matrix of \({{{\mathcal {N}}}}\). The set of quantum states with a nonnegative Wigner function is denoted as
A quantum operation \({{{\mathcal {E}}}}\) is CPWP if the following holds for any system R with odd dimension [3]
Definition 36
(Mana). The mana of a quantum state \(\rho \) is defined as [103]
The mana of a quantum channel \({{{\mathcal {N}}}}_{A\rightarrow B}\) is defined as^{Footnote 25} [3]
Definition 37
(Thauma). Let \({\varvec{D}}\) be a generalized quantum divergence. The generalized Thauma of a quantum state \(\rho \) is defined as [102]
where \({{{\mathcal {W}}}}\equiv \{\sigma \,\, {\mathbb {M}}(\sigma ) \le 0, \sigma \ge 0\}\) is the set of subnormalized states with nonpositive mana. The generalized Thauma of a quantum channel \({{{\mathcal {N}}}}_{A\rightarrow B}\) is defined as [3]
where \(\varvec{{{{\mathcal {V}}}}}_{{\mathbb {M}}}\equiv \{{{{\mathcal {E}}}}\in {\mathrm{CP}}(A:B) {\mathbb {M}}({{{\mathcal {E}}}}) \le 0\}\) is the set of subchannels with nonpositive mana.
In particular, the maxThauma of a channel is induced by the maxrelative entropy [3]
7.4 Geometric Rényi Thauma of a channel
In this section, we investigate the generalized Thauma induced by the geometric Rényi divergence:
The authors in [3] proved that the mana and maxThauma of a channel possess several nice properties, as listed in Table 2. Here we aim to show that all the desirable properties are also held by the geometric Rényi Thauma as well. These basic properties will be utilized in the next two sections for improving the converse bound on magic state generation capacity and the efficiency of quantum channel synthesis.
Lemma 38
The following properties hold for the geometric Rényi Thauma of a channel when \(\alpha \in (1,2]\):

(Reduction to states): Let \({{{\mathcal {N}}}}(\rho ) = {\text {Tr}}[\rho ] \sigma \) be a replacer channel with fixed \(\sigma \) for any \(\rho \). Then
$$\begin{aligned} {{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}}) = {{\widehat{\theta }}}_{\alpha }(\sigma ). \end{aligned}$$(172) 
(Monotonicity): Let \({{{\mathcal {N}}}}\) be a quantum channel and \(\Gamma \) be a CPWP superchannel. Then
$$\begin{aligned} {{\widehat{\theta }}}_{\alpha }(\Gamma ({{{\mathcal {N}}}})) \le {{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}}). \end{aligned}$$(173) 
(Faithfulness): \({{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}})\) is nonnegative for any quantum channel \({{{\mathcal {N}}}}\) and
$$\begin{aligned} {{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}}) = 0 \quad \text {if and only if} \quad {{{\mathcal {N}}}}\in {\mathrm{CPWP}}. \end{aligned}$$(174)
Proof
The first two properties directly follow from the argument for the generalized Thauma in [3, Proposition 9 and 10]. The third property follows from the argument in [3, Proposition 11] and the fact that the geometric Rényi divergence is continuous and strongly faithful (i.e, \({{\widehat{D}}}_\alpha (\rho \Vert \sigma ) \ge 0\) in general and \({{\widehat{D}}}_{\alpha }(\rho \Vert \sigma ) = 0\) if and only if \(\rho = \sigma \)).
\(\quad \square \)
Lemma 39
(Amortization). For any quantum state \(\rho _{RA}\), any quantum channel \({{{\mathcal {N}}}}_{A\rightarrow B}\) and the parameter \(\alpha \in (1,2]\), it holds
Proof
The proof follows the similar steps as Proposition 16. We only need to show that for any subnormalized state \(\sigma _{RA} \in {{{\mathcal {W}}}}\) and any subchannel \({{{\mathcal {E}}}}\in \varvec{{{{\mathcal {V}}}}}_{{\mathbb {M}}}\), it holds \(\gamma _{RB}\equiv {{{\mathcal {E}}}}_{A\rightarrow B}(\sigma _{RA}) \in {{{\mathcal {W}}}}\). This can be checked as follows:
The first line is the definition of the Wigner trace norm in (162). The second line is a chain relation in [3, Lemma 1]. The third line follows from the triangle inequality of the absolute value function. The fourth line follows by grouping the components with respect to index \({\varvec{v}}_B\). The fifth line follows since \({{{\mathcal {E}}}}\in \varvec{{{{\mathcal {V}}}}}_{\mathbb {M}}\) and thus \(\sum _{{\varvec{v}}_B} \big  W_{{{{\mathcal {E}}}}}({\varvec{v}}_B{\varvec{y}}_A)\big  \le \max _{{\varvec{y}}_A}\sum _{{\varvec{v}}_B} \big  W_{{{{\mathcal {E}}}}}({\varvec{v}}_B{\varvec{y}}_A)\big  \le 1\). The last line follows since \(\sigma _{RA} \in {{{\mathcal {W}}}}\). Thus we can conclude that \(\gamma _{RB} \in {{{\mathcal {W}}}}\). This completes the proof. \(\quad \square \)
Lemma 40
(Subadditivity). For any two quantum channels \({{{\mathcal {N}}}}_1\), \({{{\mathcal {N}}}}_2\) and \(\alpha \in (1,2]\), it holds
Proof
Suppose the optimal solution of \({{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}}_1)\) and \({{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}}_2)\) are taken at \({{{\mathcal {E}}}}_1\) and \({{{\mathcal {E}}}}_2\), respectively. By the subadditivity of the mana under composition, we have \({\mathbb {M}}({{{\mathcal {E}}}}_2\circ {{{\mathcal {E}}}}_1) \le 0\) (see [3, Proposition 5]). Thus \({{{\mathcal {E}}}}_2\circ {{{\mathcal {E}}}}_1\) is a feasible solution for \({{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}}_2\circ {{{\mathcal {N}}}}_2)\) and we have
where the second inequality follows from Lemma 8, the last equality follows from the optimality assumption of \({{{\mathcal {E}}}}_1\) and \({{{\mathcal {E}}}}_2\). \(\quad \square \)
Lemma 41
(SDP formula). For any quantum channel \({{{\mathcal {N}}}}\) and \(\alpha (\ell ) = 1+2^{\ell }\) with \(\ell \in {\mathbb {N}}\), it holds
Proof
This directly follows from Lemma 9 and the definition of mana in (167). Note that the absolute value conditions can be written as semidefinite conditions by introducing slack variables. \(\quad \square \)
Lemma 42
(Additivity). The geometric Rényi Thauma at \(\alpha =2\) is additive under tensor product. That is, for any two quantum channels \({{{\mathcal {N}}}}_1\), \({{{\mathcal {N}}}}_2\), it holds
Proof
We first prove the subadditivity. Suppose the optimal solution of \({{\widehat{\theta }}}_2({{{\mathcal {N}}}}_1)\) and \({{\widehat{\theta }}}_2({{{\mathcal {N}}}}_2)\) are taken at \({{{\mathcal {E}}}}_1\) and \({{{\mathcal {E}}}}_2\) respectively. Then we have \({\mathbb {M}}({{{\mathcal {E}}}}_1\otimes {{{\mathcal {E}}}}_2) = {{{\mathcal {M}}}}({{{\mathcal {E}}}}_1) + {{{\mathcal {M}}}}({{{\mathcal {E}}}}_2) \le 0\) since mana is additive under tensor product [3, Proposition 4]. This implies that \({{{\mathcal {E}}}}_1\otimes {{{\mathcal {E}}}}_2\) is a feasible solution for \({{\widehat{\theta }}}_2({{{\mathcal {N}}}}_1\otimes {{{\mathcal {N}}}}_2)\). Thus we have
where the first equality follows from Lemma 6 and the second equality follows from the optimality assumption of \({{{\mathcal {E}}}}_1\) and \({{{\mathcal {E}}}}_2\).
We now show the superadditivity by utilizing the dual formula of (184). According to the Lagrangian method, we have the dual problem as
It is easy to check that the strong duality holds. Note that if we replace K as xK with \(x = 1\), the optimization is unchanged. Thus we can choose scalar \(x = {\text {Tr}}(J_{{{{\mathcal {N}}}}}K)^\dagger / {\text {Tr}}J_{{{{\mathcal {N}}}}}K\) to make the term \({\text {Tr}}J_{{{{\mathcal {N}}}}} (xK) = {\text {Tr}}J_{{{{\mathcal {N}}}}}K\) to a real scalar. Thus optimization (187) is equivalent to
Again, by replacing \({{\widetilde{K}}} = K/w\), \({{\widetilde{Z}}} = Z/w^2\) and \({{\widetilde{f}}}_{{\varvec{u}}} = f_{{\varvec{u}}}/w^2\), we have
For any fixed \({\text {Tr}}J_{{{{\mathcal {N}}}}} {{\widetilde{K}}} \) and \(\sum _{{\varvec{u}}} {{\widetilde{f}}}_{{\varvec{u}}}\), we can quickly check that the optimal solution of the objective function is always taken at \(w = {\text {Tr}}J_{{{{\mathcal {N}}}}} {{\widetilde{K}}} / (\sum _{{\varvec{u}}} {{\widetilde{f}}}_{{\varvec{u}}})\) with the optimal value \({\text {Tr}}J_{{{{\mathcal {N}}}}} {{\widetilde{K}}} ^2/ (\sum _{{\varvec{u}}} {{\widetilde{f}}}_{{\varvec{u}}})\). Thus the optimization (189) is equivalent to
Suppose the optimal solution of \({{\widehat{\theta }}}_2({{{\mathcal {N}}}}_1)\) and \({{\widehat{\theta }}}_2({{{\mathcal {N}}}}_2)\) are taken at \(\{K_1,Z_1, f^1_{{\varvec{u}}},\rho _1\}\) and \(\{K_2,Z_2, f^2_{{\varvec{v}}},\rho _2\}\) respectively. We can check that their tensor product \(\{K_1\otimes K_2, Z_1 \otimes Z_2, f^1_{{\varvec{u}}} f^2_{{\varvec{v}}},\rho _1\otimes \rho _2\}\) forms a feasible solution for \({{\widehat{\theta }}}_2({{{\mathcal {N}}}}_1\otimes {{{\mathcal {N}}}}_2)\) in (190). Thus we have
which completes the proof. \(\quad \square \)
Remark 8
Based on numerical observations, we expect that the additivity of the geometric Rényi Thauma holds for general \(\alpha \in (1,2]\). However, the current proof seems to only work for \(\alpha = 2\).
7.5 Magic state generation capacity
In [3], the authors study an information task which uses a quantum channel to produce magic states and quantifies the “magic of channel” by the amount of magic state generated per channel use. Here, we simply dub it as the magic state generation capacity^{Footnote 26} as it characterizes the capability of a channel to generate magic states. The most general protocol to produce a magic state can be proceeded as follows (see Fig. 12).
First, we start from preparing a quantum state \(\rho ^{\scriptscriptstyle {(1)}}_{R_1 A_1}\) via a free CPWP operation \({{{\mathcal {F}}}}^{\scriptscriptstyle {(1)}}_{\emptyset \rightarrow R_1A_1}\). Then we apply the given channel \({{{\mathcal {N}}}}\) on system \(A_1\) and obtain a quantum state \(\sigma _{R_1B_1}^{\scriptscriptstyle {(1)}} = {{{\mathcal {N}}}}_{A_1\rightarrow B_1}(\rho ^{\scriptscriptstyle {(1)}}_{R_1 A_1})\). After this, we perform another free CPWP operation \({{{\mathcal {F}}}}^{\scriptscriptstyle {(2)}}_{R_1 B_1 \rightarrow R_2A_2}\) and then apply the channel \({{{\mathcal {N}}}}\) again. These processes can be conducted iteratively n times, and we obtain a quantum state \(\sigma ^{\scriptscriptstyle {(n)}}_{R_n B_n}\). At the end of such a protocol, a final free CPWP operation \({{{\mathcal {F}}}}^{\scriptscriptstyle {(n+1)}}_{R_nB_n \rightarrow S}\) is performed, producing a quantum state \(\omega _S\).
For any error tolerance \(\varepsilon \in [0,1]\), the above procedure defines an \((n,k,\varepsilon )\) \(\psi \)magic state generation protocol, if the final state \(\omega \) has a sufficiently high fidelity with k copies of the target magic state \(\psi \),
A rate r is achievable if for all \(\varepsilon \in (0,1]\) and \(\delta > 0\) and sufficiently large n, there exists an \((n,n(r\delta ),\varepsilon )\) \(\psi \)magic state generation protocol as depicted above. Then the \(\psi \)magic state generation capacity of the channel \({{{\mathcal {N}}}}\) is defined as the supremum of all achievable rates and is denoted as \(C_{\psi }({{{\mathcal {N}}}})\). On the other hand, \(r_0\) is called a strong converse rate if for every \(r > r_0\), the fidelity \(1\varepsilon \) of any generation protocol will decays to zero as the number of rounds n increases. The strong converse capacity, denoted as \(C_{\psi }^{\dagger }({{{\mathcal {N}}}})\) is the infimum of all strong converse rates. By definition, we have \(C_{\psi }({{{\mathcal {N}}}}) \le C_{\psi }^{\dagger }({{{\mathcal {N}}}})\) in general.
Based on the amortization inequality in Lemma 39, a similar argument as [3, Proposition 20] gives us the following improved bound on the magic state generation capacity:
Theorem 43
(Application 6). For any quantum channel \({{{\mathcal {N}}}}\) and \(\alpha \in (1,2]\), it holds
where \(\theta _{\min }(\psi ) = \min _{\sigma \in {{{\mathcal {W}}}}} D_{\min }(\psi \Vert \sigma )\) is the minThauma of the magic state \(\psi \).
Proof
The first inequality holds by definition. The last inequality is a direct consequence of the relation \({{\widehat{D}}}_{\alpha }(\rho \Vert \sigma ) \le D_{\max }(\rho \Vert \sigma )\) proved in Lemma 4. It remains to show the second inequality. The main ingredient to prove this is the amortization property of the geometric Rényi Thauma in Lemma 39. Consider n round magic state generation protocol as shown in Fig. 12. For each round, denote the input state of the channel \({{{\mathcal {N}}}}\) as \(\rho ^{\scriptscriptstyle (i)}_{R_iA_i}\) and the output state as \(\sigma ^{\scriptscriptstyle (i)}_{R_iB_i}\). The final state after n round operations is denoted as \(\omega _S\). Thus we have
The first and third lines follow from the monotonicity of the geometric Rényi Thauma of a quantum state under CPWP operations. The second line follows since \({{\widehat{\theta }}}_{\alpha }(\rho ^{\scriptscriptstyle (1)}_{R_1A_1}) = 0\). The last line follows from Lemma 39.
Note that any \(\psi \)magic state generation protocol is characterized by a triplet \((n,k,\varepsilon )\) with the number of rounds n, the number of target magic states k and the error tolerance \(\varepsilon \). Denote \(r = k/n\). By definition, we have \({\text {Tr}}\left[ \omega _S \psi \rangle \langle \psi ^{\otimes k}\right] \ge 1\varepsilon \) . Moreover, for any \(\sigma _S \in {{{\mathcal {W}}}}\), it holds \({\text {Tr}}\left[ \sigma _S \psi \rangle \langle \psi ^{\otimes k}\right] \le 2^{nr \theta _{\min }(\psi )}\) [102]. Without loss of generality, we can assume that \(\varepsilon \le 12^{nr \theta _{\min }(\psi )}\). Otherwise, any value above the capacity would already be a strong converse rate since \(1\varepsilon < 2^{nr \theta _{\min }(\psi )}\). Thus for any \(\sigma _S \in {{{\mathcal {W}}}}\), we have the inequalities
Consider a quantum channel \({{{\mathcal {N}}}}(\gamma ) = \left[ {\text {Tr}}\psi \rangle \langle \psi ^{\otimes k} \gamma \right] 0\rangle \langle 0 + \left[ 1  {\text {Tr}}\psi \rangle \langle \psi ^{\otimes k} \gamma \right] 1\rangle \langle 1\). Due to the dataprocessing inequality, we have
where \(\delta _\alpha (p\Vert q)\equiv \frac{1}{\alpha 1} \log \big [p^\alpha q^{1\alpha } + (1p)^\alpha (1q)^{1\alpha }\big ]\). The last inequality follows from the monotonicity property that \(\delta _\alpha (p'\Vert q) \le \delta _\alpha (p\Vert q)\) if \(p \le p' \le q\) and \(\delta _\alpha (p\Vert q') \le \delta _\alpha (p\Vert q)\) if \(p \le q' \le q\) [62]. Then we have
Combining Eqs. (198) and (201), we have
which is equivalent to
This implies that if the generation rate r is strictly larger than \({{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}})/\theta _{\min }(\psi )\), the fidelity of the generation protocol \(1\varepsilon \) decays exponentially fast to zero as the number of rounds n increases. Or equivalently, we have \(C_\psi ^{\dagger }({{{\mathcal {N}}}}) \le {{{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}})}/{\theta _{\min }(\psi )}\) and completes the proof. \(\quad \square \)
Remark 9
If the target magic state is T state \(T\rangle :=(\xi 0\rangle +1\rangle +\xi ^{1}2\rangle )/\sqrt{3}\) with \(\xi =e^{2\pi i /9}\) or \(H_+\) state (the eigenstate of a qutrit Hadamard gate with corresponding eigenvalue \(+1\)), we have \(\theta _{\min }(T)= \log (1+2\sin (\pi /18))\) and \(\theta _{\min }(H_+) = \log (3  \sqrt{3})\), respectively [102, Proposition 2].
Consider a qutrit quantum channel \({{{\mathcal {D}}}}_p\circ T\) composed by a Tgate with a qutrit depolarizing noise \({{{\mathcal {D}}}}_p\). The above Fig. 13 compares different converse bounds on the Tmagic state generation capacity of the channel \({{{\mathcal {D}}}}_p\circ T\). It is clear that our new bound based on the geometric Rényi divergence is significantly tighter than the others.
7.6 Quantum channel synthesis
Another fundamental question in the resource theory of magic asks how many instances of a given quantum channel \({{{\mathcal {N}}}}'\) are required to simulate another quantum channel \({{{\mathcal {N}}}}\), when supplemented with free CPWP operations. Such a general scheme is illustrated in Fig. 14. Denote \(S({{{\mathcal {N}}}}'\rightarrow {{{\mathcal {N}}}})\) as the smallest number of \({{{\mathcal {N}}}}'\) channels required to implement the target channel \({{{\mathcal {N}}}}\) exactly.
The following result establishes new fundamental limits on the quantum channel synthesis problem by employing the geometric Rényi Thauma of the resource and target channels, respectively.
Theorem 44
Let \({{{\mathcal {N}}}}'\) and \({{{\mathcal {N}}}}\) be two qudit quantum channels. Then the number of uses of a channel \({{{\mathcal {N}}}}'\) required to implement \({{{\mathcal {N}}}}\) is bounded from below as
Proof
Suppose the optimal simulation protocol requires to use the resource channel \(n = S({{{\mathcal {N}}}}'\rightarrow {{{\mathcal {N}}}})\) times and the protocol is given by
with \({{{\mathcal {F}}}}^{\scriptscriptstyle (i)}\) being CPWP operations. Using the subadditivity of the geometric Rényi Thauma in Lemma 40 iteratively, we have
where the equality follows from the faithfulness of the geometric Rényi Thauma in Lemma 38. Therefore, we have \(S({{{\mathcal {N}}}}' \rightarrow {{{\mathcal {N}}}}) = n \ge {{{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}})}/{{{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}}')}\), which concludes the proof. \(\quad \square \)
Together with the previous result in [3, Proposition 23], we have
Corollary 45
Let \({{{\mathcal {N}}}}'\) and \({{{\mathcal {N}}}}\) be two qudit quantum channels. Then the number of uses of a channel \({{{\mathcal {N}}}}'\) required to implement \({{{\mathcal {N}}}}\) is bounded from below as
Remark 10
Note that each lower bound is given by a quotient of two quantities. It is thus not known which one is tighter in general, despite that \({{\widehat{\theta }}}_{\alpha }({{{\mathcal {N}}}}) \le \theta _{\max }({{{\mathcal {N}}}}) \le {\mathbb {M}}({{{\mathcal {N}}}})\).
8 Conclusions
We have established several fundamental properties of the geometric Rényi divergence as well as its channel divergence. We further demonstrated the usefulness of these properties in the study of quantum channel capacity problems, strengthening the previously bestknown results based on the maxrelative entropy in general. We expect that the technical tools established in this work can find a diverse range of applications in other research areas, such as quantum network theory and quantum cryptography. For example, we illustrate one more application of the geometric Rényi divergence in the task of quantum channel discrimination in Appendix D.
There are also some interesting problems left for future investigation. The Umegaki relative entropy is the most commonly studied quantum divergence because of its operational interpretation as an optimal error exponent in the hypothesis testing problem (known as the quantum Stein’s lemma) [37, 104]. One open question is to know whether the geometric Rényi divergence as well as the Belavkin–Staszewski relative entropy have any operational interpretation.
Notes
For unassisted and twoway assisted private capacities, the new bounds are efficiently computable for general qubit channels.
It is also called the maximal Rényi divergence (see e.g. [35, Section 4.2.3]) as it is the largest possible quantum Rényi divergence satisfying the dataprocessing inequality. We here use the term “geometric” as its closedform expression is given by the matrix geometric means, depicting the nature of this quantity.
Note that all of the properties can be trivially extended to general completely positive maps. We only state these properties for quantum channels or subchannels as they are most frequently used in quantum information theory.
This is clear from the minimization formula of \({{\widehat{D}}}_\alpha \) in [15, Eq. (11)] and the monotonicity of classical Rényi divergence.
We refer to the work [11, Section II] for rigorous definitions of the unassisted quantum capacity and its strong converse rate.
We refer to [22, Section 4] for rigorous definitions of the PPT/twoway assisted quantum capacity and its strong converse rate.
It makes no difference by considering \(\Vert \rho _{AB}^{{\mathsf {T}}_B}\Vert _1 = 1\) or \(\Vert \rho _{AB}^{{\mathsf {T}}_B}\Vert _1 \le 1\).
The name follows from the Theta set \(\varvec{{{{\mathcal {V}}}}}_\Theta \) where \(\Theta \) was originally used as the transpose map in the Holevo–Werner bound.
We refer to the work [23, Page 23] for rigorous definitions of the PPT/twoway assisted quantum capacity of a bidirectional channel and its strong converse rate.
All the data and codes can be found on the GitHub page https://github.com/fangkunfred.
We refer to the work [12, Section V.A] for rigorous definitions of the unassisted private capacity and its strong converse rate.
We refer to [12, Section V.A] for rigorous definitions of the twoway assisted private capacity and its strong converse rate.
These are two main limitations of converse bounds established by using the continuity of the channel capacities.
We refer to [26, Section IV.A] for rigorous definitions of the unassisted classical capacity and its strong converse rate.
This can be easily seen from the radius characterization of the Holevo capacity \(\chi ({{{\mathcal {N}}}}) = \min _{\sigma } \max _{\rho } D({{{\mathcal {N}}}}(\rho )\Vert \sigma )\) [86].
It makes no difference by considering \(\beta (J_{{{{\mathcal {M}}}}}) \le 1\) or \(\beta (J_{{{{\mathcal {M}}}}}) = 1\).
The MATLAB codes we use are given from [34].
The definition here is sightly different from some literatures. We adopt the same notion as in [102].
This can be seen as an analog of Holevo–Werner bound for quantum capacity of a channel or lognegativity of a quantum state.
It is named as “distillable magic of a channel” in [3]. We here call it magic state generation capacity as it is consistent with the name of entanglement/coherence generation capacity in the existing literature.
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Acknowledgements
We would like to thank Omar Fawzi for bringing to our attention the open question by Berta et al. [20, Eq. (55)] and for suggesting the chain rule for the geometric Rényi divergence, which helped us simplify and unify the proofs of Proposition 16, 20, 28, 39. We also thank David Sutter for encouraging us to make some proof steps more precise. We thank Sergey N. Filippov, Stefano Pirandola and Mark Wilde for their helpful feedback to this manuscript. KF and HF acknowledge the support of the University of Cambridge Isaac Newton Trust Early Career grant RG74916.
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Appendices
Technical Lemmas
In this section, we present several technical lemmas that are used in the main text.
Lemma 46
([14]). For any positive semidefinite operators X and Y with \(X \ll Y\), Hermitian operator M and \(\alpha (\ell ) = 1 + 2^{\ell }\) with \(\ell \in {\mathbb {N}}\), the matrix inequality \(G_{1\alpha }(X, Y) \le M\) holds if and only if
When \(\ell = 0\), the conditions in the loop are taken as trivial. Here the short notation that \(\llbracket X \rrbracket _{{\mathsf {P}}}\), \(\llbracket X \rrbracket _{{\mathsf {E}}}\) and \(\llbracket X \rrbracket _{{\mathsf {H}}}\) represent the positive semidefinite condition \(X \ge 0\), the equality condition \(X = 0\) and the Hermitian condition \(X = X^\dagger \), respectively.
The following lemma proves a transformer inequality of the weighted geometric matrix means. Here we require this result to hold for a specific range of the weighting parameter that to the best of our knowledge has not been stated properly before.
Lemma 47
(Transformer inequality). Let X and Y be two positive operators, K be any linear operator, and \(\alpha \in (1,2]\). Then it holds
Furthermore, if K is invertible the statement above holds with equality.
Proof
Before proving the assertion of the lemma we need to collect some basic properties. We start by recalling the known result [41] that for \(\beta \in [0,1]\), we have
As a next preparatory fact we show that the desired statement is correct for \(\alpha = 2\), i.e.,
To see this we recall that by Schur’s complement [105, Theorem 1.3.3] we have
Choosing \(M = G_{1}(X,Y)\) thus gives
which then implies
because \(Z \mapsto K Z K^\dagger \) is a positive map [105, Exercise 3.2.2]. Using (212) again then implies (211). Because the maps \(t \mapsto t^{1}\) is operator antimonotone [106, Table 2.2] we have
As a final property we recall a fact from [56, Equation 19] stating that
Now we are ready to prove the assertion of the lemma. For any \(\beta \in [1,0)\), using (216) we have
where the first inequality step follows from (210) and (215). The second inequality is implied by (211). The final step uses (216) again.
The fact that the transformer inequality holds with equality in case K is invertible follows by applying the inequality twice as
which proves that the two inequalities above actually hold with equality. \(\quad \square \)
Corollary 48
Let X and Y be two positive operators, K be any linear operator. Let \(D_{op}(X,Y) = X^{\frac{1}{2}} \log \big (X^{\frac{1}{2}} Y^{1} X^{\frac{1}{2}}\big ) X^{\frac{1}{2}}\) be the operator relative entropy. Then the \(D_{op}\) satisfies the transformer inequality:
Furthermore, if K is invertible the statement above holds with equality.
Proof
Due to the fact that \(\lim _{\gamma \rightarrow 0} \frac{1}{\gamma } (x^{\gamma }  1) = \log (x)\), we have the limit identity
Then we have
where the first and last equalities follow from Eq. (225), the inequality follows from Lemma 47. \(\quad \square \)
A Hierarchy for ConstantBounded Subchannels
In this section we discuss the set of constantbounded subchannels
Denote \({{{\mathcal {N}}}}_\sigma \) as the constant map induced by the state \(\sigma \). For any \({{{\mathcal {M}}}}\in \varvec{{{{\mathcal {V}}}}}_{cb}\) the condition \({{{\mathcal {M}}}}(\rho )\le \sigma \) for all \(\rho \) is equivalent that \({{{\mathcal {N}}}}_\sigma  {{{\mathcal {M}}}}\) is a positive map. In terms of their Choi matrices, we have \({\mathbb {1}}_A\otimes \sigma _B  J_{{{{\mathcal {M}}}}} \in \mathcal {BP}(A:B)\) where \(\mathcal {BP}(A:B)\) is the cone of block positive operators. Thus we have
Due to the difficulty of finding a semidefinite representation for \(\mathcal {BP}\) [107], we do not expect that there is a semidefinite representation for the set \(\varvec{{{{\mathcal {V}}}}}_{cb}\). Nevertheless, the cone \(\mathcal {BP}\) can be approximated by a complete hierarchy from the inside as
where \(\mathcal {DPS}_k^*\) is the dual cone of the wellknown DPS hierarchy [108, 109] and is given by the semidefinite representation [110]
where the index \([s_1:s_2]\equiv \{s_1,s_1+1,\ldots s_2\}\), \([s]\equiv [1:s]\) and \(\Pi _k\) is the projector on the symmetry subspace of \({{{\mathcal {H}}}}_B^{\otimes k}\). Then we can construct a complete semidefinite hierarchy for the set \(\varvec{{{{\mathcal {V}}}}}_{bc}\) as
with each level given by
Consider the first level of the hierarchy
Denote \(R = {\mathbb {1}}_A \otimes \sigma _B  W_0\), we obtain
By symmetrizing the conditions \(X  Y \ge 0\) to \(X \pm Y \ge 0\), we will retrieve the set \(\varvec{{{{\mathcal {V}}}}}_\beta \). Similarly, by using a different way of variable replacement \(V = {\mathbb {1}}_A \otimes \sigma  W_1\), we have
Then symmetrizing the conditions, we will obtain a set of subchannels \(\varvec{{{{\mathcal {V}}}}}_\zeta \) which is exactly the zero set of the strong converse bound \(C_\zeta \).
The restriction to symmetric conditions ensures that the set is closed under tensor product, which is a key ingredient to proving the subadditivity of the geometric Rényi Upsiloninformation in Proposition 33. Thus a further improvement of the result in the main text can be made by considering a symmetric restriction of a higher level set \(\varvec{{{{\mathcal {V}}}}}_{cb}^k\). But we also note that the size of the SDP will exponentially increase in the level of k.
A Detailed Comparison for Generalized Amplitude Damping Channels
In this section we give a detailed comparison of our new strong converse bounds with previously known results for the generalized amplitude damping (GAD) channels. This class of channels has been systematically investigated in [70], with several converse bounds being established therein based on the dataprocessing inequality and the continuity of channel capacities as well as a few different techniques. The following plots of previous weak converse bounds are generated via the MATLAB codes provided in the arXiv submission of [70].
Due to the covariance symmetry of the GAD channels under the Pauliz operator, the quantities introduced in this work do not provide advantage over the Rains information R (resp. the relative entropy of entanglement \(E_R\)) in terms of the unassisted quantum (resp. private) capacity. In the following, we will focus our comparison for the twoway assisted scenarios where both R and \(E_R\) are not known as valid converse bounds. The comparison result for the twoway assisted quantum capacity is given in Fig. 15. The red solid line is the previously bestknown converse bound composed by several different quantities. It is clear that our new strong converse bound \({{\widehat{R}}}_{\alpha {\scriptscriptstyle (10)},\Theta }\) can be much tighter in most cases. Note that for the twoway assisted private capacity, we will obtain exactly the same result as Fig. 15 by replacing \(R_{\max }\) with \(E_{\max }\) and \({{\widehat{R}}}_{\alpha {\scriptscriptstyle (10)},\Theta }\) with \({{\widehat{E}}}_{\alpha {\scriptscriptstyle (10)},\Sigma }\) respectively.
The comparison result for the classical capacity is given in Fig. 16. The red solid line is the previously bestknown converse bound composed by several different quantities. It is clear that our new strong converse bound \({{\widehat{\Upsilon }}}_{\alpha {\scriptscriptstyle (10)}}\) can make further improvement at some parameter range, particularly for low to medium amplitude damping noise. In the range of high noise, the GAD channel becomes entanglementbreaking. Thus the \(\varepsilon \)entanglement breaking upper bound \(C_{\mathrm{EB}}\) will be the tightest one, as expected. We do not show the plot for \(N=0.5\), because the channel becomes a qubit unital channel and thus its Holevo information is already tight [112]. We do not explicitly depict the weak converse bound in [113] since it is not tight to the cases we present.
Application in Quantum Channel Discrimination
A fundamental problem in quantum information theory is to distinguish between two quantum channels \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\). In the asymmetric hypothesis testing setting (Stein’s setting), we aim to minimize the type II error probability, under the condition that the type I error probability does not exceed a constant \(\varepsilon \in (0,1)\). More precisely, for any given two quantum channels \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\), denote the corresponding type I and type II error of the adaptive protocol \(\{Q,{{{\mathcal {A}}}}\}\) as \(\alpha _n(\{Q,{{{\mathcal {A}}}}\})\) and \(\beta _n(\{Q,{{{\mathcal {A}}}}\})\). Then the asymmetric distiguishibility is defined as
Its asymptotic quantities are defined as
The bestknown singleletter strong converse bound on \({\overline{\zeta }}(\varepsilon ,{{{\mathcal {N}}}},{{{\mathcal {M}}}})\) is given by the channel’s maxrelative entropy \(D_{\max }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\) [20, Corollary 18], i.e.,
In the following, we sharpen this upper bound by the geometric Rényi channel divergence in general. This gives a more accurate estimation of the fundamental limits of channel discrimination under adaptive strategies.
Theorem 49
Let \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\) be two quantum channels and \(\varepsilon \in (0,1)\), \(\alpha \in (1,2]\). It holds
Moreover, \({{\widehat{D}}}_\alpha ({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\) is also a strong converse bound.
Proof
The first two inequalities follow from (241). The last inequality follows from the relation that \({{\widehat{D}}}_{\alpha }(\rho \Vert \sigma ) \le D_{\max }(\rho \Vert \sigma )\) in Lemma 4. In the following, we show that \({\overline{\zeta }}(\varepsilon ,{{{\mathcal {N}}}},{{{\mathcal {M}}}}) \le {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\). This follows a similar step as [20, Proposition 17].
Let \(\{Q,{{{\mathcal {A}}}}\}\) be an arbitary adaptive protocol for discriminating \({{{\mathcal {N}}}}\) and \({{{\mathcal {M}}}}\). Let p and q denote the final decision probabilities. As argued in [20, Proposition 17], we can take \(\alpha _n(\{Q,{{{\mathcal {A}}}}\}) = \varepsilon \). Then
By applying the metaconverse in [20, Lemma 14] as well as the chain rule of the geometric Rényi divergence, we have
Since Eq. (244) holds for any channel discrimination protocol, we have \({\overline{\zeta }}(\varepsilon ,{{{\mathcal {N}}}},{{{\mathcal {M}}}}) \le {{\widehat{D}}}_{\alpha }({{{\mathcal {N}}}}\Vert {{{\mathcal {M}}}})\). \(\quad \square \)
Note that our new strong converse bound is also singleletter and efficient computable via semidefinite program (it even admits a closedform expression as shown in Lemma 5). The following example of the GAD channels \({{{\mathcal {A}}}}_{0.8,N_1}\) and \({{{\mathcal {A}}}}_{0.7,N_2}\) demonstrates that \({{\widehat{D}}}_{\alpha }\) is much tighter than \(D_{\max }\).
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Fang, K., Fawzi, H. Geometric Rényi Divergence and its Applications in Quantum Channel Capacities. Commun. Math. Phys. 384, 1615–1677 (2021). https://doi.org/10.1007/s00220021040644
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DOI: https://doi.org/10.1007/s00220021040644