Convergence Rates for Quantum Evolution and Entropic Continuity Bounds in Infinite Dimensions
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Abstract
By extending the concept of energyconstrained diamond norms, we obtain continuity bounds on the dynamics of both closed and open quantum systems in infinite dimensions, which are stronger than previously known bounds. We extensively discuss applications of our theory to quantum speed limits, attenuator and amplifier channels, the quantum Boltzmann equation, and quantum Brownian motion. Next, we obtain explicit logLipschitz continuity bounds for entropies of infinitedimensional quantum systems, and classical capacities of infinitedimensional quantum channels under energyconstraints. These bounds are determined by the high energy spectrum of the underlying Hamiltonian and can be evaluated using Weyl’s law.
1 Introduction
Infinitedimensional quantum systems play an important role in quantum theory. The quantum harmonic oscillator, which is the simplest example of such a system, has various physical realizations, e.g. in vibrational modes of molecules, lattice vibrations of crystals, electric and magnetic fields of electromagnetic waves, etc. Even though much of quantum information science focusses on finitedimensional quantum systems, the relevance of infinitedimensional (or continuous variable) quantum systems in quantum thermodynamics, quantum computing, and various other quantum technologies, has become increasingly apparent (see e.g. [SL, E06] and references therein).
In this paper we make a detailed analysis of the time evolution of timeindependent, infinitedimensional quantum systems. The dynamics of such a system is described by a quantum dynamical semigroup (QDS) \((T_t)_{t \ge 0}\) under the Markovian approximation, which is valid under the assumption of weak coupling between the system and its environment. In the Schrödinger picture, this is a oneparameter family of linear, completely positive, tracepreserving maps (i.e. quantum channels) acting on states of the quantum system. In the Heisenberg picture, the dynamics of observables is given by the adjoint semigroup \((T_t^*)_{t \ge 0}\) where \(\forall \, t \ge 0\), \(T_t^*\) is a linear, completely positive, unital map on the space of bounded operators acting on the system.^{1}
There are different notions of continuity of QDSs. The case of uniformly continuous QDSs is the simplest, and is easy to characterize (see Sect. 2.1 for a compendium on semigroup theory). A semigroup is uniformly continuous if and only if the generator is bounded. In this paper, we focus on the analytically richer case of strongly continuous semigroups, which appear naturally when the generator is unbounded.
QDSs are used to describe the dynamics of both closed and open quantum systems.^{2} Open quantum systems are of particular importance in quantum information theory since systems which are of relevance in quantum informationprocessing tasks undergo unavoidable interactions with their environments, and hence are inherently open. In fact, any realistic quantummechanical system is influenced by its interactions with its environment, which typically has a large number of degrees of freedom. A prototypical example of such a system is an atom interacting with its surrounding radiation field. In quantum informationprocessing tasks, interactions between a system and its environment leads to loss of information (encoded in the system) due to processes such as decoherence and dissipation. QDSs are useful in describing these processes. The theory of open quantum systems has also found applications in various other fields including condensed matter theory and quantum optics.
Infinitedimensional closed quantum systems to which our results apply are e.g. described by timeindependent Schrödinger operators \(H =  \Delta + V\), which are ubiquitous in the literature. Examples of infinitedimensional open quantum systems, to which our results apply, include, among others, amplifier and attenuator channels, the Jaynes–Cummings model of quantum optics, quantum Brownian motion, and the quantum Boltzmann equation (which describes how the motion of a single test particle is affected by collisions with an ideal background gas). These will be discussed in detail in Sect. 5.
1.1 Rates of convergence for quantum evolution
In this case, a weaker norm, namely the energyconstrained diamond norm, (or ECD norm, in short), introduced independently by Shirokov [Shi18, (2)] and Winter [W17, Definition 3], proves more useful for studying convergence properties of QDSs in the Schrödinger picture (see Example 1). It is denoted as \(\left\ \bullet \right\ _{\diamond }^{E}\), where E characterizes the energy constraint.
In this paper, we introduce a oneparameter family of ECD norms, \(\left\ \bullet \right\ _{\diamond ^{2\alpha }}^{S,E}\); here S denotes a positive semidefinite operator, E is a scalar taking values above the bottom of the spectrum of S, and \(\alpha \in (0,1]\) is a parameter (see Definition 2.3). We refer to these norms as \(\alpha \)ECD norms. They reduce to the usual ECD norm for the choice \(\alpha =1/2\), when S is chosen to be the Hamiltonian of the system. A version of the \(\alpha =1/2\)ECD norm, for S being the number operator, was first introduced in the context of bosonic channels by Pirandola et al. [PLOB17, (98)].
To illustrate the power of the \(\alpha \)ECD norms over the standard diamond norm, and even over the usual ECD norm, we discuss the example of the (single mode bosonic quantumlimited) attenuator channel with timedependent attenuation parameter \(\eta (t):=e^{t}\) (see Example 5 for details):
Example 1
1.1.1 Quantum speed limits.

open quantum systems with unbounded generators,

states with infinite expected energy, and

systems whose dynamics is generated by an operator which is different from that which penalizes the energy.
1.2 Explicit convergence rates for entropies and capacities
It is wellknown that on infinitedimensional spaces, the von Neumann entropy is discontinuous [We78]. Hence, in order to obtain explicit bounds on the difference of the von Neumann entropies of two states, it is necessary to impose further restrictions on the set of admissible states. In [W15], continuity bounds for the von Neumann entropy of states of infinitedimensional quantum systems were obtained by imposing an additional energyconstraint condition on the states, and imposing further assumptions on the class of admissible Hamiltonians. The latter are assumed to satisfy the socalled (Gibbs hypothesis). Under the energyconstraint condition and the Gibbs hypothesis it is true that for any energy E above the bottom of the spectrum of the Hamiltonian H, the Gibbs state \(\gamma (E)=e^{\beta (E) H}/Z_H(\beta (E))\)^{5} is the maximum entropy state of expected energy E [GS11, p. 196]. Bounds on the difference of von Neumann entropies stated in [W15] are fully explicit up to the occurrence of the entropy of a Gibbs state of the form \(\gamma (E/\varepsilon ),\) where \(\varepsilon \) is an upper bound on the trace distance of the two states.
Since entropic continuity bounds are tight in the limit \(\varepsilon \downarrow 0,\) we study (in Sect. 7) the entropy of such a Gibbs state in this limit. Note that for the Gibbs state \(\gamma (E/{\varepsilon })\), the limit \(\varepsilon \downarrow 0\) translates into a high energy limit. By employing the socalled Weyl law [I16], which states that certain classes of timeindependent Schrödinger operators \(H=\,\Delta +V\) have asymptotically the same high energy spectrum, we show that the asymptotic behaviour of the entropy of the Gibbs state is universal for such classes of operators. This in turn yields fully explicit convergence rates both for the von Neumann entropy and for the conditional entropy (see Proposition [Entropy convergence]).
In finite dimensions, continuity bounds on conditional entropies have found various applications, e.g. in establishing continuity properties of capacities of quantum channels [LS09] and entanglement measures [CW03, YHW08], and in the study of socalled approximately degradable quantum channels [SSRW15]. Analogously, in infinite dimensions, continuity bounds on the conditional entropy for states satisfying an energy constraint [W15], were used by Shirokov [Shi18] to derive continuity bounds for various constrained classical capacities of quantum channels.^{6} These bounds were once again given in terms of the entropy of a Gibbs state of the form \(\gamma (E/\varepsilon )\). Here, \(\varepsilon \) denotes the upper bound on the ECD norm distance between the pair of channels considered, and E denotes the energy threshold appearing in the energy constraint. Our result on the high energy asymptotics of Gibbs states yields a refinement of Shirokov’s results, by providing the explicit behaviour of these bounds for small \(\varepsilon \).
The bounds that we obtain on the dynamics of closed and open quantum systems (see Proposition 3.2 and Theorem 1) also allow us to identify explicit time intervals over which the evolved state is close to the initial state. Since entropic continuity bounds require such a smallness condition for the trace distance between pairs of states, we can then bound the entropy difference between the initial state and the timeevolved state (see Example 12).
We start the rest of the paper with some mathematical preliminaries in Sect. 2. These include a discussion of QDSs, definition and properties of the \(\alpha \)ECD norms, and some basic results from functional analysis that we use. In Sect. 3 we state our main results. These consist of (i) rates of convergence for quantum evolution in both closed and open quantum systems, and (ii) explicit convergence rates for entropies and certain constrained classical capacities of quantum channels. The results concerning (i) are proved in Sects. 4 and 5, while those on (ii) are proved in Sect. 7. In Sect. 6 we discuss some interesting applications of our results, in particular to generalized relative entropies and quantum speed limits. We end the paper with some open problems in Sect. 8. Certain auxiliary results and technical proofs are relegated to the appendices.
2 Mathematical Preliminaries
Notation In the sequel, all Hilbert spaces \({\mathcal {H}}\) are infinitedimensional, separable and complex. We denote the space of trace class operators on a Hilbert space \({\mathcal {H}}\) by \({\mathcal {T}}_1({\mathcal {H}})\), that of Hilbert–Schmidt operators by \({\mathcal {T}}_2({\mathcal {H}})\), and the qth Schatten norm by \(\left\ \bullet \right\ _q,\) see e.g. [RS1, Sect. VI.6]. The set of all quantum states (i.e. positive semidefinite operators of unit trace) on a Hilbert space \({\mathcal {H}}\) is denoted as \({\mathscr {D}}({\mathcal {H}}).\) We denote the spectrum of a selfadjoint operator H by \(\sigma (H)\), and its spectral measure by \({\mathcal {E}}^H\) [RS1, p. 224]. For the state \(\rho _{AB}\) of a bipartite system AB with Hilbert space \({{\mathcal {H}}}_A \otimes {{\mathcal {H}}}_B\), the reduced state of A is given by \(\rho _A = {{\,\mathrm{tr}\,}}_B \rho _{AB}\), where \({{\,\mathrm{tr}\,}}_B\) denotes the partial trace over \({{\mathcal {H}}}_B\). Occasionally, we also write \(\rho _{{\mathcal {H}}_A}\) instead of \(\rho _A.\) The form domain of a positive semidefinite operator S, i.e. \(\left\langle Sx,x \right\rangle \ge 0\) for all \(x \in D(S),\) is denoted by \({\mathfrak {D}}(S):=D(\sqrt{S}).\) We denote the space of bounded linear operators between normed spaces X, Y as \({\mathcal {B}}(X,Y)\), and as \({\mathcal {B}}(X)\) if \(X=Y.\)
Theorem
(Baire). Let \(X \ne \emptyset \) be a complete metric space and \((A_n)_{n\in {{\mathbb {N}}}}\) a family of closed sets covering X, then there is \(k_0 \in {{\mathbb {N}}}\) for which \(A_{k_0}\) has a nonempty interior.
2.1 Quantum dynamical semigroups (QDS)
A quantum dynamical semigroup (QDS) \((T_t)_{t \ge 0}\) in the Schrödinger picture is a oneparameter family of bounded linear operators \(T_t:{\mathcal {T}}_1({\mathcal {H}}) \rightarrow {\mathcal {T}}_1({\mathcal {H}})\) on some Hilbert space \({\mathcal {H}}\) with the property that \(T_0={\text {id}}\) (where \({\text {id}}\) denotes the identity operator between operator spaces and I the identity acting on the underlying Hilbert space), and \(T_tT_s=T_{t+s}\) for all \(t,s \ge 0\) (the semigroup property).^{7} In addition, they are completely positive (CP) and tracepreserving (TP). The adjoint semigroup is denoted as \((T^*_t)\), where for each \(t \ge 0\), \(T^*_t\) is a bounded linear operator on \({\mathcal {B}}({\mathcal {H}})\), which is CP and unital, i.e. \(T_t^*(I)= I \) for all \(t \ge 0\). Moreover, \(T^*_t\) is the adjoint of \(T_t\) with respect to the Hilbert Schmidt inner product. Due to unitality, the QDS \((T^*_t)\) is said to be a quantum Markov semigroup (QMS).

uniform continuity if \(\lim _{t \downarrow 0} \sup _{x \in X; \left\ x \right\ =1} \left\ S_txx \right\ =0,\)

strong continuity if for all \(x \in X: \lim _{t \downarrow 0}S_tx =x,\) and

\(\hbox {weak}^*\) continuity if for all \(y \in X_*\), where \(X_{*}\) is the predual Banach space of X, and \(x \in X\) the map \(t \mapsto (S_tx)(y)\) is continuous.
2.2 Functional analytic intermezzo
The dynamics of a closed quantum system is described by strongly continuous oneparameter QDSs^{8} according to the following definition:
Definition 2.1
Since the selfadjoint timeindependent Hamiltonian H fully describes the above QDSs, we will refer to both \(T_t^S\) and \(T_t^{{\text {vN}}}\) as Hassociated QDSs.
2.3 A generalized family of energyconstrained diamond norms
Motivated by the ECD norm introduced in [Shi18, W17] we introduce a generalized family of such energyconstrained norms labelled by a parameter \(\alpha \in (0,1]\), which coincides with the ECD norm for \(\alpha = 1/2\). We refer to these norms as \(\alpha \)energyconstrained diamond norms, or \(\alpha \)ECD norms in short. The notion of a regularized trace is employed in the definition of these norms.
Definition 2.2
Definition 2.3
Of particular interest to us will be (i) the 1/2ECD norm \(\left\ \bullet \right\ _{\diamond ^{1}}^{S,E},\) which reduces to the ECD norm \(\left\ \bullet \right\ _{\diamond }^{E}\) considered in [Shi18, W17] when S is chosen to be the underlying Hamiltonian, as well as (ii) the 1ECD norm \(\left\ \bullet \right\ _{\diamond ^{2}}^{S,E}\), since they penalize the first and second moments of the operator S, respectively. Although the operator S in the ECD norm is not necessarily an energy observable (i.e. Hamiltonian), we will refer to the condition \(E^{2\alpha } \ge {\text {tr}}(S^{2\alpha } \rho _{\mathcal {H}} )\) as an energyconstraint.
We show that by studying the entire family of norms, we obtain a more refined analysis for convergence rates of QDSs. Moreover, we allow the generator of the dynamics of the QDS to be different from the operator penalizing the states in the condition \(E^{2\alpha } \ge {\text {tr}}(S^{2\alpha } \rho _{\mathcal {H}} )\). This does not only allow greater flexibility but also enables us to study open quantum systems since the generator of the dynamics of an open quantum system is not selfadjoint in general and therefore also not positive.

The \(\alpha \)ECD norm \(\left\ \bullet \right\ _{\diamond ^{2\alpha }}^{S,E}\) defines a norm on the space of hermitian preserving superoperators.
 The \(\alpha \)ECD norm \(\left\ \bullet \right\ _{\diamond ^{2\alpha }}^{S,E}\) is increasing in the energy parameter E and satisfies for \(E'\ge E> \inf (\sigma (S))\)$$\begin{aligned} \left\ \bullet \right\ _{\diamond ^{2\alpha }}^{S,E} \le \left\ \bullet \right\ _{\diamond ^{2\alpha }}^{S,E'} \le \left( \frac{E'}{E}\right) ^{2\alpha } \left\ \bullet \right\ _{\diamond ^{2\alpha }}^{S,E}. \end{aligned}$$
 In the limit \(E \rightarrow \infty \) we recover the actual diamond norm$$\begin{aligned} \sup _{E> \inf (\sigma (S))} \left\ \bullet \right\ _{\diamond ^{2\alpha }}^{S,E} = \left\ \bullet \right\ _{\diamond }. \end{aligned}$$
 The following calculation shows that the topology, for \(\alpha \le \beta \), induced by the \(\diamond ^{\beta }\) norm is not stronger than the topology induced by \(\diamond ^{\alpha }\), i.e. \(\left\ T \right\ _{\diamond ^{2\beta }}^{S,E} \lesssim \left\ T \right\ _{\diamond ^{2\alpha }}^{S,E}\)We used the spectral decomposition Open image in new window in (1), applied Hölder’s inequality such that \(1 = \frac{\alpha }{\beta }+\frac{(\beta \alpha )}{\beta }\) in (2), and rearranged in (3).^{9}$$\begin{aligned} {\text {tr}} \left( S^{2\alpha } \rho \right)&{\mathop {=}\limits ^{(1)}}&\int \nolimits _{\sigma (S)} \sum _{i=1}^{\infty } (s^{2\beta } \lambda _i)^{\frac{\alpha }{\beta }} \lambda _i^{\frac{(\beta \alpha )}{\beta }} d\langle {\mathcal {E}}^S_{s} \varphi _i,\varphi _i \rangle \nonumber \\&{\mathop {\le }\limits ^{(2)}}&\left( \int \nolimits _{\sigma (S)} \sum _{i=1}^{\infty } s^{2\beta } \lambda _i \ d\langle {\mathcal {E}}^S_{s} \varphi _i,\varphi _i \rangle \right) ^{\alpha /\beta } {\mathop {=}\limits ^{(3)}}{\text {tr}} \left( S^{2\beta } \rho \right) ^{\alpha /\beta }. \end{aligned}$$(2.7)
3 Main Results
3.1 Rates of convergence for quantum evolution
Proposition 3.1
The bound (3.2) shows that the dynamics governed by the Schrödinger equation is \(\alpha \)Hölder continuous in time on sets of Open image in new window with uniformly bounded Open image in new window The bound is also tight, at least for \(\alpha =1\), as the prefactor becomes exactly one as \(\alpha \rightarrow 1\) which is illustrated in Fig. 1. From the bound on the dynamics of the Schrödinger equation in Proposition 3.1, we obtain an analogous result for the dynamics of the von Neumann equation (2.6). The latter result generalizes and improves the bound in [W17, Theorem 6], by providing a bound with rate \(t^{1/2}\) rather than \(t^{1/3}\) for the ECD norm, which implies faster convergence to zero [see (3.6) of the following Proposition and Fig. 2]:
Proposition 3.2
In Fig. 2 we see that estimate (3.5) globally improves the estimate stated in [W17, Theorem 6]. For times larger than the time interval [0, 1 / 4] that is shown in Fig. 2 the estimates [W17, Theorem 6] and (3.5) exceed the maximal diamond norm distance two of two quantum channels and therefore only provide trivial bounds. The bound on the pure states (3.7) however, is especially an improvement over the other two (3.5) for large times.
The above results which are proved in Sect. 4 provide estimates on the dynamics of closed quantum systems. In Sect. 5 we develop perturbative methods to obtain bounds on the evolution of open quantum systems which have the same timedependence, i.e. \(\alpha \)Hölder continuity in time, as the estimates on the dynamics of closed quantum systems stated in Proposition 3.2.
Theorem 1
 1.Assume that K is relatively Hbounded with Hbound a and bound b. If \(G:=KiH\) on D(H) is the generator of a contraction semigroup, then for energies \(E>\inf (\sigma (\leftH \right))\) the QDS \((\Lambda _t)\) of the open system in the Schrödinger picture, generated by \({\mathcal {L}}\) as in (5.6), satisfies, for any \(c>0\) the \(\alpha \)Hölder continuity estimateFor \(\alpha =1/2\) the above inequality reduces to$$\begin{aligned} \left\ \Lambda _t \Lambda _s \right\ _{\diamond ^{2\alpha }}^{\leftH \right, E} \le \omega _H(\alpha ,a,b,c,E) \vert t s \vert ^{\alpha }. \end{aligned}$$For \(\alpha =1\) one can take \(c \downarrow 0\) to obtain \(\omega _H(1,a,b,0,E) = 4(3b+(1+3a)E).\)$$\begin{aligned} \left\ \Lambda _t \Lambda _s \right\ _{\diamond ^{1}}^{\leftH \right, E} \le \ \ 8\sqrt{2} {\text {max}} \left\{ 2\sqrt{c},\tfrac{3b}{\sqrt{c}}+(1+3a) \sqrt{\tfrac{E}{2}} \right\} \ \sqrt{\vert t s \vert }. \end{aligned}$$(3.9)
 2.Assume that H is relatively Kbounded with Kbound a and bound b. If \(G:=KiH\) on D(K) is the generator of a contraction semigroup, then for energies \(E>\inf (\sigma (\leftK \right))\) the QDS \((\Lambda _t)\) of the open system in the Schrödinger picture satisfies, for any \(c>0\), the \(\alpha \)Hölder continuity estimate$$\begin{aligned} \left\ \Lambda _t \Lambda _s \right\ _{\diamond ^{2\alpha }}^{\leftK \right, E} \le \ \omega _K(\alpha ,a,b,c,E) \vert t s \vert ^{\alpha }. \end{aligned}$$
While many open quantum systems describe the effect of small dissipative perturbations on Hamiltonian dynamics which is the situation of framework (1) of Theorem 1, there are also examples of open quantum systems which do not have a Hamiltonian dynamics such as the attenuator channel discussed in Example 5. These systems can be analyzed by case (2) in Theorem 1. From these bounds on the dynamics, one can then derive new quantum speed limits which outperform and extend the currently established quantum speed limits in various situations (see also Remark 1):
Theorem 2
 (A)Consider a closed quantum system with selfadjoint Hamiltonian H and fix \(E >\inf (\sigma (\leftH \right))\) and \(\alpha \in (0,1]\).
 The minimal time needed for an initial state Open image in new window , for which \(E^{2\alpha } \ge {\text {tr}}(\leftH \right^{2\alpha }\vert \varphi _0 \rangle \langle \varphi _0 \vert )\), to evolve under the Schrödinger equation (2.5) to a state \( \varphi (t) \rangle \) with relative angle \(\theta := \arccos \left( {{\,\mathrm{Re}\,}}\langle \varphi (0) \varphi (t) \rangle \right) \in [0,\pi ]\), satisfiesFor \(\alpha =1/2\) this expression reduces to$$\begin{aligned} t_{\text {min}} \ge \left( \frac{22\cos (\theta )}{g_{\alpha }^2}\right) ^{1/(2\alpha )}\frac{1}{E}. \end{aligned}$$(3.10)$$\begin{aligned} t_{\text {min}} \ge (1\cos (\theta ))/2 \frac{1}{E}. \end{aligned}$$(3.11)
 Consider an initial state \(\rho (0)=\rho _0\) to the von Neumann equation (2.6) with \(E^{2\alpha } \ge {\text {tr}}(\leftH \right^{2\alpha }\rho _0)\). The minimal time for it to evolve to a state \(\rho (t)\) which is at a Bures anglerelative to \(\rho (0)\), satisfies$$\begin{aligned} \theta := \arccos \left( \left\ \sqrt{\rho (0)}\sqrt{\rho (t)} \right\ _1 \right) \in [0,\pi /2] \end{aligned}$$(3.12)$$\begin{aligned} t_{\text {min}} \ge \left( \frac{1\cos (\theta )}{g_{\alpha }}\right) ^{1/\alpha } \frac{1}{E}. \end{aligned}$$(3.13)

 (B)Consider an open quantum system governed by a QDS \((\Lambda _t)\) satisfying the conditions of Theorem 1. Let \(\rho _0\) denote an initial state, with purity \(p_{\text {start}} = {\text {tr}}(\rho _0^2)\), for which \(E^{2\alpha } \ge {\text {tr}}(\leftH \right^{2\alpha }\rho _0)\) (or \(E^{2\alpha } \ge {\text {tr}}(\leftK \right^{2\alpha }\rho _0)\)). Then the minimal time needed for this state to evolve to a state with Bures angle \(\theta \), satisfies either for \(\omega _{H}\) or \(\omega _{K}\) as in (3.8), where the choice of \(\omega _{\bullet }\) depends on whether one considers the situation (1) or (2) in Theorem 1,Moreover, the minimal time to reach a state with purity \(p_{\text {fin}}\) satisfies$$\begin{aligned} t_{\text {min}} \ge \left( \frac{22\cos (\theta )}{\omega _{\bullet }}\right) ^{1/\alpha }. \end{aligned}$$(3.14)$$\begin{aligned} t_{\text {min}} \ge \left( \frac{\vert p_{\text {start}}p_{\text {fin}} \vert }{2\omega _{\bullet }}\right) ^{1/\alpha }. \end{aligned}$$(3.15)
3.2 Explicit convergence rates for entropies and capacities
Our next set of results comprises explicit convergence rates for entropies of infinitedimensional quantum states and several classical capacities of infinitedimensional quantum channels, under energy constraints. See Sect. 7 for definitions, details and proofs. The Hamiltonian arising in the energy constraint is assumed to satisfy the Gibbs hypothesis. Continuity bounds on these entropies and capacities rely essentially on the behaviour of the entropy of the Gibbs state \(\gamma (E):=e^{\beta (E) H}/Z_H(\beta (E)) \in {\mathscr {D}}({\mathcal {H}})\) (where \(Z_H(\beta (E))\) is the partition function, for some positive semidefinite Hamiltonian H) in the limit \(E \rightarrow \infty \). This asymptotic behaviour is studied in Theorem 3, and discussed for standard classes of Schrödinger operators in Example 11.
Assumption 1
(Gibbs hypothesis). A selfadjoint operator H satisfies the Gibbs hypothesis, if for all \(\beta >0\) the operator \(e^{\beta H}\) is of trace class such that the partition function \(Z_H(\beta (E)) = {\text {tr}}(e^{\beta H})\) is welldefined.
The asymptotic behaviour of the entropy of the Gibbs states allows us then to obtain explicit convergence rates for entropies of quantum states and capacities of quantum channels.
We obtain the following explicit convergence rates for the von Neumann entropy \(S(\rho )\) of a state \(\rho \), and the conditional entropy \(S(AB)_{\rho }\) of a bipartite state \(\rho _{AB}\) [defined through (7.2)]. For \(x \in [0,1]\), we define \(h(x):=\,x\log (x)(1x)\log (1x)\) (the binary entropy), \(g(x):=(x+1)\log (x+1)x\log (x),\) and \(r_{\varepsilon }(t) = \frac{1+\tfrac{t}{2}}{1\varepsilon t}\) a function on \((0,\frac{1}{2\varepsilon }]\), with \(\varepsilon \in (0,1)\).
Proposition
(Entropy convergence). Let H be a positive semidefinite operator, with \(E_H:=\inf (\sigma (H)) \ge 0\), on a quantum system A satisfying the Gibbs hypothesis and assume that the limit \(\xi :=\lim _{\lambda \rightarrow \infty } \frac{N_{H}^{\uparrow }(\lambda ) }{N_{H}^{\downarrow }(\lambda )}>1\) exists such that \(\eta :=\left( \xi 1\right) ^{1}\) is welldefined.
 1.
\( \leftS(\rho )S(\sigma ) \right\le 2 \varepsilon \eta \log \left( (EE_H)/\varepsilon \right) (1+o(1)) + h(\varepsilon )\) as \(\varepsilon \downarrow 0.\)
 2.Let \(\varepsilon < \varepsilon '\le 1\) and \(\delta =\frac{\varepsilon '\varepsilon }{1+\varepsilon '}\), then as \(\varepsilon \downarrow 0\)$$\begin{aligned} \leftS(\rho )S(\sigma ) \right\le (\varepsilon '+2 \delta ) \eta \log \left( (EE_H)/\delta \right) (1+o(1)) + h(\varepsilon ')+h(\delta ).\qquad \end{aligned}$$(3.16)
 3.For states \(\rho ,\sigma \in {\mathscr {D}}({\mathcal {H}}_A \otimes {\mathcal {H}}_B)\) with \({\text {tr}}(\rho _A H), {\text {tr}}(\sigma _A H)\le E\), \(\tfrac{1}{2}\left\ \rho \sigma \right\ \le \varepsilon \), and \(\varepsilon '\) and \(\delta \) as in (2), the conditional entropy (7.2) satisfies as \(\varepsilon \downarrow 0\)$$\begin{aligned} \leftS(A \vert B)_{\rho }S(A \vert B)_{\sigma } \right\le & {} 2(\varepsilon '+4 \delta ) \eta \log \left( (EE_H)/\delta \right) (1+o(1))\nonumber \\&+(1+\varepsilon ') h(\tfrac{\varepsilon '}{1+\varepsilon '})+2h(\delta ). \end{aligned}$$(3.17)
For the constrained productstate classical capacity \(C^{(1)}\), whose expression is given by (7.17), and the constrained classical capacity C, defined through (7.18), we obtain the following convergence results:
Proposition
(Capacity convergence). Consider positive semidefinite operators \(H_{A}\) on a Hilbert space \({\mathcal {H}}_A\) and \(H_{B}\) on a Hilbert space \({\mathcal {H}}_B\), where \(H_B\) satisfies the Gibbs hypothesis. Moreover, let \(E_{H_B}:=\inf (\sigma (H_B))\). We also assume that the limit \(\xi :=\lim _{\lambda \rightarrow \infty } \frac{N_{ H_{B} }^{\uparrow }(\lambda ) }{N_{ H_{B} }^{\downarrow }(\lambda )}>1\) exists such that \(\eta :=\left( \xi 1\right) ^{1}\) is welldefined.
4 Closed Quantum Systems
In this section we study the dynamics of closed quantum systems in \(\alpha \)ECD norms.
From Proposition A.1 in the appendix it follows that if a state \(\rho = \sum _{i=1}^{\infty } \lambda _i \vert \varphi _i \rangle \langle \varphi _i \vert \) satisfies the energy constraint \({\text {tr}}(S^{2\alpha }\rho )<\infty \) for some positive operator S, then all Open image in new window for which \(\lambda _i \ne 0,\) are contained in the domain of \(S^{\alpha }.\) However, the expectation value \({\text {tr}}(S\rho )\) of an operator S in a state \(\rho \) can be infinite even if all the eigenvectors of \(\rho \) are in the domain of S. This is shown in the following example.
Example 2
Consider the free Schrödinger operator \(S:=\frac{d^2}{dx^2}\) on the interval \([0,\sqrt{1/8}]\) with Dirichlet boundary conditions modeling a particle in a box of length \(1/\sqrt{8}\). This operator possesses an eigendecomposition with eigenfunctions \((\psi _i)\) such that \(\frac{d^2}{dx^2} = \sum _{i=1}^{\infty } i^2 \vert \psi _i \rangle \langle \psi _i \vert .\) However, the state \(\rho =\sum _{i=1}^{\infty } \frac{1}{i(i+1)} \vert \psi _{i} \rangle \langle \psi _{i} \vert \), here \(\sum _{i=1}^{\infty } \frac{1}{i(i+1)}=1,\) satisfies \({\text {tr}}(S\rho )=\infty .\)
Proposition 3.2 implies that any group \(T^{{\text {vN}}}_t(\rho )= e^{itH}\rho e^{itH}\), with selfadjoint operator H, is continuous with respect to the ECD norm induced by \(\leftH \right\) without any further assumptions on H besides selfadjointness. Before proving this result, we start with the definition of the Favard spaces [EN00, Chap. 2., Sect.5.5.10] or [BF15, Sect. 4] and an auxiliary lemma:
Definition 4.1
In order to link Favard spaces to QDSs, we require a characterization of these spaces in terms of the resolvent of the associated generator.
Lemma 4.2
Proof
Proof of Proposition 3.1
Before extending the above result to the dynamics of the von Neumann equation (2.6) for states on the product space \({\mathcal {H}} \otimes {\mathbb {C}}^n\), we need another auxiliary Lemma on the action of the Schrödinger dynamics on states:
Lemma 4.3
Proof
From estimate (4.7) we can then state the proof of Proposition 3.2:
Proof of Proposition 3.2
The preceding Propositions 3.1 and 3.2 show that the quantum dynamics of closed quantum systems generated by some selfadjoint operator H is always continuous with respect to the \(\alpha \)ECD norm induced by the absolute value of the same operator H.
We now do a perturbation analysis for the convergence in \(\alpha \)ECD norm:
Proposition 4.4
Proof
Consider a density matrix with spectral decomposition \(\rho = \sum _{i=1}^{\infty } \lambda _i \ \vert \varphi _i \rangle \langle \varphi _i \vert .\) If any of the Open image in new window then \({\text {tr}}(S^{2\alpha }\rho ) = \infty \) as in Proposition A.1. Thus, we may assume that all Open image in new window Therefore, if \({\text {tr}}(S^{2\alpha }\rho ) \le E^{2\alpha }\) then also \({\text {tr}}(\leftH \right^{2\alpha } \rho ) \le a {\text {tr}}( S^{2\alpha }\rho ) + b \le a E^{2\alpha }+b\) which proves the Proposition, since the estimate follows from Proposition 3.2. \(\quad \square \)
The previous result allows us to study QDSs generated by complicated Hamiltonians using more accessible operators penalizing the states in the ECD norms. We illustrate this in the following example where we see that it suffices to penalize the kinetic energy of a state and still obtain convergence for the semigroup of the Coulomb Hamiltonian.
Example 3
(Coulomb potential). If H is relatively S bounded and both H and S are positive, then it follows from [RS2, Theorem X.18] that H is also S formbounded. This is to say that \(\sqrt{H}\) is also \(\sqrt{S}\) bounded. Iterating this idea, we find that \(H^{2^{n}}\) is relatively \(S^{2^{n}}\) bounded for all \(n \in {\mathbb {N}}_0.\) Let \(H:=\,\Delta +\frac{1}{\leftx \right}\) and \(S:=\,\Delta \) on \(L^2({\mathbb {R}}^3)\), then H is relatively Sbounded, see for example [RS2, Theorem X.15]. Thus, the semigroup \(T_t^{{\text {vN}}}(\rho ):=e^{itH} \rho e^{itH}\) is \(2^{n}\)Hölder continuous in time with respect to \(\left\ \bullet \right\ _{\diamond ^{2^{1n}}}^{S,E}\).
We provide a simple example showing that it is impossible to select arbitrary unbounded selfadjoint operators to penalize the energy in the diamond norm and still have the same convergence rates in time:
Example 4
(Harmonic oscillator). Let \(H_{{\text {osc}}}:=\,\Delta +\leftx \right^2\) be the dimensionless Hamiltonian of the harmonic oscillator on \(D(H_{{\text {osc}}}):=\left\{ \varphi \in L^2({\mathbb {R}}^d); \Delta \varphi , \ \leftx \right^2\varphi \in L^2({\mathbb {R}}^d) \right\} .\) The oneparameter group of the harmonic oscillator \(T^{{\text {vN}}}_t(\rho ):=e^{itH_{{\text {osc}}}} \rho e^{itH_{{\text {osc}}}}\) does not obey a uniform linear timerate in the 1ECD norm induced by the negative Laplacian \(\Delta \) for any \(E>0 = \inf (\sigma (\,\Delta )).\) To see this, it suffices to study the dynamics generated by the Schrödinger equation (2.5) with Hamiltonian \(H_{{\text {osc}}}\). Then, the Favard space \(F_1\) coincides with the operator domain \(D(H_{{\text {osc}}}),\) as stated in [EN00, Corollary 5.21]. However, the domain of the Laplacian penalizing the energy is \(D(\,\Delta )=\left\{ f \in L^2({\mathbb {R}}^d); \Delta f \in L^2({\mathbb {R}}^d) \right\} \) which is strictly larger than \(F_1=D(H_{{\text {osc}}}),\) as for \(f \in D(\,\Delta )\) one does not require that \(\vert x \vert ^2 f \in L^2({\mathbb {R}}^d).\)
The perturbation result, Proposition 4.4, essentially relies on operator boundedness and provides explicit bounds to compare the two different \(\alpha \)ECD norms induced by the perturbed and unperturbed operator. This result is a special case of a more abstract result, stated as Proposition B.1 in Appendix B, that relies on the special geometry of the space of trace class operators. It yields the same rate \(t^{\alpha }\) for the convergence with respect to the perturbed and unperturbed norms. However, it does not provide an explicit prefactor.
5 Open Quantum Systems
We start with an auxiliary Lemma that provides sufficient conditions under which a perturbation of the generator of a contraction semigroup leaves its Favard spaces invariant:
Lemma 5.1
Proof
The most general form of the generator of a uniformly continuous QMS is the socalled GKLS representation, named after Lindblad [Lin76] and Gorini, Kossakowski and Sudarshan [GKS76].
Theorem GKLS
This construction has been generalized by Davies [Da77] to unbounded generators which is discussed below:
5.1 Extension of GKLS theorem to unbounded generators [Da77]
Lemma 5.2
Proof
We are now able to prove Theorem 1 which shows that the uniform continuity for the \(\alpha \)ECD norm which we obtained for closed quantum systems in Proposition 3.2 applies to open quantum systems as well:
Proof of Theorem 1
We start by proving the first part of the theorem: That G is the generator of a contraction semigroup if \(a<1\) follows from [EN00, Theorem 2.7].
First, we observe that \( K \otimes I_{{\mathbb {C}}^n}\) is still relatively \(H \otimes I_{{\mathbb {C}}^n}\)bounded with the same bound a [Si15, Theorem 7.1.20].
Corollary 5.3
Proof
We continue with a discussion of applications of Theorem 1. Let us start by continuing our study of the quantumlimited attenuator and amplifier channels that we started in Example 1:
Example 5
Example 6
(Linear quantum Boltzmann equation [A02, HV09]). Since this example describes scattering effects, that depend on the ratio of mass parameters, we exceptionally include physical constants in this example. Consider a particle of mass M whose motion without an environment is described by the selfadjoint Schrödinger operator \(H_0 = \frac{\hbar ^2}{2M}\Delta + V.\) The linear quantum Boltzmann equation describes the motion of the particle in the presence of an additional ideal gas of particles with mass m distributed according to the Maxwell–Boltzmann distribution \(\mu _{\beta }(p) = \frac{1}{\pi ^{3/2} p_{\beta }^3} e^{ \leftp \right^2 /p_{\beta }^2}\) where \(p_{\beta } = \sqrt{2m/\beta }.\)
Here, we discuss for simplicity the linear quantum Boltzmann equation under the Born approximation of scattering theory [HV09]: Let \(m_{\text {red}} = mM/(m+M)\) be the reduced mass and \(n_{\text {gas}}\) the density of gas particles. We assume that the scattering potential between the gas particles and the single particle is of shortrange and smooth such that \(V \in {\mathscr {S}}({\mathbb {R}}^3)\) where \({\mathscr {S}}({\mathbb {R}}^3)\) is the Schwartz space [RS1]. In the Born approximation the scattering amplitude becomes \(f(p) = \frac{m_{\text {red}}}{2\pi \hbar ^2} {\mathcal {F}}(V)(p/\hbar ),\) where \({\mathcal {F}}\) is the Fourier transform.
By combining the attenuator channel with the amplifier channel, and using an operator proportional to the number operator N as the Hamiltonian part, we obtain the example of a damped and pumped harmonic oscillator which found, for example, applications in quantum optics, to describe a single mode of radiation in a cavity [A02]:
Example 7
Next, we study the evolution of quantum particles under Brownian motion which is obtained as the diffusive limit of the quantum Boltzmann equation that we discussed in Example 6 [HV09, Sect. 5].
Example 8
(Quantum Brownian motion [AS04, V04]). Consider the Hamiltonian of a harmonic oscillator \(H=\frac{d^2}{dx^2} +x^2\) and Lindblad operators for \(j \in \left\{ 1,2\right\} \) given by \(L_j:= \gamma _j x + \beta _j \frac{d}{dx} \) where \(\gamma _j, \beta _j \in {\mathbb {C}}.\) In particular, choosing \(\gamma _j=\beta _j \) turns \(L_j\) into the annihilation operator\(L_j = \gamma _j \left( \frac{d}{dx}+x\right) \) and \(L^*\) into the creation operator\(L_j^* = \gamma _j \left( \frac{d}{dx}+x\right) \) which have been considered in the previous example.
The field of quantum optics is a rich source of open quantum systems to which the convergence Theorem 1 applies and we discuss a few of them in the following example:
Example 9
Hence, the operators \(\frac{1}{2} L_k^*L_k\) are selfadjoint and dissipative and for a large class of Hamiltonians H the asymptotics of Theorem 1 can be applied.
6 Generalized Relative Entropies and Quantum Speed Limits
We start with some immediate consequences of Propositions 3.1, 3.2, and Theorem 1 on certain generalized relative entropies and distance measures which are dominated by the trace norm:
Definition 6.1
As a Corollary of Proposition 3.2 for closed quantum systems and Theorem 1 for open quantum systems, we obtain:
Corollary 6.2
 The Bures distance and Bures angle satisfy$$\begin{aligned} d_{B}(T^{{\text {vN}}}_t(\rho ),T^{{\text {vN}}}_s(\rho )) \le \sqrt{2 g_{\alpha } E^{\alpha }\ \vert t s \vert ^{\alpha }} \text { and }\\ \theta (T^{{\text {vN}}}_t(\rho ),T^{{\text {vN}}}_s(\rho )) \le \arccos \left( {\text {max}}\left\{ 1 g_{\alpha } E^{\alpha }\ \vert t s \vert ^{\alpha },1 \right\} \right) . \end{aligned}$$
 For the 1 / 2divergences we obtain$$\begin{aligned}&D^{{\text {Tsallis}}}_{1/2}(T^{{\text {vN}}}_t(\rho ) \vert \vert T^{{\text {vN}}}_s(\rho )) \le 2 g_{\alpha }E^{\alpha }\ \vert t s \vert ^{\alpha },\\&D^{\text {R}\acute{{\mathrm{e}}}\text {nyi}}_{1/2}(T^{{\text {vN}}}_t(\rho ) \vert \vert T^{{\text {vN}}}_s(\rho )) \le 2\log \left( \left( 1 g_{\alpha }E^{\alpha }\ \vert t s \vert ^{\alpha }\right) _{+}\right) , \text { and } \\&{\tilde{D}}^{\text {R}\acute{{\mathrm{e}}}\text {nyi}}_{1/2}(T^{{\text {vN}}}_t(\rho ) \vert \vert T^{{\text {vN}}}_s(\rho )) \le 2\log \left( \left( 1 g_{\alpha }E^{\alpha }\ \vert t s \vert ^{\alpha }\right) _{+}\right) \end{aligned}$$
 For the 1 / 2divergences it follows that$$\begin{aligned}&D^{{\text {Tsallis}}}_{1/2}(\Lambda _t(\rho ) \vert \vert \Lambda _s(\rho )) \le \omega _{\bullet } \vert ts \vert ^{\alpha },\\&D^{\text {R}\acute{{\mathrm{e}}}\text {nyi}}_{1/2}(\Lambda _t(\rho ) \vert \vert \Lambda _s(\rho )) \le 2\log \left( \left( 1\tfrac{\omega _{\bullet }}{2}\vert ts \vert ^{\alpha }\right) _{+}\right) , \text { and } \\&{{\tilde{D}}}^{\text {R}\acute{{\mathrm{e}}}\text {nyi}}_{1/2}(\Lambda _t(\rho ) \vert \vert \Lambda _s(\rho )) \le 2\log \left( \left( 1\tfrac{\omega _{\bullet }}{2}\vert ts \vert ^{\alpha }\right) _{+}\right) \end{aligned}$$
 For the Bures distance and Bures angle, we obtain$$\begin{aligned}&d_B(\Lambda _t(\rho ), \Lambda _s(\rho )) \le \ \sqrt{\omega _{\bullet } \vert ts \vert ^{\alpha }}\text { and}\\&\quad \theta (\Lambda _t(\rho ),\Lambda _s(\rho )) \le \arccos \left( {\text {max}}\left\{ 1\tfrac{\omega _{\bullet }}{2}\vert ts \vert ^{\alpha },1 \right\} \right) . \end{aligned}$$
Proof
It suffices to show that all quantities can be estimated by the trace norm. For the 1 / 2sandwiched Rényi divergences, this is already shown in (6.4). Proposition 3.2 then provides the upper bounds for closed systems and Theorem 1 yields the bounds for open systems. For estimates on Bures distances and Bures angles an application of the Fuchsvan de Graaf inequality [FG99], (6.3), shows that \(d_{B}(\rho ,\sigma )^2 \le \left\ \rho  \sigma \right\ _{1}\) and \(\theta (\rho ,\sigma ) \le \arccos \left( 1 \tfrac{\left\ \rho  \sigma \right\ _1}{2} \right) .\) The Powers–Størmer inequality [PS70, Lemma 4.1] implies that \(D^{{\text {Tsallis}}}_{1/2}(\rho \vert \vert \sigma ) \le \left\ \rho \sigma \right\ _{1}.\)\(\quad \square \)
Remark 1
Consider a closed system with Hamiltonian \(S=\frac{d^2}{dx^2}\) on \({\mathbb {R}}\). The state \(\psi \in L^2({\mathbb {R}})\) with Fourier transform \({\mathcal {F}}(\psi )(x) = \frac{c}{(1+x^2)^{1/2}}\) where \(c>0\) is such that \(\psi \) is of unit norm. Then, \(\langle S \psi , \psi \rangle = \infty \) whereas \({\text {tr}}(S^{\alpha }\rho )<\infty \) for \(\alpha <1/4\). Thus, the above bounds (6.5) and (6.6) reduce to the trivial bound \(t_{\text {min}} \ge 0.\)
For infinitedimensional open quantum systems, the first term in the bound on the purity (6.7) reduces to zero if the Lindblad operators are not Hilbert–Schmidt, which is the case for all examples presented in Sect. 5. In particular, if the Lindblad operators are unbounded, then the bound simplifies to \( t_{\text {min}} \ge 0.\)
We can now state the proof of Theorem 2:
Proof of Theorem 2
7 Entropy and Capacity Bounds
In this section, we obtain explicit continuity bounds for different families of entropies of quantum states, and various constrained classical capacities of quantum channels in infinite dimensions.
We now want to compare the delicate continuity properties of the von Neumann entropy with the properties of the Tsallis\((T_q)\) and Rényi\((S_q)\) entropies:
Definition 7.1
Unlike the von Neumann entropy, our next Proposition shows that the Tsallis and Rényi entropies are Lipschitz continuous, without any assumptions on the expected energy of the state or the Hamiltonian:
Proposition 7.2
Proof
Remark 2
Before entering the general theory, let us study the fully explicit case of the harmonic oscillator first:
Example 10
Our aim in this section is to show that, in some sense, the logarithmic divergence of the entropy of the Gibbs state, as \(E \rightarrow \infty \), is not a special feature of the harmonic oscillator but universal for many classes of Hamiltonians. This result allows us then to state explicitly a rate of convergence in continuity bounds on entropies and capacities.
We start with some preliminary related ideas:
Theorem 3
Proof
Example 11
Calculation
Instead of just referring to Example 10 for the harmonic oscillator, we apply Theorem 3:
In Fig. 4a we compare the true inverse temperature \(\beta (E)\) of the Gibbs state for the quantum harmonic oscillator as in Example 10 with the asymptotic law \(\beta (E)\approx \frac{1}{E}\) obtained from Theorem 3. In Fig. 4b we compare the inverse temperature of the Gibbs state for the Hamiltonian describing a particle in a box of length \(\frac{1}{\sqrt{8}}\) with the asymptotic law \(\beta (E) \approx \frac{1}{2E}\) we obtained in Example 11. The following Proposition, which relies on Theorem 3, shows that for large generic classes of Schrödinger operators with compact resolvent, the entropy of the Gibbs states obeys a universal high energy asymptotic behaviour.
The Proposition [Entropy convergence] then follows as an application of Theorem 3, which provides an explicit rate of convergence for entropies on infinitedimensional Hilbert spaces:
Proof
The two Corollaries C.1 and C.2 of Theorem 3 that are stated in Appendix C provide convergence rates on QMI and hence on \(C_{\text {ea}}\).
We continue our discussion of attenuator and amplifier channels, that were defined in Example 5 by studying their convergence of entropies.
Example 12
(Entropy bounds for attenuator and amplifier channels). We start by discussing how the expected energy of output states of these channels with timedependent attenuation and amplification parameters behave as a function of time.
7.1 Capacity bounds
Another application of the high energy asymptotics of the entropy of the Gibbs state are bounds on capacities of quantum channels. Concerning these bounds, we need to introduce, before stating our result, the definition of an ensemble, its barycenter, and the Holevo quantity [Shi18].
Definition 7.3
Remark 3
Proof
8 Open Problems
Concerning the first part of the paper, it would be desirable to study extensions of our work to nonautonomous systems, such as systems described by a Schrödinger operator with timedependent potentials. For the Schrödinger equation, an application of the variation of constants formula yields a bound for such systems as well (see Proposition 3.1). This should also work, under suitable assumptions, for nonautonomous open quantum systems. However, more mathematical care may be needed for the latter.
To answer the important questions: (i) “How fast can entropy increase?”for any infinitedimensional open quantum system whose dynamics is governed by a QDS, and (ii) “How fast can information be transmitted?”through any quantum channel (obtained by freezing the time parameter in the QDS), it seems necessary to find bounds on the evolution of the expected energy for the state of the underlying open quantum system over time (as has been done for the case of the attenuator and amplifier channels in Example 12.^{15}) To our knowledge, such bounds have not been obtained in full generality yet. See also [BN88, DKSW18, OCA] for related results on question (i).
The first step to answer these two questions was provided by Winter [W15] and Shirokov [Shi18], who derived continuity bounds on entropies and capacities, respectively. Our paper provides, as a second step, a timedependent bound on the evolution of the expected energy of the state of the open quantum system, which enters these continuity bounds through the energy constraint. Understanding the behaviour of this expected energy as a function of time is needed in order to infer, from the continuity bounds, how fast entropies and capacities can change.
It would be furthermore desirable to extend Theorem 3 to higherorder terms. In Fig. 4a we see that the leadingorder approximation for the inverse temperature provided by Theorem 3 is almost indistinguishable from the true solution for the harmonic oscillator whereas the leadingorder approximation in Fig. 4b for the particle in a box seems to converge somewhat slower than the true solution. A better understanding of higher order terms should be able to capture these behaviours more precisely.
Footnotes
 1.
\(T_t^*\) is the adjoint of \(T_t\) with respect to the Hilbert–Schmidt inner product.
 2.
For closed quantum system, the QDS consists of unitary operators \(T_t\). Since \(T_{t}=T_t^*\) this semigroup extends to a group with \(t \in {\mathbb {R}}\).
 3.
This is because if \((T_t)_{t \ge 0}\) is a QDS, then for any t, \(T_t\) is a quantum channel.
 4.
We use dimensionless notation in this paper.
 5.
Here \(Z_H\) denotes the partition function and \(\beta \) denotes the inverse temperature.
 6.
For a discussion of these capacities see Sect. 7.
 7.
For notational simplicity, we will henceforth suppress the subscript \(t \ge 0\) in denoting a QDS.
 8.
As mentioned earlier, since a QDS for a closed system consists of unitary operators, it extends to a group.
 9.
We assume here that all vectors \(\varphi _i\) are in the operator domain \(D(S^{\beta })\), as otherwise the traces are infinite by Proposition A.1.
 10.
\(\forall x \in D(K): \langle Kx,x\rangle \le 0\).
 11.
For notational simplicity, we suppress the tensor products with the identity on all other factors.
 12.
To simplify the nomenclature, we henceforth suppress the word constrained when referring to the different capacities.
 13.
 14.
A sufficient condition for \(H\ge 0\) to define a Gibbs state is that the resolvent of H is a Hilbert–Schmidt operator.
 15.
In fact, in Example 12, explicit expressions, and not just bounds, have been obtained.
Notes
Acknowledgements
Support by the EPSRC Grant EP/L016516/1 for the University of Cambridge CDT, the CCA is gratefully acknowledge (S.B.). N.D. is grateful to Pembroke College and DAMTP for support.
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