Eventually Entanglement Breaking Markovian Dynamics: Structure and Characteristic Times

We investigate entanglement breaking times of Markovian evolutions in discrete and continuous time. In continuous time, we characterize which Markovian evolutions are eventually entanglement breaking, that is, evolutions for which there is a finite time after which any entanglement initially present has been destroyed by the noisy evolution. In the discrete-time framework, we consider the entanglement breaking index, that is, the number of times a quantum channel has to be composed with itself before it becomes entanglement breaking. The PPT conjecture is that every PPT quantum channel has an entanglement breaking index of at most 2; we prove that every faithful PPT quantum channel has a finite entanglement breaking index, and more generally, any faithful PPT CP map whose Hilbert–Schmidt adjoint is also faithful is eventually entanglement breaking. We also provide a method to obtain concrete bounds on this index for any faithful quantum channel. To obtain these estimates, we use a notion of robustness of separability to obtain bounds on the radius of the largest separable ball around faithful product states. We also extend the framework of Poincaré inequalities for non-primitive semigroups to the discrete setting to quantify the convergence of quantum semigroups in discrete time, which is of independent interest.


Introduction
Distributing entangled quantum states using noisy quantum channels reliably and efficiently is one of the fundamental challenges in quantum information theory, both from an experimental and theoretical point of view. Entanglement breaking channels, i.e., quantum channels that only output separable states when acting on one half of a bipartite quantum state, are useless for such non-classical communication protocols. If half of an entangled quantum state passes through a noisy channel several times, at what point does it lose its entanglement with the other half? In this work, we will prove several upper and lower bounds on this time in terms of properties of the channel; in other words, we establish bounds on the entanglement breaking time of the channel. This question arises naturally in the context of quantum repeaters [6,17] and such bounds limit their power to implement non-classical communication protocols.
Mathematically, the situation can be modeled by means of continuous or discrete-time quantum Markov semigroups (QMS). The classification of the ability of QMS to preserve entanglement both at finite time and asymptotically has recently received considerable attention. The authors of [38] (see also [35]) took a more qualitative and asymptotic point of view, completely characterizing the class of discrete-time QMS that do not become entanglement breaking asymptotically. Additionally, [49] showed that certain classes of channels lead to eventually entanglement breaking QMS.
We further contribute to this classification by providing a simple characterization of the finite-time and asymptotic entanglement behavior of continuous-time QMS based on the structure of their decoherence-free (DF) subalgebra, building upon the work of [38]. In particular, we show that the class of eventually entanglement breaking continuous-time QMS is precisely the one of primitive QMS, that is of those which possess a unique full-rank invariant state. On the other hand, continuous-time QMS that break entanglement asymptotically are precisely those possessing a non-trivial, yet commutative DF subalgebra. Finally, QMS with a non-commutative DF subalgebra never break entanglement. We also manage to obtain quantitative lower and upper bounds on the entanglement breaking time of a QMS: The former are obtained by exploring the connection between the spectrum of a quantum channel and whether its Choi matrix is of positive partial transpose (PPT). On the other hand, upper bounds are derived by exploiting the rapid-decoherence property of Markovian evolutions, which will provide us with a method to estimate how far the Choi matrix of the QMS at a given time is to its asymptotics. This property is usually obtained through functional inequalities for the underlying quantum channel, and we relate our results to the most widely used in the literature, like the Poincaré inequality [53] or the logarithmic Sobolev inequality [3,5,33]. This combined with some knowledge about the geometry of the convex set of separable states [27] allows us to obtain quantitative bounds on the entanglement breaking times.
The situation becomes more complicated in the discrete time setting. The asymptotic picture was developed by [38], where discrete-time quantum Markov semigroups which preserve entanglement asymptotically were characterized. We continue this path by characterizing faithful quantum channels which are entanglement breaking in finite time. To do so, we first study the ability of irreducible quantum channels [21,22] to preserve entanglement. We show that there are irreducible quantum channels that only become entanglement breaking asymptotically and, in the case of quantum channels with an invariant state of full rank we obtain a converse, relating quantum channels that do not preserve entanglement asymptotically to irreducible channels. Moreover, again under the faithfulness assumption, i.e., that of the existence of an invariant state of full rank, we show that quantum PPT channels become entanglement breaking after a finite number of iterations by exploring results on the structure of the spectrum of irreducible maps. We then extend this result to faithful completely positive maps by a suitable similarity transformation. This generalizes the results of [49], where the authors showed the statement for doubly stochastic channels.
However, in light of the PPT squared conjecture [6], it is also desirable to obtain quantitative bounds to when a channel becomes entanglement breaking. So far, the conjecture was only proved for low-dimensional cases [16,18] or for some particular families of quantum channels [20,35]. In [18], the authors obtain upper bounds on the number of iterations in terms of the Schmidt number of a channel. Here instead, we once again adopt an approach based on functional inequalities for discrete-time QMS. Unfortunately, there is not a lot of work on functional inequalities for non-primitive evolutions in discrete time [9,52], which is particularly important in our setting. To start amending this gap in the literature and increase the class of examples our techniques apply to, we generalize the framework of Poincaré inequalities to discrete-time non-primitive QMS. We believe these techniques are of independent interest and will find applications elsewhere. Lower bounds are again found by use of the PPT criterion.
The lower and upper bounds we obtain with our techniques are tight up to constants for some classes of examples. Moreover, we use similar techniques to consider the similar problems of when a pair of QMS becomes entanglement annihilating [43], how often one has to apply a doubly stochastic primitive quantum channel until it becomes mixed unitary and when the output of quantum Gibbs samplers are approximate quantum Markov networks.

Structure of the Paper and Contributions
• In Sect. 2, we investigate the entanglement breaking properties of continuous-time quantum Markov semigroups. We prove the set of primitive QMS coincides with the set of eventually entanglement breaking continuous-time QMS in Theorem 2.1 and establish upper and lower bounds on the entanglement breaking times in Propositions 2.7 and 2.11. To establish the upper bounds, we use a notion of "robustness of separability" in Sect. 2.1.2 and use strong decoherence bounds which we discuss in Sect. 2.1.1. The lower bounds use the PPT criterion. • In Sect. 3, we investigate the structural properties of discrete-time QMS.
We identify classes of eventually entanglement breaking channels that are dense in the set of quantum channels in Proposition 3.1 and discuss an application of this to the PPT 2 conjecture in Corollary 3.2. We discuss irreducible evolutions in Sect. 3.1, prove they are asymptotically entanglement breaking in the remarks following Proposition 3.4, and establish necessary conditions for an irreducible map to be PPT, as well as sufficient conditions for an irreducible map to be entanglement breaking, in Theorem 3.7. We then prove that PPT quantum channels with a full-rank invariant state are eventually entanglement breaking in Theorem 3.10. We use these results to characterize faithful eventually entanglement breaking discrete-time QMS in Theorem 3.11. In Sect. 3.3, we discuss the "phase subspace" of asymptotically entanglement breaking channels, and in particular characterize irreducible quantum channels in terms of their phase subspace. In Sect. 3.4, we relate completely positive non-trace-preserving maps to quantum channels, and in Theorem 3.14 establish weak conditions under which a completely positive PPT map is eventually entanglement breaking. • In Sect. 4, we establish finite-time properties of discrete-time quantum Markov semigroups. In Sect. 4.1, we investigate the so-called decoherencefree subalgebra of discrete-time evolution, and the contraction properties of the evolution with respect to this subalgebra, which is more subtle than in the continuous-time case. In Theorem 4.4, we establish a discrete-time Poincaré inequality, which we use in Sect. 4.2 to establish entanglement breaking times, including for PPT channels with a full-rank invariant state (Theorem 4.8). We also provide a method to compute entanglement breaking times for any faithful quantum channel in Remark 15. In Sect. 4.4, we apply these techniques to establishing times for doubly stochastic discrete-time evolutions to become mixed unitary. We also discuss another application of continuous-time decoherence bounds, namely to approximate quantum Markov networks, in Sect. 4.3. Similarly, we will denote by id H , or simply id, the identity superoperator on B(H). We denote by D(H) the set of positive semidefinite, trace one operators on H, also called density operators, and by D + (H) the subset of full-rank density operators. We denote by |Ω the maximally entangled state on H ⊗ H: given any orthonormal basis |i of H,

Notation, Basic Definitions and Preliminaries
Given a bipartite system H ⊗ K, we denote by SEP(H : K) the convex subset of separable states in D(H ⊗ K).
For p ≥ 1, the Schatten p-norm of an operator A ∈ B(H) is denoted by A p := (Tr|A| p ) simply denote by Φ p→q the operator norm of a superoperator Φ from Schatten class p to Schatten class q. Moreover, given a density matrix σ ∈ D + (H), we define the σ-weighted p-norm to be given by A p,σ = Tr(|σ In the case p = 2, the . 2,σ norm derives from an inner product A, B σ := Tr(σ The norms of superoperators between two such spaces are denoted by Φ p,σ→q,ω . These norms provide a natural framework to study the convergence of semigroups and we refer to, e.g., [46] for a review of some of their properties. Quantum Channels, Markovian Evolutions and Their Spectrum Next, a quantum channel is a completely positive, trace-preserving map Φ : B(H) → B(H). Given a linear map Φ : B(H) → B(H), its spectrum is denoted by sp(Φ). We will exploit extensively the special structure of the spectrum of quantum channels and its connection to the semigroup's asymptotic behavior, which we will now review in detail. Φ * corresponds to the dual map with respect to the Hilbert Schmidt inner product A, B HS := Tr(A † B). Like any linear operator on B(H), a quantum channel Φ admits a Jordan decomposition: where λ k are the eigenvalues of Φ, P k the associated (not necessarily orthogonal) eigenprojections, and N d k k = 0, where d k := Tr(P k ), so that K k=1 P k = I. Note that we split the Jordan decomposition into two parts, Φ P and Φ Q , and implicitly used the fact that all eigenvalues of quantum channels are contained in the unit disk of the complex plane. The operator Φ P corresponds to the eigenvalues of modulus 1, referred to as the peripheral spectrum. For any k ∈ [K], λ k P k + N k constitutes the kth Jordan block of Φ. Here, we have used that the peripheral eigenvalues of a quantum channel are semi-simple, so there is no associated nilpotent part [57].
In particular, since Φ is hermiticity preserving (and in particular positive), the eigenvalues of Φ either are real, or come in conjugate pairs. Since, moreover, Φ is positive unital (Φ(I) = I) or trace preserving (TrΦ(A) = Tr(A) for all A ∈ B(H)), 1 ∈ sp(Φ), all the other eigenvalues of Φ lie in the unit disk of the complex plane, and the eigenvalues lying on the peripheral spectrum are associated to one-dimensional Jordan blocks.
Given a quantum channel Φ : B(H) → B(H), the sequence {Φ n } n∈N is called a discrete-time quantum Markov semigroup (discrete-time QMS). Here, semigroup refers simply to the property that Φ n+m = Φ n • Φ m . Analogously, a continuous-time quantum Markov process (continuous-time QMS) corresponds to a family (Φ t ) t≥0 of quantum channels Φ t : B(H) → B(H) that satisfies the following conditions: Φ 0 = id, Φ t+s = Φ t •Φ s for any s, t ∈ R + 0 , and Φ t depends continuously on t. Any continuous-time QMS can be written as Φ t = e tL for a generator L : B(H) → B(H).
We primarily consider discrete-or continuous-time QMS which are faithful, that is, those that have a full-rank invariant state. Let us recall basic ergodic properties of these evolutions. The simplest case is that of primitive evolutions. We call a quantum dynamical semigroup (or the Liouvillian generator) primitive if it has a unique full-rank fixed point σ. In this case for any initial state ρ ∈ D(H), we have ρ t = e tL (ρ) → σ as t → ∞ (see [8,Theorem 14]). We refer to [8,57] for other characterizations of primitive channels and sufficient conditions for primitivity.
A notion closely related to primitivity is that of irreducibility. A positive linear map Φ : B(H) → B(H) is said to be irreducible if, for any orthogonal projection P ∈ B(H), Φ(P B(H)P ) ⊂ P B(H)P implies that P = 0 or P = I. For a positive trace-preserving map, this property is equivalent to the existence of a unique invariant state σ > 0 such that for every ω ∈ D(H), we have . Another useful criterion is that for a quantum channel, irreducibility is equivalent to 1 being a non-degenerate eigenvalue with the corresponding eigenvector (being proportional to) a faithful quantum state. This makes irreducibility a good candidate for a property that implies that the quantum channel is EEB, although we will show that is generally not the case. It is then natural to introduce the phase subspaceÑ (Φ) to study evolutions that are not necessarily primitive. It is defined as i.e., the linear span of the peripheral points and denote by P the projection onto it.
More generally, given a faithful continuous-time quantum Markov semigroup (Φ t ) t≥0 (i.e., for any t ≥ 0, Φ t (σ) = σ for a given full-rank state σ), there exists a completely positive, trace-preserving map E N , such that for any [11]) and so that there exists a self-adjoint operator H such that, Φ t • E N (ρ) = e iHt E N (ρ)e −iHt . In words, the evolution asymptotically behaves like a unitary evolution. On the other hand, observables under the action adjoint semigroup ( where the evolution is unitary. The completely positive map E * N is a projection ) and X ∈ B(H). In the case of a non-necessarily faithful quantum channel Φ : B(H) → B(H), the phase subspace is similarly known to possess the following structure (Theorem 6.16 of [57], Theorem 8 of [58]): there exists a decomposition of H as H (1.4) where p i is the orthogonal projector onto the ith subspace, for some fixed states τ i ∈ D + (K i ). Moreover, there exist unitaries U i ∈ H i , and a permutation π ∈ S K which permutes within subsets of {1, . . . , K} for which the corresponding H i 's have equal dimension such that for any element X ∈Ñ (Φ), decomposed . This is equivalent to the Choi matrix of the channel being separable (see [30] for the proof in the trace-preserving case; standard arguments can extend the result to the general case), where the Choi matrix of Φ is defined as   , H B ), whereas the class of 2-locally entanglement annihilating channels on H is denoted by LEA 2 (H). Note that any quantum channel that is entanglement breaking is also 2-locally entanglement annihilating: EB(H) ⊂ LEA 2 (H).
In this paper, we study the entanglement properties of quantum Markovian evolutions in discrete and continuous time. A discrete-time QMS {Φ n } n∈N , resp. a continuous-time QMS (Φ t ) t≥0 , is said to be eventually entanglement breaking (EEB) if there exists n 0 ∈ N, resp. t 0 ≥ 0, such that for any n ≥ n 0 , resp. t ≥ t 0 , Φ n , resp. Φ t , is entanglement breaking. The class of eventually entanglement breaking Markovian evolutions is denoted by EEB(H), leaving the choice of framework of discrete or continuous time implicit. We also say a quantum channel Φ is eventually entanglement breaking if the discrete QMS {Φ n } ∞ n=1 is eventually entanglement breaking. On the other hand, Markovian evolutions which are not entanglement breaking at any finite time are called entanglement saving, using language introduced by Lami and Giovannetti [38]; the class of entanglement saving Markovian evolutions is denoted by ES(H). Thus, the set of all Markovian evolutions (in either discrete time or in continuous time) decomposes into two disjoint classes,

EEB(H) ES(H).
(1.8) Lami and Giovannetti also introduce the notion of asymptotically entanglement saving evolutions in the discrete-time case. They showed that every discretetime QMS has at least one limit point, and either all of the limit points of a discrete-time QMS {Φ n } ∞ n=1 are entanglement breaking, or none of them are. They term the latter case as asymptotically entanglement saving, and we denote the set of asymptotically entanglement saving evolutions on H as AES(H). In analogy, we call the former case by asymptotically entanglement breaking, denoted AEB(H). Thus, the set of discrete-time QMS on H decomposes into the disjoint classes AES(H) AEB(H).
(1.9) It is interesting to compare (1.8) and (1.9) is entangled for every n, and the discrete-time QMS is entanglement saving. So we see AES(H) ⊂ ES(H). However, a priori, an entanglement saving QMS could be asymptotically entanglement breaking: at any finite n, J(Φ n ) could be entangled, but in the limit, J(Φ n ) could be in the set of separable states (though necessarily on the boundary). We therefore define EB ∞ (H) = AEB(H) ∩ ES(H), the set of discrete-time QMS which are asymptotically entanglement breaking, but not entanglement breaking for any finite n. With this notion, we may relate (1. (1.10) Remark 1. The isomorphism between bipartite states and CP maps given by the Choi matrix and the equivalence between EB maps and separable states discussed before allows us to directly translate results on the entanglement Figure 1. Relations between classes of quantum Markov semigroups, in which arrows represent subsets; for example, primitive discrete-time quantum Markov semigroups are a subset of irreducible discrete-time quantum Markov semigroups. The dashed arrows indicate relations which only hold for quantum Markov semigroups {Φ n } ∞ n=1 associated to a PPT channel Φ. The arrows without annotations are standard and are discussed in the text loss of CP maps to statements about the separability of a bipartite state. We will mostly state our results in the picture of CPTP maps (see also Sect. 3.4 for a method to remove the TP assumption), but it should be straightforward to derive the corresponding statements for bipartite states.
Some of the definitions introduced in this chapter, along with a preview of some of the results, are depicted diagrammatically in Fig. 1.
Let us now consider an example to illustrate these definitions and show that all sets arise naturally in physical systems. Example 1.1. Consider a discrete-time quantum Markov semigroup {Φ n } n∈N associated to a repeated interaction system, in which a system S interacts with a chain of identical probes E k , one at a time, for a duration τ . During the interaction, the dynamics of the system are modeled by a Hamiltonian evolution, and at the times (kτ ) k≥1 , the evolution forms a semigroup. In this example, the system and each probe are 2-level systems, with associated Hilbert spaces where a/a * , resp. b/b * , are the annihilation/creation operators for S, resp. E, and E ∈ R >0 (resp. E 0 ∈ R >0 ) corresponds to the energy of the excited state of S (resp. E). We can express these operators in the (ground state, excited state) basis of each system by We consider the initial state of each probe to be a thermal state, where β ∈ [0, ∞] represents the inverse temperature (setting Boltzmann's constant to one). In the case of zero-temperature (β = ∞), we take We consider an interaction modeling the two systems coupling through their dipoles, in the rotating wave approximation. In this setting, the system and each probe interact via the potential λv RW ∈ B(H S ⊗ H E ), where λ ≥ 0 is a coupling constant, and where u 1 is a constant, which we take to be equal to 1 with units of energy. This is a common approximation in the regime |E − E 0 | min{E, E 0 } and λ |E 0 |. The system begins in some state ρ i , couples with the first probe (in thermal state ξ β ), and evolves for a time τ > 0 according to the unitary operator That is, ρ i ⊗ ξ β evolves to U (ρ i ⊗ ξ β ) U * . Then we trace out the probe to obtain This process is repeated, and at the end of the kth step, the system is in the state where Φ is the quantum channel given by What class in the decomposition (1.10) does the discrete-time QMS {Φ n } n∈N lie in? To answer that question, we first compute the eigenvectors and eigenvalues of Φ, yielding for ν:= (E 0 − E) 2 + λ 2 , and, defining the rescaled inverse temperature β * := E0 E β, To analyze whether or not Φ n is entanglement breaking, it suffices to check if J(Φ n ) is PPT, as Φ is a qubit channel. To that end, define the Gibbs factor g = exp(−βE 0 ) and partition function Z = 1 + g. Then we have Thus, taking (|0 , |1 ) to be the (ground state, excited state) basis for each system, which has eigenvalues for g ∈ (0, 1], and 1, 1, ±|γ| 2n for g = 0. In either case, the eigenvalues only depend on the independent parameters |γ| ∈ [0, 1] and g ∈ [0, 1]. Since (id ⊗ T )J(Φ n ) ≥ 0 is equivalent to Φ n ∈ EB(H), we find • If |γ| = 1, {Φ n } n∈N ∈ AES(H). This occurs when ντ ∈ 2πZ; in this case, Φ is a unitary channel. • If |γ| = 0, then Φ ∈ EB(H). This occurs when E = E 0 , and ντ ∈ π +2πZ.

Continuous-Time Quantum Markov Semigroups
In this section, we exclusively investigate the simpler case of continuous-time quantum Markov processes. It turns out that in this case, there is a simple characterization of EEB(H): In order to prove the above result, we first need the following straightforward extension of Theorem 1 of [27].
Proof [Theorem 2.1]. First assume that (Φ t ) t≥0 is primitive. Then there exists a full-rank state σ ∈ D(H) such that, for any ρ ∈ D(H), Φ t (ρ) → σ as t → ∞. Therefore, H I consisting of separable states. Conversely, assume that (Φ t ) t≥0 ∈ EEB(H), i.e., there exists n ∈ N such that Φ n = Φ n 1 is entanglement breaking. Note that for any t ≥ 0 the map Φ t is invertible. In [38,Theorem 11], the authors show that for invertible maps, being AES equivalent to having more than one eigenvalue in the peripheral spectrum. As we assume Φ t to be EEB, we conclude that Φ t has a unique stationary state. In Proposition 7.5 of [57] this is shown to imply that the evolution is primitive, which concludes the proof.
We note that the results of [59] show that imposing the stronger notion that the underlying Markovian semigroup is asymptotically decoupling, i.e., that all outputs are product states in the limit t → ∞ is equivalent to the quantum channel being mixing, i.e., converges asymptotically to a unique, not necessarily faithful, quantum state. Their results also hold in discrete time.
Furthermore, for non-primitive continuous-time quantum Markov semigroups, one can provide a characterization of AES(H) in a similar fashion as what is done in Theorem 24 of [38] in the discrete-time case. We can then show: In [29] the authors show that such "measure and prepare" quantum channels are entanglement breaking. If now there exists i such that d Hi > 1, choose as j=1 |j |j is the maximally entangled state on H i ⊗ H i , and the result follows from the fact that id Hi ⊗ E N (ρ) is entangled.
) is commutative and non-trivial, meaning d Hi = 1 for all i ∈ J , and |J | > 1.

Remark 2. The dephasing semigroup provides a simple example of an element in EB ∞ (H).
Proof. Since Proposition 2.3 characterizes when (Φ t ) t≥0 ∈ AEB, it remains to exclude eventually entanglement breaking maps. However, the semigroup is eventually entanglement breaking if and only if it is primitive, by Theorem 2.1, and is primitive if and only if |J | = 1, which completes the proof. Proposition 2.1 justifies the introduction of the following characteristic times in the continuous-time primitive case: let (Φ t ) t≥0 be a primitive continuous-time quantum Markov semigroup with invariant state σ ∈ D + (H). The entanglement breaking time t EB (Φ) of (Φ t ) t≥0 , analogous to the entanglement breaking index in [37], is defined as follows: Similarly, given a continuous-time quantum Markov semigroup (Γ t ) t≥0 over a bipartite Hilbert space H A ⊗ H B , we define the entanglement annihilation time t EA (Γ) as follows In the case when Γ t = Φ t ⊗ Φ t , for Φ t : B(H) → B(H), this time is called the 2-local entanglement annihilation time, and is denoted by The quantity t LEA2 can be seen as a quantitative version of the notion of asymptotic decoupling for Markovian quantum dynamics [59]. Entanglement breaking, entanglement annihilation, and 2-local entanglement annihilation times of quantum Markov semigroups in discrete time can be similarly defined. In the next two subsections, we provide bounds on t EB , t EA and t LEA2 : the upper bounds found in Sect. 2.1 use the strong decoherence property of Markovian evolutions together with estimates on the radius of open balls around any full-rank product state. On the other hand, lower bounds found in Sect. 2.2 mainly use the inclusion EB(H) ⊂ PPT(H).

Upper Bounds on Entanglement Loss Via Decoherence
First, we briefly review in Sect. 2.1.1 the notion of strong decoherence of a quantum Markovian evolution which leads to the derivation of bounds on the time it takes for any state evolving according to a continuous-time quantum Markov semigroup to come ε-close to equilibrium. As a second step, in Sect. 2.1.2, we get quantitative bounds on the radius of balls surrounding any full-rank separable state on a bipartite Hilbert space H A ⊗ H B . Upper bounds on entanglement loss times follow by simply choosing ε as the radius of the separable ball around the adequate state found in Sect. 2.1.2. This is done in Sect. 2.1.3.

Strong Decoherence.
At the beginning of this section, we briefly recalled the convergence of quantum Markovian evolutions toward their decoherencefree subalgebra. Moreover, any finite-dimensional, faithful continuous-time quantum Markov semigroup (Φ t ) t≥0 satisfies the so-called strong decoherence property (SD): there exist constants K, γ > 0, possibly depending on d H , such that for any initial state ρ 1 : In the primitive case, good control over the constants K and γ can be achieved from so-called functional inequalities (see, e.g., [33,46,53]). These techniques were recently extended to the non-primitive case in [3,5]. Some of them have also been adapted to the discrete case (see [44,45,53]). In this section, we briefly review these tools that we use in the next section in order to derive upper bounds on the various entanglement loss times previously defined.
Poincaré Inequality Perhaps the simplest functional inequality is the Poincaré inequality (or spectral gap inequality): In the case of a continuous-time quantum Markov semigroup (Φ t ) t≥0 with associated generator L, its Poincaré constant is defined as [3]: and L * is the generator of the dual semigroup (Φ * t ) t≥0 acting on observables. The Poincaré constant turns out to be the spectral gap of the operator L * +L 2 , whereL is the adjoint of L * with respect to ., . σTr , namely minus its second largest distinct eigenvalue. Moreover, to the inner product ., . σTr , andÊ N is the conditional expectation onto the decoherence-free subalgebra ofΦ t .

Modified Logarithmic Sobolev Inequality The prefactor
Tr ∞ obtained from the Poincaré method is known to be suboptimal in some situations. A stronger inequality that one can hope for is the so-called modified logarithmic Sobolev inequality (MLSI): given a continuous-time quantum Markov semigroup (Φ t ) t≥0 , its associated modified logarithmic Sobolev constant α 1 (L) is defined as follows [3]: where EP L (ρ):=−Tr(L(ρ)(ln ρ−ln σ)) is the so-called entropy production of the semigroup (Φ t ) t≥0 . Its name is justified by the fact that d dt D(Φ t (ρ) E N (ρ)) = − EP L (ρ). This directly leads to the following exponential decay in relative entropy: This together with the bound D(ρ E N (ρ)) ≤ ln σ −1 Tr ∞ and the quantum Pinsker inequality implies (SD) with K = (2 ln( σ −1 Tr ∞ )) 1/2 and γ = α 1 (L). In particular, this new prefactor K constitutes a considerable improvement over the one derived from the Poincaré method. Similar decoherence times were recently obtained via decoherence-free hypercontractivity in [5] and other similar techniques have recently been developed to obtain convergence bounds similar to the ones obtained by MLSI inequalities [4,24].
Estimates in Diamond Norm By Proposition 2.1, we know that in continuous time, it only makes sense to talk about entanglement breaking times for primitive quantum Markov semigroups. As we will see in the next subsection, these times can be derived from the strong decoherence property of a primitive semigroup (Φ t ) t≥0 on B(H), of invariant state σ, when tensorized with the identity map on a reference system H R H. The resulting semigroup (Φ t ⊗ id HR ) t≥0 is faithful and non-primitive, and its associated conditional expectation E N takes the form σ ⊗ Tr H (·). In this case, for any ρ ∈ D(H ⊗ H R ): where · denotes the diamond norm. Hence, the strong decoherence bound on (Φ t ⊗ id HR ) t≥0 can be simply derived from a strong mixing bound for the primitive evolution (Φ t ) t≥0 at the cost of a multiplicative factor d 1/2 H . The same factor would appear from the spectral gap estimate, since in that case H . However, in finite dimensions, it was shown in [3] that the modified logarithmic Sobolev constant of (Φ t ⊗ id HR ) t≥0 is positive and in some situations can be of the same order as the one of (Φ t ) t≥0 (see also [24]). In this case, K = (2 ln(d H σ −1 ∞ )) 1/2 constitutes an improvement over the constant ∞ )) 1/2 that one would get from the diamond norm estimate, after using the strong mixing bound provided by MLSI for (Φ t ) t≥0 . Other related forms of convergence measures for (Φ t ⊗ id HR ) t≥0 have also been investigated in [25].

Separable Balls Around Separable States.
In this subsection, we consider the problem of finding separable balls around separable states. We restrict our discussion to full rank separable states, as separable states that are not faithful lie on the boundary of the set of separable states [38]. Note, however, that there are faithful separable states that lie on the boundary of the set of separable states and the boundary of the set of separable quantum states is still a subject of active research [15,36]. Thus, one way of quantifying how much the state lies in the interior of the set of separable states is the following measure of "robustness of separability," inspired by the robustness of entanglement introduced in [54] (see also Chapter 9 of [1]).

Definition 2.5 (Robustness of separability).
Let ρ AB > 0 be separable on the bipartite Hilbert space H ≡ H AB . We define its robustness of separability w.r.t. the maximally mixed state, R(ρ AB ), as Proposition 2.6 (Properties of the robustness of separability). Let ρ AB ∈ SEP(A : B). Then we have the following properties.

5)
and equality holds in the second inequality for product states: Proof. The upper bound in (2.5) follows from the fact that For product states, we may explicitly evaluate and by expanding ρ A in its eigenbasis, I A in the same basis, ρ B in its eigenbasis, and I B in the same basis.
For the second point, we first note that if R(ρ AB ) = 0, then for any λ ∈ (0, 1), This quantity is in the affine hull of SEP and can be made arbitrarily close to ρ AB by taking λ small, which proves ρ AB is not in the relative interior of SEP. The other implication follows from the bound (2.7), which is proven as follows. By the definition of R and the closedness of SEP(A : B), we may write is a separable state (cf. Theorem 1 of [27]), from which it follows that σ AB is separable as well, as a convex combination of separable states. For the last point, it suffices to prove the statement in the case of two states. If λ = min(R(ρ AB ), R(σ AB )), then and so for any t ∈ [0, 1], Remark 3. Together, (2.7) and (2.6) recover Lemma 2.2 in the case of density matrices. Note Proposition 3 of [38] shows that ρ A ⊗ ρ B ∈ int SEP(A : B) when ρ A and ρ B are full rank. Lemma 2.2 strengthens this result by giving a quantitative bound: where B p (r, ρ AB ) is the closed ball in p-norm of radius r around ρ AB . Admittedly, it is not a priori clear how to obtain good lower bounds on the robustness of separability for general separable states ρ AB and we leave this for future work.
Remark 4 (Separable balls in relative entropy). In the previous section, we commented on how to obtain better convergence estimates based on showing the exponential decay of other distance measures, like the relative entropy. It is natural to wonder if obtaining separable balls in the relative entropy and applying the convergence bound directly might lead to better results. Unfortunately, to the best of our knowledge, the radius of separable balls in the relative entropy is not known. However, such results would only lead to a constant improvement on the bounds for t EB we obtain with our methods. To see why this is the case, note for ρ ∈ D and σ ∈ D + , where the first inequality is Pinsker's inequality and second inequality follows simply from [2, Theorem 1]. Thus, the radius of the largest ball in trace distance and relative entropy can only differ by square roots and a factor which is polynomial in σ −1 ∞ . As we have a logarithmic dependence on the radius and on σ −1 ∞ in our bounds (see, e.g., Proposition 2.7), we could only hope to improve our bounds by a constant factor if we were able to derive optimal balls in relative entropy.

Upper Bounds.
Here, we combine the tools gathered in the last two subsections, namely estimates on the radius of balls surrounding tensor product states, as well as the strong decoherence property, in order to estimate from above the entanglement loss in the different situations defined at the beginning of this section. Proposition 2.7. Let (Φ t ) t≥0 be primitive with full-rank invariant state σ and generator L. Assuming that (SD) holds for (Φ t ⊗ id) t≥0 : Remark 5. Recall that strong decoherence always holds for some K, λ > 0 in the finite-dimensional setting considered here.
Proof. By Lemma 2.2 and the fact that 1-norm upper bounds the 2 norm, we know that the · 1 -norm around The proof of Theorem 4.10 can be adapted to get upper bounds on the entanglement annihilating time of a tensor product of semigroups. For example: with respect to ω). Assume that the spectral gaps are both lower bounded by λ > 0. Then, Proof. Since λ is a lower bound on the spectral gap of (Φ t ⊗ Ψ t ) t≥0 , it follows from Eq. (2.7) that choosing t as in the statement is enough to ensure that all the outputs of the semigroup are contained in the separable ball around σ ⊗ ω,

Lower Bounds Via the PPT Criterion
Here, we derive a lower bound on the time it takes a Markov semigroup (Φ t ) t≥0 to become entanglement breaking based on spectral data. The idea is simply to use the useful fact that the set of PPT states includes the set of separable states. We recall that a state ρ ∈ D(H A ⊗ H B ) is said to have a positive partial transpose (PPT) if the operator id ⊗ T (ρ) is positive, where the superoperator T denotes the transposition with respect to any basis (see Proposition 2.11 of [1]). We will prove lower bounds for the semigroup to become 2-locally entanglement annihilating, but note that also implies that it is not entanglement breaking.

Sufficient Conditions for Entanglement Loss.
In the next lemma, given a channel Φ we find necessary conditions on k for Φ k to be 2-locally entanglement annihilating.
Proof. We have |i j| ⊗ |j i| is the flip operator. Therefore, any witness Taking X AB = P asym := I⊗I−F 2 , the condition becomes using that Φ * is unital and Tr[F ] = d H . The right-hand side can be rewritten as which can be seen by taking A = Φ −k (B). Thus, it follows that and we obtain the claim.

Remark 7.
Since EB(H) ⊂ LEA 2 (H), these also constitute conditions for Φ k to be entanglement breaking.
Lower Bounds. In the next proposition, we derive a lower bound on t EB for a continuous-time quantum Markov semigroup using Lemma 2.9.

Proposition 2.11 (Lower bound for. t EB and t LEA2 ). For any continuous-time quantum Markov semigroup
j=1 are the eigenvalues of L, the generator of the QMS. In the case that (Φ t ) t≥0 is reversible with respect to a faithful state σ, L is self-adjoint with respect to · , · σ , and max j |Re(λ j (L))| = L 2,σ→2,σ is the largest eigenvalue (in modulus) of L.
maxj |Re(λj (L))| using that Φ t is the exponential of −tL and that (Φ t ) t≥0 is trace-preserving, so L must have a zero eigenvalue. By Lemma 2.9 it follows that 1 + (d 2 H − 1)e −2t maxj |Re(λj (L))| ≥ d H is a sufficient condition for the semigroup not to be entanglement annihilating at time t and the bound t LEA2 > 1 2 t 0 follows after rearranging the terms. The bound t EB > t 0 follows in the same way from (2.11).
Example 2.12 (The depolarizing channel ). The quantum depolarizing channel Φ p of parameter p is a unital quantum channel modeling isotropic noise, Its output can be interpreted as the time-evolved state under the so-called quantum depolarizing semigroup: This continuous-time QMS is primitive, with invariant state I dH , and has generator L(ρ) = I dH Tr[ρ] − ρ, which is the difference of a rank-1 projection and the identity map and therefore has eigenvalues 0 and −1. The spectral gap of 1 provides the following strong decoherence via (2.2): Propositions 2.11 and 2.7 therefore yield lower and upper bounds on the entanglement breaking time, namely These bounds match up to constant factors. In fact, the lower bound is tight: as calculated in, e.g., [39]. Likewise, considering time evolution under Φ t ⊗ Φ t , Proposition 2.11 and Eq. (2.9) yield the bounds which can be compared to the exact result,

Structural Properties of Discrete-Time Evolutions
In the previous section, we showed that the only continuous-time quantum Markov semigroups which become entanglement breaking in finite time were primitive quantum Markov semigroups. Now, we investigate the discrete-time case, where the situation becomes much more subtle. First, while every primitive discrete-time QMS is eventually entanglement breaking, not all eventually entanglement breaking discrete-time QMS are primitive. For example, the quantum channel has a rank-1 invariant state (and thus is not primitive), but the associated discrete-time QMS {Φ n } ∞ n=1 is entanglement breaking. Note that Ψ 2 τ = Ψ τ . For ∈ (0, 1), we define the channel Φ :=(1 − )Φ + Ψ I d which maps quantum states to strictly positive operators. This is known to be equivalent to the irreducibility of Φ [57, Theorem 6.2], which in particular implies the existence of a stationary state σ ∈ D + (H) of full rank. As ε → 0, we have Φ → Φ, which shows the density of faithful channels. Next, we reuse the same Φ and consider the quantum channel Φ :=(1 − )Φ + Ψ σ . As ε → 0, Φ → Φ, and moreover for each ε ∈ (0, 1), Φ is eventually entanglement breaking. To see this, note that as Φ has σ as a stationary state, it follows that Φ Ψ σ = Ψ σ Φ = Ψ σ and, thus, Clearly, lim n→∞ (Φ ) n = Ψ σ , which is primitive. We have As we saw in Proposition 2.6, separable states of the form σ 1 ⊗σ 2 with σ 1 , σ 2 ∈ D + (H) full rank are in the interior of the set of separable states. Thus, the Choi matrix of (Φ ) n converges to a separable state in the interior of the set of separable states and will be separable for some finite n. This also allows us to conclude the density of primitive channels, since they are in particular EEB. The corresponding results for PPT channels are obtained by noting that if Φ is PPT, then Φ and Φ are PPT as well.
One potentially useful observation is that to prove the PPT 2 conjecture, one may restrict to a dense set. More generally, define for k ∈ N the PPT k conjecture as the claim that any PPT quantum channel Φ has Φ k ∈ EB(H). Proof. Let Φ be a PPT quantum channel. By Proposition 3.1, there is a sequence Φ n → Φ with each Φ n being PPT and primitive. By continuity of x → x k , we have Φ k = lim n→∞ Φ k n . Since the set of entanglement breaking channels is closed, Φ k ∈ EB(H).
As mentioned before, our goal in this paper is to estimate the time after which a quantum system undergoing a quantum Markovian evolution has lost all its entanglement. In order to better characterize discrete-time evolutions for which asking this question makes sense, we first need to leave aside those evolutions for which the phenomenon does not occur, that is, evolutions that either destroy entanglement after an infinite amount of time (EB ∞ ), or even those of never-vanishing output entanglement (AES).
A big part of this question was already answered in the discrete-time case by [38]. In this paper, (Theorem 21), the authors showed that, given a quantum channel Φ : and only if either it has a non-full-rank positive fixed point, or the number of peripheral eigenvalues is strictly greater than 1, which itself is equivalent to the existence of 1 ≤ n ≤ d H such that Φ n has a non-full-rank positive fixed point. In the same paper, the authors showed that if Φ has more than d H peripheral eigenvalues, then {Φ n } n∈N is asymptotically entanglement saving.
These interesting results clearly show the link between the spectral properties of the quantum channel Φ and the entanglement properties of the corresponding discrete-time quantum Markov semigroup {Φ n } n∈N . In the next subsections, we further develop this intuition. First, we prove the following simple consequence of a result from [38].
The result follows from the fact that a quantum channel Ψ : B(H) → B(H) such that Ψ 1 > d H is not entanglement breaking (see [38]).

Irreducible Evolutions
The previous result suggests looking at channels with less than d H peripheral eigenvalues counted with multiplicity in order to characterize EEB. Since primitive channels are eventually entanglement breaking, a natural next step is to consider the wider class of irreducible channels introduced in Sect. 1.1.
The study of the peripheral spectral properties of irreducible maps is the subject of the non-commutative Perron-Frobenius theory for irreducible completely positive maps; see [21], or [57,Sect. 6.2]. See also [28, Appendix A] for a summary of this theory and extensions to deformations of irreducible CPTP maps. Together with the Jordan decomposition (see, e.g., [34,Sect. 1.5.4]), this theory provides a useful decomposition of irreducible quantum channels. In the next proposition, we recall this decomposition and provide a minimal set of quantities needed to construct such a map. This structural result will be used later to construct irreducible quantum channels that are not EEB. Moreover, it will allow us to show that irreducibility combined with the PPT property is enough to ensure that a quantum channel is EEB. (c) We have where P n (·) = Tr[u −n · ]u n σ for u:= z−1 k=0 θ k p k and θ:= exp(2iπ/z). Then Φ is an irreducible quantum channel. On the other hand, any irreducible quantum channel Φ can be decomposed as (3.2) for some choices of z, {p n } z−1 n=0 , σ, and Φ Q as in (1)- (4). Moreover, in either case, σ is the unique fix point 3 of Φ; P n (·) are its peripheral eigenprojections, associated to eigenvalues θ n and eigenvectors u n σ; and, for any j, k = 0, . . . , z − 1, we have the intertwining relations where the subscripts are interpreted modulo z. Additionally, for Φ P : = z−1 n=0 θ n P n , Proof. Let us note that (4c) is equivalent to the property that To see this, note that the generalized discrete Fourier transform F : ). All indices are taken mod z. Next, using the definition of the P n , (3.5) is equivalent to , and F has trivial kernel, (3.5) implies (4c). The converse follows similarly. Equation (3.3) follows from a simple computation. The fact that an irreducible map can be decomposed as (3.2) for some choices of z, {p n } z−1 n=0 , σ and Φ Q as in (1)-(4b) and (3.5) is not new, and we refer to [57, Sect. 6.2] for more details. We believe however that the forward implication is, and postpone its proof to Appendix A for sake of clarity.

Remarks.
• The matrixĴ k is separable, and thus, Φ k P is entanglement breaking, for any k ≥ 1.
• If z > 1, then 1 dHĴ k does not have full support. Thus, 1 dHĴ k is on the boundary of the set of density matrices and thus on the boundary of SEP and PPT as well. In fact, we can say something stronger than this: whenever z > 1, there exist entangled density matrices arbitrarily close to each 1 dHĴ k , k = 0, 1, . . . , z − 1. To see this, note that 1 dHĴ k ∈ L k D(H ⊗ H)L k . However, we can construct entangled states in L j D(H ⊗ H)L j for any j = k. For instance, let |0 ∈ p 0 H, |1 ∈ p 1 H, |j 0 ∈ p −j H, and |j 1 ∈ p 1−j H be normalized vectors. Then is (local-unitarily equivalent to) a Bell state and has is an entangled density matrix and can be made arbitrarily close to 1 dHĴ k by sending t → 0. Moreover, the limit points of { 1 dH J(Φ n )} ∞ n=0 are exactly { 1 dHĴ k } z−1 k=0 , which follows from the mixing-time results in the sequel (e.g., Theorem 4.4). This shows the analysis of Sects. 2.1 and 2.2 in the primitive case does not carry over to the irreducible discrete-time case, because the aforementioned limit points are neither in the interior of the set of separable states nor the interior of the set of PPT states.
• The above proposition shows that the peripheral eigenvectors of irreducible channels Φ commute. In Theorem 32 of [38], the authors show that asymptotically entanglement saving channels are characterized by the fact that they possess at least two non-commuting phase points. This implies {Φ n } n∈N ∈ AEB(H), which generalizes Corollary 6.1 of [49] to the non-unital case. There, the authors show that a unital irreducible quantum channel is AEB if and only if its phase space is commutative. • The intertwining property (3.3) holds for Φ and Φ P (which itself is an irreducible map) and therefore for Φ Q . This implies J(Φ Q ) = L 1 J(Φ Q )L 1 , i.e., the Choi matrix of Φ Q is supported on the same subspace as that of Φ P . • Given a map Φ Q which intertwines with {p n } z−1 n=0 , a sufficient condition for Φ Q ≥ −z (σ ⊗ I) L 1 ≡ −J(Φ P ) is given by since in that case The intertwining property (3.3) of irreducible maps is very useful for understanding their entanglement breaking properties. In fact, a slight generalization of this property will prove useful.
. Given two orthogonal resolutions of the identity, -block preserving. Using this notion, the following result shows that PPT channels which are block preserving in the above sense must annihilate off-diagonal blocks. We note that results of a similar flavor were shown in [12,13].
Theorem 3.7. Let Φ be an irreducible CPTP map, k ≥ 1, and let us adopt the notation of Proposition 3.4. Assume Φ is not primitive (i.e., z ≥ 2). Then On the other hand, if and additionally, for each j such that rankp j ≥ 2, In the case z = d H , we may write p j = |j j| for j = 0, . . . , z − 1. In this case, Φ k Q (|i j|) = 0 for all i = j, if and only if Φ k ∈ EB(H).

Proof. (3.8) follows immediately from the fact that if Φ is irreducible, then
-block preserving, and Lemma 3.6. Next, assume (3.9) holds. Let {|i } dH−1 i=0 be an orthonormal basis of H such that there are disjoint index sets {I n } z−1 n=0 such that for each n, |i ∈ p n H for i ∈ I n . Taking the Choi matrix in the |i basis, using that Φ k (|i j|) = 0 whenever i and j do not share an index set I n , which follows from (3.9). Then (see, e.g., (A.3)):

We note that i,j∈In Φ k Q (|i j|) ⊗ |i j| ∈ p n−k ⊗ p n B(H ⊗ H)p n−k ⊗ p n , since both Φ and Φ P map p n B(H)p n to p n−1 B(H)p n−1 [see Eqs. (3.3) and (A.2)]. Since by assumption
is separable on that space. We may embed this state in B(H ⊗ H) (without changing the tensor product structure) yielding that state on B(H ⊗ H). Summing over n then yields the fact that J(Φ k ) is separable, so Φ k ∈ EB(H).
For the case z = d, we simply note that rankp j = 1 for all j, and hence the statement follows from the above two results.
Since the limit points of {J(Φ n )} ∞ n=1 are separable but arbitrarily close to entangled states (as shown in the remarks following Proposition 3.4) the question arises of whether or not there are quantum channels that are both irreducible and in EB ∞ (H). In the case that Φ has maximal period z = d H , Lemma 3.3 resolves this affirmatively as long as Φ Q is not nilpotent. In the case when the period is much less than the dimension; say z = 2 < d H , then the underlying argument (relying on the reshuffling criterion via (2.11)) provides little help: Φ n 1 = z + o(n) < d H for large n. However, using Theorem 3.7, we can design EB ∞ (H) irreducible channels rather easily, as shown in the following example.  (3.6), and Φ is an irreducible CPTP map of period 2. Moreover, Φ n Q (|e 0 f 0 |) = λ n |f 0 e 0 | n odd λ n |e 0 f 0 | n even which is nonzero for any n. Thus, Φ ∈ EB ∞ (H) by Theorem 3.7. Additionally, it was proved in Theorem 21 of [38] that if the number of zero eigenvalues of Φ is strictly less than 2(d H − 1), the fact that Φ has at least two peripheral eigenvalues implies that it is entanglement saving. However, in the present example, Φ has four nonzero eigenvalues (two peripheral, and ±|λ|), and therefore d 2 H −4 zero eigenvalues. Thus, for d H ≥ 3, Theorem 11 of [38] does not apply to Φ. Theorem 3.7 implies the following corollary, which is extended to the nonirreducible case in the next section. Estimates on the entanglement breaking index, which we recall is the first n ∈ N such that Φ n is entanglement breaking, are provided in Sect. 4.2. Proof. Since the channel is PPT, we have (3.9). Then setting := spr(Φ Q ) < 1, by Gelfand's formula we have that for any ε ∈ (0, 1 − ), there exists n 0 > 0 such that for all k ≥ n 0 , and (3.10) holds. Hence, Φ k ∈ EB(H).

Beyond Irreducibility
A non-irreducible channel can be decomposed into irreducible components, on which it acts irreducibly. More specifically, we may decompose the identity I of H into maximal subspaces with corresponding orthogonal projections D, P 1 , . . . , P n s.t. Φ restricted to P i B(H)P i is irreducible, and DH is orthogonal to the support of every invariant state of Φ [10]. In particular, D = 0 is equivalent to Φ being a faithful quantum channel (that is, possessing a fullrank invariant state). In general, Φ may act non-trivially on P i B(H)P j for i = j. The following proposition shows that this is not the case for PPT maps Φ. Remark 10. This result extends that of [35] where it was shown that PPT maps are AEB. It also completes Theorem 4.4 of [49] where it was shown that any unital PPT channel is EEB. Additionally, Theorem 4.8 provides a quantitative version of this result.  Combining Theorem 3.10 with that observation that for an irreducible channel Φ of period z, the channel Φ z is the direct sum of primitive channels leads to the following structural result. Proof. Since primitive channels are eventually entanglement breaking, 4 the direct sum of primitive channels Φ = i Φ i is also eventually entanglement breaking, since is separable once each J(Φ n i ) is separable. Thus, (3) implies (1). To prove (2)  It remains to show (1) implies (2). Let Φ ∈ EEB(H) be a faithful quantum channel. Without loss of generality, assume Φ is not irreducible. Recall from the proof of Theorem 3.10 that there exist P 1 , . . . , for all i and j, and Φ i = Φ| PiB(H)Pi is irreducible, with some period z i , and

Perron-Frobenius projections {p
k=0 . Let Z be the least common multiple of the periods z i . Then for each i, j, k, . Moreover, since Φ is eventually entanglement breaking, Φ nZ is PPT for some n ∈ N. Hence, by Lemma 3.6 and (3.12), unless j = k and i = . For j = k and i = , is therefore a nilpotent linear endomorphism (on the vector space p

Criteria for AEB Channels to be Irreducible
In Sect. 3.1, we saw that all irreducible channels are asymptotically entanglement breaking. In this section, we derive a weak converse for this claim and give criteria for when AEB channels are irreducible. Recall the decomposition of the phase space of a quantum channel given in (1.5). Below, we show how this result relates to irreducible quantum channels, following the same notation as in (1.5): Proposition 3.12. The following are equivalent: 1. There exists a decomposition (1.4) such that Φ satisfies (1.5) with K 0 = {0} and d i = 1 for each i, and π is a K-cycle 2. Φ is irreducible. Therefore, any asymptotically entanglement breaking channel with corresponding K-cyclic permutation and K 0 = {0} is irreducible.
Let X ∈Ñ (Φ) be given by X = z−1 j=0 λ j u j σ for some λ j ∈ C. Since z−1 n=0 p n = I, we have where σ| pn is σ restricted to the subspace p n H, and the direct sum decomposition is with respect to the decomposition H = z−1 n=0 p n H. Note moreover, z−1 j=0 λ j θ nj is the nth coefficient of the discrete Fourier transform F z : C z → C z of the vector λ:=(λ j ) z−1 j=0 . Since the Fourier transform is invertible, as λ ranges over C z , the vector of Fourier coefficients range over C z as well. We therefore findÑ On the other hand, assume we are given such a decomposition, as suggested by the name. On the other hand, assume X ∈ B(H) is also invariant under Φ. Then X ∈Ñ (Φ), so X = K i=0 λ i τ i for some λ i ∈ C. But then Thus, X is proportional to σ, and Φ must have a unique invariant state. Since σ is additionally faithful (by construction), we conclude that Φ is irreducible (cf. Sect. 1.1 or Theorem 6.4 of [57], which shows that a positive map is irreducible if and only if its spectral radius is a non-degenerate eigenvalue with positivedefinite left and right eigenvectors).

Reduction to the Trace-Preserving Case
When considering quantum channels as a form of time evolution, tracepreservation is a natural assumption as an analogue of conserving total probability. When considering maps resulting from the inverse Choi isomorphism J −1 applied to bipartite states, however, the map J −1 (ρ AÃ ) is trace-preserving if and only if the first marginal is completely mixed: ρ A = IA dA . Thus, a priori, results about quantum channels only provide results about a restricted class of bipartite states. However, a large class of CP maps are (up to normalization) similar to CPTP maps. Moreover, this similarity transformation preserves many properties.
In the following, spr(Φ) denotes the spectral radius of a map Φ, defined as spr(Φ) = max λ |λ|, where λ ranges over the eigenvalues of Φ. The spectral radius of a quantum channel is 1. Proposition 3.13. Let Φ be a CP map such that for some X > 0, Φ * (X) = spr(Φ)X and for some σ > 0, Φ(σ) = spr(Φ)σ. Then has the following properties: • This transformation is not novel; the map Γ σ • Φ * • Γ −1 Φ(σ) is known as the Petz recovery map [47] and is usually considered in the case of CPTP maps Φ. Replacing Φ with Φ * and σ with X yields (3.14) in the case that Φ * (X) = spr(Φ)X. This transformation has also been considered in [28,Appendix A] in order to study the spectral properties of deformed CP maps as a function of the deformation. See [57, Theorem 3.2] for a similar but slightly different approach (using Φ * (I) instead of X).
• By the Perron-Frobenius theory (see [21,Theorem 2.5]), if Φ is a positive map, then the spectral radius spr(Φ) is an eigenvalue of Φ, and Φ admits a positive semidefinite eigenvector σ, which can be normalized to have unit trace. Since spr(Φ) = spr(Φ * ), the same logic applied to Φ * yields X ≥ 0 such that Φ * (X) = spr(Φ)X. Hence, the assumption of Proposition 3.13 is that both of these eigenvectors are full rank. • This transformation cannot be applied in general to a CPTP map Φ in order to obtain a CPTP unital mapΦ, since trace-preservingness will be lost. An intuitive way to see this is that a similarity transformation corresponds to a choice of (non-orthogonal) basis, and by fixing Φ * (I) = I, we choose a particular basis for Φ in which the dual eigenvector for spr(Φ) is represented by the identity matrix. Thus, in general, we cannot simultaneously choose a basis to fix a representation for the eigenvector of spr(Φ). Proof.
1. P X (Φ) is the positive multiple of the composition of CP maps and hence is CP. Since we have that P X (Φ) * is unital: and hence P X (Φ) is TP. Lastly, ρ:= ΓX (σ) Tr[...] is a full-rank invariant state for P X (Φ): up to normalization, This transformation allows us to establish our most general result on the entanglement breakingness of PPT maps. Theorem 3.14. Let Φ be a CP map such that for some X > 0, Φ * (X) = spr(Φ)X and for some σ > 0, Φ(σ) = spr(Φ)σ. Then if Φ is PPT, it is EEB.

Discrete-Time Decoherence-Free Poincaré Inequality
To derive quantitative bounds on entanglement breaking times for faithful discrete-time evolutions it suffices to restrict to primitive evolutions due to Theorem 3.11, since the entanglement breaking time of a direct sum of channels is the maximum of the entanglement breaking times of the component channels. Consider a primitive channel Φ. Since (Φ n ⊗ id)(Ω) where σ is the unique invariant state of Φ, we can use Lemma 2.2 to show that any state sufficiently close to σ ⊗ I dH is separable. Thus, it remains to establish a mixing time for Φ ⊗ id. The argument given in (2.4) shows that in fact it suffices to obtain a mixing time for Φ itself.
However, in the discrete-time case, mixing-time bounds even for primitive evolutions have some subtleties. Consider the qubit quantum channel Φ defined in the Pauli basis as: It is easy to see Φ defines a valid unital quantum channel by a direct inspection of the Choi matrix. Moreover, Φ 2 (X) = Tr(X) I 2 , which implies that Φ is primitive (e.g., [57,Theorem 6.7]). One can also easily check that: We see that both Φ * Φ(I) = I and Φ * Φ(σ z ) = σ z , implying the second largest singular value (with multiplicity) of Φ is 1 and results such as [53,Theorem 9] based on deriving mixing times from the singular values yield trivial estimates for the convergence of the map. This example is revisited in Remark 13. In this section, we remedy this by deriving mixing-time bounds which are always non-trivial. In fact, we will remove the primitivity assumption and work directly with general faithful quantum channels. To the best of our knowledge, the only framework that so far allows for convergence statements in the nonprimitive case is that of [52]. In that work, the underlying constants that govern the convergence of the quantum channel are a function of the complete spectrum of the channel. This approach does not provide a simple variational formulation of the constants that govern the evolution (such as (2.1)), which is usually vital for applications. Here, we provide a new bound in the spirit of [53] by exploiting the relation between the singular values of Φ and the Poincaré constant of the generator of a well-chosen continuous-time quantum Markov semigroup. Furthermore, we give necessary and sufficient conditions for the distance between the image of the quantum channel and its limit to be strictly contractive in terms of properties of this semigroup. Thus, we believe that these techniques will find applications outside of our context of estimating when a channel becomes entanglement breaking.
In Sect. 1.1, we briefly described the general asymptotic structure of quantum channels. In the Heisenberg picture, i.e., for completely positive, unital channels Φ * , the situation is similar: assuming that K 0 = {0}, the range of the projector P * , which is called the decoherence-free subalgebra of {Φ n } n∈N and denoted by N (Φ * ), has the form where p i is the projection onto the ith term in the direct sum, and each τ i ∈ D(K i ) is a full rank state (see, e.g., [38,Theorem 34]). This occurs when Φ is faithful (it possesses a full-rank invariant state). Recalling (1.4), under the faithfulness assumption, we likewise have in the Schrödinger picture, whereÑ (Φ) is the phase subspace of Φ and P is the projection ontoÑ (Φ). For any such discrete-time quantum Markov semigroup, the Jordan decomposition (1.1) immediately yields the following convergence result: for any X ∈ B(H), Here, the map P * plays the role of the conditional expectation E * N associated to quantum Markov semigroup in continuous time discussed in Sect. 1.1. Note, we work in the Heisenberg picture here because P * projects onto N (Φ * ) which is an algebra, while P projects ontoÑ (Φ) which is not.
Next, the following holds for any ρ ∈ D(H) and X = σ The relationship (4.3) allows the convergence in the Schrödinger picture to be controlled by the convergence ofΦ. This has the following advantages: the norm · 2,σTr is induced by an inner product (unlike · 1 ), andP is a conditional expectation as shown by the following lemma (unlike P ). (i)P is adjoint preserving: for any X ∈ B(H),P (X † ) =P (X) † ; (ii) for any X ∈ B(H) and Y, Z ∈ N (Φ),P (Y XZ) = YP (X)Z; (iii) for any X ∈ B(H), Tr(σ Tr X) = Tr(σ TrP (X)); (iv)P is self-adjoint with respect to σ Tr . Moreover, for any X, Y ∈ B(H).
(i) follows from the fact that P itself is adjoint-preserving. This can be seen from the fact that P = lim N →∞ 1 N N n=1 Φ n is the Cesaro mean of Φ and hence inherits the trace-preserving and complete positivity from Φ itself (see, e.g., Proposition 6.3 of [57]).
(iii) follows from a simple computation: since Φ is trace preserving, so is P (as discussed in the first point), and hence Tr(σ TrP (X)) = Tr(P • Γ σTr (X)) = Tr(Γ σTr (X)) = Tr(σ Tr X) (iv) is a consequence of (i)-(iii): Next, we show how to obtain the strong decoherence constants of a quantum Markov chain {Φ n } ∞ n=1 in terms of the Poincaré constant of a semigroup defined via Φ. However, in the discrete-time case the situation is a bit more subtle, as discrete-time semigroups are not necessarily strictly contractive. That is, there might be X =P (X) such that Φ (X) −Φ •P (X) 2,σTr = X −P (X) 2,σTr .
Thus, before showing how to obtain functional inequalities that characterize the strict contraction of the channel, we first discuss some necessary conditions for the contraction to occur.
Cycles in the Phase Algebra Here, we show that quantum channels might not be contractive for times not proportional to the order of the permutation in the phase algebra. More precisely, recall that for for some permutation π. We will now construct quantum channels which are not contractive for times smaller than the largest cycle in π. To this end, define in a fixed orthonormal basis for some 0 < < 1 and let π ∈ S d be the d-cycle (0 1 2 · · · d − 1). A is clearly a positive matrix with 1-s on its diagonal. Next, we define the quantum channel Φ π,A (X) := U π (A • X) U † π where U π : |i → |π(i) is a unitary representation of π and • denotes the Hadamard product.
which shows that these channels are not contractive at intermediate steps.

Remark 13. Note that in general we have that
) is a strict inclusion, even if the action of the channel Φ on the phase space does not contain any cycle. As an example, consider the qubit quantum channel Φ defined at the start of the section. Recall and that Φ is a primitive unital quantum channel such that Φ 2 (X) = Tr(X) I 2 . Since σ Tr = I/2,Φ = Φ and Φ * • Φ(σ z ) = σ z , we have σ z ∈ N ((Φ * t ) t≥0 ) but σ z ∈ N (Φ) = CI. Not surprisingly, it can be checked that the state 1 2 (I + σ z ) does not contract under this channel.
The previous discussion suggests that, in general, it is necessary to wait until the algebras coincide and the quantum channel has no cycle to ensure strict contraction. The next theorem, which generalizes Theorem 9 of [53] to the non-primitive case, shows that this is also sufficient. Moreover, for any ρ ∈ D(H) and any n ∈ N: where 1 − λ(L * ) < 1 corresponds to the second largest singular value of Φ, counted without multiplicities. (ii) More generally, let (Φ (k) t ) t≥0 be the continuous-time quantum Markov semigroup whose corresponding generator is given (in the Heisenberg picture) by Then, for any k larger than the size m Then, for any ρ ∈ D(H) and any n ∈ N: Moreover, there are quantum channels for which the minimal integer k such that the decoherence-free algebras coincide is given by m.
Finally, the condition of non-cyclicity of the action of Φ on its phase space can be removed by choosing k to be proportional to the least common denominator of the cycles of the semigroup.
Proof. (i) That 1 − λ(L * ) is the second largest singular value of Φ follows from the definition of the semigroup and of its Poincaré constant. This is because L * is, by definition, a s.a. operator with spectrum in [−2, 0]. In this case, the conditional expectation w.r.t. L * is just the projection onto the eigenspace corresponding to the eigenvalue 1 of Φ * •Φ, which just correspond to right singular vectors of the singular value 1 of Φ * . An application of the min-max theorem for eigenvalues then yields the claim. Then, notice that where we simply used (iv) of Lemma 4.1 and the commutativity ofP witĥ Φ for the second identity. One can similarly prove that X −P (X) 2 2,σTr = X 2 2,σTr − P (X) 2 2,σTr . Next, the conditional expectation E * N associated with the continuous-time quantum Markov semigroup (Φ * t ) t≥0 of generator L * is equal toP . To see this, note that σ Tr (defined by the discrete-time QMS) is invariant under the semigroup, since L(σ Tr ) = (Φ) * • Φ(σ Tr ) − σ Tr = (Φ) * (σ Tr ) − σ Tr = 0, and therefore is invariant under E * N . So bothP and E * N are conditional expectations onto the same algebra N ((Φ * t ) t≥0 ) = N (Φ) which both preserve σ Tr . Next, [11, Theorem 9] shows that there exists a unique conditional expectation onto N (Φ) which preserves a given faithful invariant state, and consequently we haveP = E * N . Next, λ(L * ) is the largest positive number λ such that λ X −P (X) 2 2,σTr ≤ − X, (Φ * •Φ − id)(X) σTr . (4.12) By a simple computation, one shows that the right-hand side of the above inequality is equal to X 2 2,σTr − Φ (X) 2 2,σTr . Similarly, the norm on the left hand side of Eq. (4.12) is equal to X 2 2,σTr − P (X) 2 2,σTr . Hence, (4.12) is equivalent to λ(L) ( X 2 2,σTr − Φ •P (X) 2 2,σTr ) ≤ X 2 2,σTr − Φ (X) 2 2,σTr , which is itself equivalent to (4.8). A simple iteration procedure of (4.12) together with (4.3) leads to (4.9) after observing that Tr ∞ . (ii) We apply a Schur decomposition of the quantum channelΦ = U * T U, where U is unitary w.r.t. to ·, · σTr and T is upper triangular. W.l.o.g. assume that the eigenvalues corresponding to the peripheral spectrum are on the diagonals T i,i for 1 ≤ i ≤ r, where r is the size of the peripheral spectrum. As the action of the channel is unitary on that subspace, the first r columns and rows of the matrix T must be orthonormal. But their diagonal entries already have norm 1, which shows that we may decompose the matrix T into a direct sum T = D phase ⊕ N , where D phase is diagonal and acts on the phase space of the channel, while N is upper triangular and has eigenvalues < 1 in modulus. Now, note that the con- is equivalent to the span of right singular vectors corresponding to the singular value 1 ofΦ k being equal to the phase space of Φ. It then follows from the decomposition of T that the two algebras coinciding corresponds to the matrix N k only having singular values strictly smaller than 1. This is equivalent to N k ∞ < 1, where by . ∞ we mean the operator norm of the matrix. Clearly, for k ≥ m the matrix N k is diagonalizable and, thus, the operator norm is bounded by the largest eigenvalue of N k in modulus. As N has spectral radius < 1, it follows that N k ∞ < 1. Finally, observe that the example given in Remark 13 saturates this bound. (4.10) and (4.11) follow similarly to (i).
Note that Theorem 4.4 always ensures a strict contraction of the quantum channel, unlike previous results in the literature, as the second largest singular value of a quantum channel ignoring multiplicities is always strictly smaller than 1. Moreover, in the case of quantum channels Φ satisfying detailed balance, it follows from the last proposition that Φ 2 is strictly contractive. To see why this is the case, not that these channels are always diagonalizable and have a real spectrum. Thus, the only possible eigenvalues for the peripheral spectrum are 1 and −1 and we can only have cycles of length 2.

Remark 14.
As in the continuous case, it is expected that is possible to further improve these convergence bounds by considering discrete versions of relative entropy convergence. In the discrete case, strict contraction in relative entropy is related to the notion of a strong data processing inequality. In certain situations, the corresponding strong data processing constant can be related to the so-called logarithmic Sobolev inequality of order 2 (see [42,44,45,48]), although all the results known in the quantum case only cover primitive evolutions. Also in this setting, it is possible to connect the contraction of the quantum channel to the contraction of the quantum semigroup in continuous time considered here.

Entanglement Breaking Times for Discrete-Time Evolutions
Similarly to the continuous-time case, we define the following entanglement loss times: the entanglement breaking time (or entanglement-breaking index, In the case when Γ n = Φ n ⊗ Φ n , Φ n : B(H) → B(H), this time is called the 2-local entanglement annihilation time, and is denoted by In analogy with Sect. 2.1.1, we say that a discrete-time quantum Markov semigroup {Φ n } n∈N on B(H) satisfies the so-called discrete strong decoherence property if there exist k ∈ N and constantsK > 0 andγ > 1, possibly depending on d H , such that for any initial state ρ ∈ D(H) and any n ∈ N: This property was shown to hold for faithful channels in Eq. (4.11) with k the size of the largest Jordan block of Φ,K = σ −1 It is now straightforward to adapt the results from the last sections to obtain bounds on when discrete-time quantum Markov semigroups become entanglement breaking.

Theorem 4.5. For a quantum channel Φ : B(H) → B(H) that is primitive with full-rank invariant state σ, we have
where k is the size of the largest Jordan block of Φ and λ(L * k ) is the spectral gap of L * k :=(Φ * ) k • (Φ) k − id. Proof. The proof follows the same reasoning as in Proposition 2.7, simply using Theorem 4.4 instead of (2.2) and noting that primitive quantum channels do not have cycles.

Remark 15.
1. In the system studied in Example 1.1, for |γ| < 1, the map Φ is primitive, with σ = ρ β * , and σ −1 ∞ = 1 + g, while 1 − λ(L * ) = |γ| 2 . Moreover, Φ has four distinct eigenvalues, and in particular, every Jordan block is of size 1. Hence, which matches the log((|γ| −1 )) −1 scaling of the exact entanglement breaking time computed in Example 1. We now derive some lower bounds on the time it takes to a bipartite channel to become 2-locally entanglement annihilating. Next proposition is a simple consequence of Corollary 2.10. Note that the lower bound given here is also a lower bound on the time it takes for a discrete-time quantum Markov semigroup to become entanglement breaking (cf. Remark 7). .
Proof. Note that we have for any linear operator Λ that We saw as a consequence of Theorem 3.10 that PPT channels with a fullrank state are eventually entanglement breaking. In the following proposition, we provide a quantitative version of that statement.
Proof. Note that it follows from the proof of Theorem 3.10 that the Choi matrix of the channel Φ is the direct sum of the Choi matrix of primitive quantum channels. The proof proceeds similarly to the one of Proposition 2.7 together with a use of Eq. (3.11), restricted to each one of these direct sums.

Approximate Quantum Markov Networks Via Strong Mixing
Another application of our results is to determine when all outputs of a quantum Gibbs sampler are close to being quantum Markov networks. A quantum Gibbs sampler (Φ Gibbs t ) t≥0 is a continuous-time quantum Markov semigroup that converges toward the Gibbs state of the Hamiltonian of a given lattice spin system. In the case when the Hamiltonian H consists of commuting terms, it was shown in [32] that at large enough temperature the spectral gap λ can be chosen independently of the size of the lattice. In particular, this means that the following bound on the trace distance between any initial state ρ and the equilibrium Gibbs state σ: On the other hand, the existence of a modified logarithmic Sobolev constant α 1 independent of the system size is still an important open problem. It is a well-known fact that Gibbs states corresponding to commuting interactions are quantum Markov networks [7,40]: given any tripartition ABC of the lattice such that B shields A away from C, σ ≡ σ ABC is a quantum Markov chain, that is, there exists a quantum channel R B→BC , usually referred to as the recovery map, such that R B→BC • Tr C (σ) = σ. Equivalently, such states are the ones with vanishing conditional quantum mutual information: More recently, [23] introduced the concept of an ε-approximate quantum Markov chain. Such tripartite quantum states ρ can be defined by requiring that their conditional quantum mutual information is small, that is, I(A : C|B) ρ ≤ ε, for ε > 0. It was shown in [23] that this is equivalent to the existence of a map R B→BC such that ρ is close to R B→BC • Tr C (ρ), say in trace distance. A simple use of the data processing inequality implies that states that are close to a quantum Markov chain are ε-approximate Markov chains. However, the opposite statement does not hold true in general [19,31].
In this section, we are interested in deriving bounds on the time it takes a state evolving according to a quantum Gibbs sampler to become an εapproximate quantum Markov network. More precisely, let ABC be a tripartition of a lattice spin system Λ as above, and let (Φ Gibbs t ) t≥0 a continuous-time quantum Gibbs sampler. A very simple bound can be achieved by rudimentary triangle inequality: letting ρ t :=Φ Gibbs t (ρ), where the time to become an ε-quantum Markov chain with respect to a clique A − B − C is defined as A much better bound can however be derived from the joint use of the following bound from [23]: for any tripartite state ρ ABC ∈ D(H A ⊗ H B ⊗ H C ), there exists a CPTP map Γ B→BC such that where F (ρ, σ) = (Tr| √ ρ √ σ|) 2 denotes the fidelity of to states ρ and σ. By the Fuchs and van de Graaf inequality, this implies that: (4.17) Together with the Alicki-Fannes-Winter continuity bounds on conditional entropies, we obtain the following result: ≤ δ(t) ln |B| + (1 + δ(t)/2) h 2 δ(t)/2 1 + δ(t)/2 ≤ δ(t) (ln |B| + 1) + δ(t) 2 2 where the second inequality follows from Lemma 2 of [56], and given 0 ≤ p ≤ 1, h 2 (p) denotes the binary entropy of the distribution {p, 1 − p}. The result directly follows.
The above proposition together with the bound coming from the Poincaré inequality yields a bound where K = σ −1 ∞ and γ = λ. Having access to a modified logarithmic Sobolev inequality would provide an even stronger bound: by Theorem 4 of [31], This together with (4.17) yields

Time to Become Mixed Unitary
Functional analytical methods can also be used in order to derive other convergence results. As an example, we provide a bound on the time required by a doubly stochastic discrete-time quantum Markov semigroup to come close to a random unitary channel. Birkhoff's theorem states that any doubly stochastic matrix may be written as a convex combination of permutation matrices. It is well known that the quantum analogue of this theorem does not hold in general, i.e., there are doubly stochastic quantum channels that cannot be written as a convex combination of unitary channels, that is, mixed unitary channels. We refer to, e.g., [41] and references therein for a discussion of this problem. However, we will show that applying the channel often enough might lead it to become a mixed unitary channel. Let The goal of this section is to estimate this time, and we will follow an approach that is similar to the last sections on entanglement breaking times. In [55,Corollary 2], the author shows that if Müller-Hermes for interesting discussions. DSF acknowledges financial support from VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059) and from the QuantERA ERA-NET Cofund in Quantum Technologies implemented within the European Union's Horizon 2020 Programme (Quan-tAlgo project) via the Innovation Fund Denmark. We would like to thank the anonymous QIP 2019 reviewers for pushing us more to consider the nonirreducible case, and an anonymous referee for suggesting the very simple proof of Lemma 2.2.
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A Proof of Proposition 3.4
Let us show that given z, {p n } z−1 n=0 , σ, and Φ Q , the decomposition (3.2) gives an irreducible quantum channel. Note, by the definition of P n , for any X ∈ B(H), Since P 0 (X) = Tr[X]σ, we have for j = 0, 0 = P 0 • P j (X) = Tr[P j (X)]σ, yielding that P j is trace-annihilating: Tr[P j (X)] = 0 for all X ∈ B(H). In the same way, using assumption 4c, Φ Q is trace-annihilating. Thus, Φ = P 0 + z−1 n=1 θ n P n + Φ Q is trace preserving. Next, we prove (3.4), which will prove Φ is CP via assumption 4b. For Φ P := be an orthonormal basis of H such that the first rank(p 0 ) elements are a basis for p 0 H, the next rank(p 1 ) elements are a basis for p 2 H, and so on. We have p 0 = rank(p0)−1 i=0 |i i|, and so forth. Thus, In particular, J(Φ P ) = z σ ⊗ I L 1 . Thus, by assumption 4b, and hence Φ is CP. Since Φ is CPTP, we can use (1.2) to prove Φ is irreducible. We have