Group Transference Techniques for the Estimation of the Decoherence Times and Capacities of Quantum Markov Semigroups

1 Group transference techniques for the estimation of the decoherence times and capacities of quantum Markov semigroups. Ivan Bardet∗, Marius Junge†, Nichcolas Laracuente†, Cambyse Rouzé‡, Daniel Stilck França§

∗Institut National de Recherche en Informatique et en Automatique, Paris, France †Department of Mathematics University of Illinois at Urbana-Champaign Champaign, IL, USA ‡Faculty of Mathematics, Technische Universität München, Munich, Germany §Centre for the Mathematics of Quantum Theory, University of Copenhagen, Copenhagen, Denmark

Abstract—Capacities of quantum channels and various information processing tasks is of central decoherence times both quantify the extent to which importance to the development of quantum technologies. quantum information can withstand degradation by interactions with its environment. However, calculating The dynamics of open systems are modeled capacities directly is known to be intractable in general. by completely positive trace preserving maps. In the Much recent work has focused on upper bounding certain Markovian approximation, continuous time evolutions capacities in terms of more tractable quantities such as are then modeled by quantum Markov semigroups specific norms from operator theory. In the meantime, there Tt t 0 of such maps. Given a concrete quantum Markov has also been substantial recent progress on estimating ( ) ≥ decoherence times with techniques from analysis and semigroup, it is then important to identify short time geometry, even though many hard questions remain open. versus long time behaviour of the evolution. For example, In this article, we introduce a class of continuous-time it is important to know how long entanglement can be quantum channels that we called transferred channels, preserved. This remains a challenging problem. Even which are built through representation theory from a for classical systems, precise decoherence time estimates classical Markov kernel defined on a compact group. In particular, we study two subclasses of such kernels: are very delicate, see [1], [2]. The aim of this paper is Hörmander systems on compact Lie-groups and Markov to obtain some ‘concrete’ estimates on the decoherence chains on finite groups. Examples of transferred channels time of such dissipative evolutions, as well as to derive include the depolarizing channel, the dephasing channel, bounds on various capacities using classical and quantum and collective decoherence channels acting on d qubits. functional inequalities. Some of the estimates presented are new, such as those for channels that randomly swap subsystems. We In the classical setting, the connection between then extend tools developed in earlier work by Gao, functional inequalities and decoherence times is very Junge and LaRacuente to transfer estimates of the well-established, (see e.g. [1], [3], [4]), and many works classical Markov kernel to the transferred channels and started to establish the connections in the quantum study in this way different non-commutative functional case in recent years. But many of the techniques only inequalities. The main contribution of this article is the application of this transference principle to the estimation work for semigroups with a unique invariant state, of decoherence time, of private and quantum capacities, and actually proving such inequalities for quantum of entanglement-assisted classical capacities as well as systems remains challenging. In this paper, we start arXiv:1904.11043v2 [quant-ph] 4 Jun 2021 estimation of entanglement breaking times, defined as the mending this gap by showing how to obtain various first time for which the channel becomes entanglement quantum functional inequalities starting from a classical breaking. Moreover, our estimates hold for non-ergodic channels such as the collective decoherence channels, an one. To make the connection between classical and important scenario that has been overlooked so far because quantum Markov semigroups, we consider mixed unitary of a lack of techniques. quantum channels in which the unitaries form a representation of a group, and their weights come from a classical Markovian process on the group. We I.INTRODUCTION call such semigroups transferred semigroups, and they include widely studied error models, such as depolarizing N quantum mechanics, the evolution of a global, or dephasing quantum channels. We are particularly I closed system leads to a unitary evolution of interested in non-ergodic semigroups of channels, that states. However, any realistic quantum system undergoes is, semigroups admitting more than one invariant state. dissipative dynamics, due to its unavoidable interaction These semigroups have been known to play an important with its surrounding environment. Understanding how role in various quantum error prevention schemes this noise limits the usefulness of these systems for [5]. Unfortunately, even the literature for non-ergodic semigroups in the commutative community is relatively as mixed unitary channels with unitaries arising from a sparse. However, we will show how to directly translate representation of a finite group play an important role in classical results to the quantum setting if the underlying quantum information theory. classical dynamics is ergodic, even if the quantum a) Layout of the paper:: In SectionII, we semigroup is not. introduce the framework of quantum Markov semigroups Besides the obvious application of decoherence and explain their connection to classical diffusions and times estimate, we use this transference principle jump processes on groups via the so-called transference to obtain entropic inequalities for these semigroups. technique. In Section III, we explain the technical tools By showing some tensorization results and that the that allow us to bound various norm estimates of a inequalities remain valid when we tensorize the quantum Markov semigroup in terms of the kernel of an underlying semigroup with the identity channel, we use associated classical process: namely, noncommutative Lp them to estimate several different capacities of these spaces and the norm transference technique. SectionIV semigroups, such as the (two-way) private and quantum is devoted to some examples of transferred semigroups capacity, classical capacity and the entanglement assisted to illustrate the technique. We then illustrate how to use classical capacity. Moreover, these bounds are in the our techniques to estimate capacities and apply them to strong converse sense and have the right asymptotics. other resource theories in SectionV. Then, in SectionVI, The geometry of the underlying space (i.e. of the we show how contractivity properties (in particular, compact Lie group or the finite group) is crucial for these ultracontractivity) of the quantum Markov semigroups a priori estimates, in particular for concrete estimates. can also provide a way to estimates some capacities in However, it should be noted that these inequalities are the quantum case, without the use of transference. not sharp in general, as they do not depend on the representation at hand. Furthermore, as we will observe later, one may obtain the same quantum channel from II.QUANTUM MARKOVSEMIGROUPSVIAGROUP transfering Markov kernels from different groups, which TRANSFERENCE can lead to more or less pertinent estimates. A quantum Markov semigroup (QMS) (Tt)t≥0 To exemplify the power of these methods, on B(H) is a uniformly continuous semigroup consider collective channels. These are quantum channels of completely positive maps such that T0 = id in which the same error occurs in different registers. The and Tt(IH) = IH for all t ≥ 0 [8]. The limit simplest examples for collective channels are derived L = limt→0(id −Tt)~t exists and is called the Lindblad from the standard Pauli matrices, and correspond to generator. We insist on our convention that consequently the same Pauli error occurring on different qubits at −tL Tt = e with a minus sign! This is not the most often the same time. These quantum channels clearly do not used convention in the quantum case but it is more have a unique invariant state and thus, it is difficult consistent with the classical situations we will consider. to quantify how fast they mix using current techniques in the literature. Nevertheless, using group transference, The QMS (Tt)t≥0 models the evolution of we will be able to estimate decoherence times of these observables in the Heisenberg picture. In the dual channels independently from the number of qubits, and Schrödinger picture, one is instead interested in the similar estimates hold for their capacities. Moreover, we evolution of states of density matrices. We recall that will derive some estimates that are new even in the a density matrix ρ ∈ B(H) on H is a trace-one positive classical literature. This will be the case for quantum semi-definite operator. We denote by D(H) the set of channels that randomly swap subsystems. density matrices on H and by D+(H) the set of invertible Inspired by the techniques of [6], we also (full-rank) density matrices. show a smorgasbord of functional inequalities for these We shall mainly (but not only) study self-adjoint (or semigroups, such as spectral gap, hypercontractivity, symmetric) QMS for the Hilbert Schmidt scalar product: logarithmic Sobolev inequalities and ultracontractive ∗ ∗ estimates. The latter three are particularly important to Tr [Tt(x ) y] = Tr [x Tt(y)] ∀x, y ∈ B(H) , ∀t ≥ 0 . obtain good estimates for small times, before the spectral This is equivalent to the fact that T T †, where T † gap kicks in and closes the deal. We also exemplify t = t is the adjoint of T with respect to the Hilbert-Schmidt why in the non-transference case, the correct notion t inner product. This also implies that the maximally for applications of the decoherence-time is a complete IH IH mixed state is an invariant state: Tt( ) = . version, i.e. where we consider the semigroup tensorized dH dH with the identity on a matrix algebra. Form now on, we always assume that the Finally, let us point out that in contrast to [6], maximally-mixed state is an invariant state. Then, the set which focuses on Lie groups and Hormänder systems of fixed-points N of the QMS becomes an algebra, as (where very good estimates are available from the fix proved by Frigerio in [9]. It is defined by: fundamental work of Rothschild and Stein [7]), we are also interested in finite groups and jump processes, Nfix = {x ∈ B(H) ; Tt(x) = x ∀t ≥ 0} .

2 Let Efix be the orthogonal projection on Nfix for the The following lemma, which is a special case of a result Hilbert-Schmidt scalar product. As proved in the same from [6], is at the heart of the transference method. In article, it is a conditional expectation in the sense of particular, it will allow us to obtain contraction properties operator algebra, that is Efix(a x b) = a Efix(x) b for of a transferred quantum Markov semigroup in terms of all a, b ∈ Nfix and x ∈ B(H). Remark also that as an the ones of the classical Markov semigroup from which † orthogonal projection, it is self-adjoint: Efix = Efix. We it is transferred. denote by D(Nfix) ≡ E[D(H)] the image of the density Lemma II.1 (Lemma 4.6 in [6]). The following relation matrices for this conditional expectation: one has ρ ∈ holds for all t 0: D(Nfix) if and only if ρ = Efix[ρ]. Selfajoint QMS are ≥ in particular ergodic, in the sense that: π ○ Tt = (St ⊗ idB(H)) ○ π . (4) Tt(x) Ð→ Efix(x) ∀x ∈ B(H) . (1) t→+∞ Proof. We recall the proof for sake of completness. We This can for instance be seen by considering the have, for any x ∈ B(H), spectrum of the QMS, which in this case is real with no π ○ Tt(x)(g) peripheral eigenvalue (see [10] for instance). ∗ −1 −1 = u(g) kt(h ) u(h ) x u(h) dµG(h) u(g) SG We now proceed to the presentation of the class −1 −1 −1 = kt(gg h ) u((hg) ) x u(hg) dµG(h) of QMS we shall study in this article. We start in SG Section II-A by introducing the general method based on −1 −1 = kt(gh ) u(h ) x u(h) dµG(h) group transference, which allows to build a QMS from a SG (classical) symmetric Markov semigroup on a group with = (St ⊗ idB(H))(π(x))(g) . right invariant kernel. In Section II-B and Section II-C we specialize this discussion to two classes of Markov From the invariance of µG, one can also easily semigroups: Hörmander diffusions and jumps. On the verify that any QMS (Tt)t≥0 transferred from (St)t≥0 is doubly stochastic: T † d−1I d−1 I for any t 0. On other hand, given a QMS, we show in Section II-D how t ( H H) = H H ≥ to find a Markov semigroup for which the QMS can be the other hand, the reversibility of (St)t≥0 is transferred transferred. This construction can be interpreted as a finer to the QMS (Tt)t≥0: version of the characterization of quantum convolution Lemma II.2. Assume that the Markov semigroup (St)t≥0 semigroups of [11]. 1 is reversible, or equivalently that kt(g) = kt(g− ) for any g ∈ G. Then any QMS (Tt)t≥0 transferred from (St)t≥0 A. General construction is self-adjoint with respect to d−1 I . H H The starting point is a compact group G, either Lie or finite, with Haar measure µG (we shall simply Proof. The result follows from the simple calculation: write µ when there is no ambiguity). Let (St)t 0 be ≥ ⟨x, Tt(y)⟩HS a Markov semigroup on the space L∞(G) of bounded, −1 ∗ ∗ measurable functions on G. We will always assume that = kt(g ) Tr(x u(g) y u(g)) dµG(g) SG St t 0 admits the following kernel representation: ( ) ≥ −1 ∗ ∗ = kt(g ) Tr((u(g) x u(g) ) y) dµG(g) SG St(f)(g) = kt(g, h) f(h) dµG(h) . (2) SG −1 ∗ ∗ = S kt(g ) Tr((u(g) x u(g)) y) dµG(g) We also assume that S is right-invariant, which G ( t)t≥0 T x , y , means that the probability to visit h from g only depends = ⟨ t( ) ⟩HS −1 on gh . This implies that µG is an invariant probability where the third line follows from the identity kt g 1 ( ) = k g, h k gh− , e e −1 distribution and that t( ) = t( ), where is kt(g ) for all g ∈ G. the neutral element of the group. We keep the same notation kt(g) for kt(g, e). Let g u g be a projective representation of G −1 ↦ ( ) Since d IH is an invariant state of (Tt)t≥0, on some finite dimensional Hilbert space . We define H H the set Nfix of fixed points is an algebra (see [9], the following convolution QMS on B(H) which we call [12], Theorem 6.12 of [10]), and is characterized as the a transferred QMS: commutant of the projective representation (see Theorem

−1 ∗ 6.13 of [10]): Tt(x) = S kt(g ) u(g) x u(g) dµG(g) . (3) G ′ Nfix = {σ ∈ B(H)S∀g ∈ G ∶ σu(g) = u(g)σ} ≡ u(G) . At the root of the transference techniques that we study (5) in this article is a factorization property between S ( t)t≥0 By deﬁnition, it is also the algebra of ﬁxed points and (Tt)t≥0, involving the standard co-representation of the ∗-automorphisms x ↦ u(g)∗ x u(g), g ∈ G. This ∗ π ∶ B(H) → L∞ (G, B(H)) , π(x)(g) = u(g) xu(g) . implies that the following commuting diagram holds:

3 Efix the vector ﬁeld V , u g t is a one-parameter family B(H) Nfix j ( j( )) of unitaries and hence π π d EµG u g t i a (7) L G, . ( j( ))V = j ∞( B(H)) B(H) dt t=0 where a is self-adjoint. This implies that, for Here B(H) in the lower right corner has to be understood j ∈ B(H) as the subalgebra of constant value functions on G with any x ∈ B(H), value in . B(H) (Vj ⊗ id ○ π(x)(g) In practice, we will only consider situations where B(H) d the classical Markov semigroup St t 0 is primitive, that π x g t g ( ) ≥ = dt ( )( j( ) )V is, µ is the unique invariant distribution and furthermore t=0 G d ∗ ∗ = u(g) u(gj(t)) x u(gj(t)) u(g)V Stf Ð→ EµG [f] = f(g) dµG(g) . dt t=0 t→+∞ SG = −iπ([aj, x])(g) . This does not imply that (Tt)t≥0 is also primitive, however, it will always be ergodic as deﬁned in Therefore we get Equation (1). (LV ⊗ idB(H)) ○ π(x) 2 We now turn our attention to two special cases = − Q i π([aj, [aj, x]]) j of the above construction. In both cases, we explicitly 2 2 construct the Lindblad generator of the QMS. = π(Q aj x + x aj − 2aj x aj) . j

B. Diffusion It means that the Lindblad generator of the transferred QMS is given by Given a Riemannian manifold , a Hörmander M 2 2 system on M is a set of vector ﬁelds V = LV (x) = Q aj x + x aj − 2 aj x aj . j {V1, ..., Vm} such that, at each point p ∈ M, there exists an integer K such that the iterated commutators Considering the more general case of Hörmander

Vi1 , [Vi2 , ⋯[Vik , ⋅]] , k = 1, ..., K, generate the systems instead of the Laplacian is motivated by tangent space Tp M. Specializing to the case of a Lie simple examples relevant to quantum information. One group G, a Hörmander system V = {V1, ..., Vm} can such example is the Lindblad generator with Kraus more simply be defined as a set of vectors in the Lie operators σx and σz. These are transferred from a algebra, i.e. the tangent space at the neutral element e, Hörmander system for the group SU(2). However, such that for some K ∈ N the iterated commutators of the third direction is missing and hence it is not a order at most K span the whole tangent space. For fixed Laplace-Beltrami Laplacian, which would involve the j ∈ {1, ..., K}, we find a geodesic gj(t) with gj(0) = e whole orthogonal basis of the Lie algebra. 1 such that for any f ∈ C (G) d Vj(f)(h) = f(gj(t) h)V . Conversely, if a Lindblad generator of a QMS dt t 0 = on B(H) has the form given by the previous equation This leads to the corresponding left invariant classical for some self-adjoint elements aj ∈ B(H), then we generator can consider the anti-self-adjoint operators i aj as 2 tangent elements of the Lie group U(H) at the identity LV ∶= − Q Vj . (6) j IH. Therefore they generate a Hörmander system. Furthermore, we can consider the Lie-subgroup G of The generator L generates a Markov semigroup V with tangent space at identity spanned by this P e−tLV on L G . Whenever V is a basis for U(H) t = ∞( ) Hörmander system. The corresponding generator L is the Lie algebra, then L ∆ is the negative of the V V ≡ − therefore the generator of a primitive Markov semigroup Laplacian and P is called the heat semigroup. ( t)t≥0 S on L G . Since the semigroup commutes with the right action ( t)t≥0 ∞( ) of the group it is implemented by a right-invariant convolution kernel as in Equation (2) and it is reversible We summarize this discussion in the next theorem. with respect to the Haar measure.

Next, considering a projective representation g ↦ u(g) of G on some finite dimensional Hilbert space H, Theorem II.3. Let g ↦ u(g) be a projective we want to find the Lindblad generator of the QMS representation of a compact Lie group G on some defined by Equation (3). We first observe that, for fixed finite dimensional Hilbert space H. Then the Lindblad −tLV j ∈ {1, ..., K} and given the geodesic gj associated to generator of the transferred QMS (Tt = e )t≥0 as

4 defined by Equation (3) is given by Tr [y u(h)]. Applying Equation (9) to fy we find 2 2 LV (ρ) = Q aj ρ + ρ aj − 2 aj ρ aj , (8) dfy(gt) = Q ch Tr [y (u(hgt− ) − u(gt− ))] dt j h∈G{e} √ h where the aj are defined by Equation (7). Conversely, + Q ch Tr [y (u(hgt− ) − U(gt− ))] dNt . let L be the Lindblad generator of a QMS on B(H) h∈G{e} which takes the form (8) for some self-adjoint elements From this we deduce aj in B(H). Then there exists a compact Lie group G, a du g c u h I u g − dt continuous projective representation u ∶ G → U(H) and ( t) = Q h ( ( ) − H) ( t ) h∈G{e} a Hörmander system V V1, ..., Vm in the Lie algebra = { } √ h ∗ c u h I u g − dN of G such that π ∶ x ↦ (g ↦ u(g) x u(g)) satisfies + Q h ( ( ) − H) ( t ) t h∈G{e} π x L id π x x . h (L( )) = ( V ⊗ B(H)) ○ ( ) ∀ ∈ B(H) − ˜ = Q (u(h) − IH) u(gt ) dNt . (10) h∈G{e} This equation is well-known in the theory of quantum C. Jumps stochastic calculus, see [13], [14].

Let now G be a finite group and let −tL Theorem II.4. Let (St = e )t≥0 be a Markov (kt(g, h))g,h∈G be a right-invariant density kernel semigroup on a finite group G with right-invariant on G. Let us denote by g the stochastic process 1 ( t)t≥0 Markov kernel. Write cg = −L(g− , e) for g ≠ e. Then the on G induced by this kernel. The corresponding Markov generator of the QMS (Tt)t≥0 defined by Equation (3) is semigroup admits a transition matrix L such that given for all x by tL ∈ B(H) St = e− for all t ≥ 0. In view of Equation (2), the ∗ connection between the Markov kernel and the transition L(x) = Q cg x − u(g) x u(g) , x ∈ B(H) . matrix is therefore given by: g∈G{e} (11) 1 −tL − kt(g, h) = SGS e (g, h) , g, h ∈ G. Furthermore, dH IH is an invariant density matrix and if (St)t 0 is reversible, then so is (Tt)t 0. Writing c L h−1, e for all h e, we then have by ≥ ≥ h = − ( ) ≠ Conversely, let L be a Lindblad generator on B(H) of right invariance that for all f ∈ L∞(G), the form m L f g ch f hg f g . ( )( ) = − Q [ ( ) − ( )] ∗ h∈G L(x) = Q ck(x − uk x uk) , x ∈ B(H), k 1 Thanks to the right-invariance, we can define a family = ˜ h for some unitary operators uk and some positive of independent Poisson processes Nt t 0 with ∈ U(H) ( ) ≥ h∈G constants ck. Assume that the group G generated by intensity ch such that for any function f ∈ L∞(G): u1, ..., um is finite and define ˜ h ˜ h f(gt) − f(gt− ) = Q (f(hgt− ) − f(gt− )) Nt − Nt− . m h G ∈ L(f)(g) = − Q ck[f(ukg) − f(g)] . Define the compensated Poisson process with intensity k=1 √ ch and jumps 1~ ch, h ≠ e: Then L is the generator of a primitive Markov semigroup −tL 1 (St = e )t≥0 on the oriented graph N h N˜ h c t . t = √c ( t − h ) h E = {(g, ukg)S k = 1, ..., m ; g ∈ G} . h h h Writing df gt f gt f gt− and dN N N − , ( ) ∶= ( ) − ( ) t ∶= t − t Furthermore, the map π ∶ B(H) → L∞(G, B(H)) we can rewrite the previous equation as the stochastic defined by differential equation: ∗ π(x)(k) = uk x uk

df gt ch f hgt− f gt− dt (9) ( ) = Q ( ( ) − ( )) extend to a ∗-representation of B(H) on L∞(G, B(H)) h∈G{e} such that L id π x π x for all x . √ h ( ⊗ B(H))○ ( ) = ○L( ) ∈ B(H) + Q ch (f(hgt− ) − f(gt− )) dNt h G e ∈ { } Proof. We begin by proving Equation (11). By definition, f hg f g dN˜ h . = Q ( ( t−) − ( t−)) t we have for all x ∈ B(H) h∈G{e} d d ∗ We are now ready to build a QMS from this Markov L(x) = − Tt V = − E (u(gt) x u(gt))]V . dt t 0 dt t 0 chain. Let g ↦ u(g) be a projective representation of G = = on some finite dimensional Hilbert space H. We want Equation (11) follows from an application of the Îto to find a stochastic differential equation for u g . formula for compensated Poisson processes. The fact that ( ( t))t≥0 −1 To this end, take y ∈ B(H) and define fy ∶ h ∈ G ↦ dH IH is an invariant density matrix is straighthforward

5 as clearly algebra that they generate, as elements of the unitary † IH group of . Consequently the result of L = 0 , U(H) H dH Section II-B applies. † A jump part, given by where L is the adjoint of L for the Hilbert-Schmidt ● m scalar product. The case where St t 0 is reversible is ( ) ≥ ∗ the content of Lemma II.2. The second part of the proof B(H) ∋ x ↦ Q κi (x − ui x ui) , i 1 is straightforward from what preceded. = where the ui are unitary operators on H. Compared Remark II.5. We should warn the reader that a selfadjoint to previously, this class is larger than the one Lindblad generator may sometimes be both transferred presented in Section II-B. Indeed, the family {ui} from a compact Lie group or a finite group. For example, spans a subgroup of the unitary group U(H), XρX+Y ρY the partial depolarizing generator L(ρ) = ρ − 2 however, in general, it will not be a finite group. is both discrete and continuous in this sense. Remark II.7. 1) Starting with a Lindblad generator, there may be an D. The general situation ambiguity on the choice of the underlying group and classical Markov semigroup leading to it. Indeed, in The two cases explored above are particular the jump scenario when the QMS is self-adjoint, it instances of convolution QMS as defined by is always possible to write the Lindblad generator as Kossakowski in [15]. Such QMS were then entirely in the diffusive case. Then either the group is large, characterized by Kümmerer and Maassen in [11], both i.e. the commutator is I , and we can treat it as in terms of their Lindblad generator and as the QMS C H an Hörmander system, or the group is small (for us having an essentially commutative dilation. We recall finite) and we can treat it as a Markov semigroup the first characterization. with jumps on the Cayley graph of the group. In

Theorem II.6 (Theorem 1.1.1 in [11]). Let (Tt)t≥0 be both cases, estimates on the decoherence time of a QMS on B(H). The two following assertions are the corresponding QMS can be found. We shall equivalent. illustrate this fact in SectionIV. 1) There exists a weak∗-continuous convolution 2) It should be clear that the construction of QMS in Section II-A is in essence different from the one of semigroup (ρt)t≥0 of probability measures on the group Aut (B(H)) of automorphisms on B(H) convolution QMS. In the former, we can start from such that any compact group with a Markov kernel. Then we shall see that from the ∗-corepresentation π and Tt(x) = S α(x) dρt(α) , x ∈ B(H) . Lemma II.1, we can transfer certain properties of Aut (B(H)) this kernel to the induced QMS. The existence of 2) The Lindblad generator of T takes the form L this ∗-corepresentation, which was absent in [11] n 1 and only discovered in [6], stands at the root of this x i h, x a2 x x a2 a x a L( ) = − [ ] + Q 2 j + j − j j transference principle. j=1 m ∗ + Q κi (x − ui x ui) , (12) i=1 E. Collective decoherence where h and the aj are self-adjoint operators in Motivated by applications in quantum information B(H), where the ui are unitary operators on H theory, we shall study a particular class of transferred and where the κi are positive real numbers. QMS. These QMS are particularly relevant in the Remark that in quantum information terms, study of fault-tolerant passive error correction as the theorem above characterizes the generators of they display non-trivial decoherence-free subsystems, quantum dynamical semigroups consisting of mixed that is, subsystems preserved from dissipative effects. unitary channels. The generators of the form given by The interesting QMS are therefore non-primitive (with Equation (12) are thus the sum of three parts: non-trivial fixed-point algebras). Let G be a group and u G a projective representation of G on The first part corresponds to a unitary evolution with ∶ → B(H) ● some finite dimensional Hilbert space . For all n 1, generator given by x i h, x where h is H ≥ B(H) ∋ ↦ [ ] this representation induces a new representation on ⊗n self-adjoint; H given by: ● A diffusive part, given by n g ↦ u(g)⊗ . n 1 x a2 x x a2 2a x a , Let S , T be defined as in Equations (2) B(H) ∋ ↦ Q 2 j + j − j j ( t)t≥0 ( t)t≥0 j=1 and (3) using the representation g ↦ u(g). We write (n) ⊗n where the aj are self-adjoint operators. Any such (Tt )t≥0 the corresponding QMS on H for the n family {aj} is a Hörmander system for the sub-Lie representation u⊗ and Ln its generator.

6 a) Diffusive case:: In the diffusive case non-trivial fixed-point algebra. As first illustrated in [16], presented in Section II-B, the generator L of (Pt)t≥0 the relevant Lp spaces in this case are the conditioned or has the following form: amalgamated Lp spaces. Furthermore, the transference techniques require to look at the amplification of the x a2 x x a2 2a x a , (13) L( ) = Q k + k − k k classical semigroup S to the algebra L G k ( t)t≥0 ∞( ) ⊗ B(H) (see Lemma II.1). This in turn makes it necessary where the ak’s are selfajdoint operators on . Then the H to consider completely bounded version of the Lp generator Ln takes the form spaces. All these notions are introduced in Section III-A. 2 2 Section III-B is dedicated to the presentation of the Ln(x) = Q ak(n) x + x ak(n) − 2 ak(n) x ak(n) , k transference techniques. We present them in a general with framework, as we believe they can also be useful in n other settings (see [6] for an other example of application ⊗j−1 n−j ak n I ak I , in quantum information theory). Finally we specialize ( ) = Q H ⊗ ⊗ H j=1 to QMS in Section III-C, where these transference where in the jth term of the above sum, ak acts on the techniques are applied to transfer estimates on the jth copy of H. classical Markov kernel to the QMS. More generally, if L is the generator of a QMS n on ⊗ of the above form, then any i ak belongs to B(H ) A. Lp norms and entropies the tangent space at identity of some unknown compact Lie group, hence the family satisfies the transference We are now going to introduce several Lp norms principle and the different results presented in this article and entropies related to von Neumann algebras. Although can be applied. As a consequence, we obtain bounds this may not be clear at first sight, it turns out that many independent of the number n of qudits. of them are just the sandwiched Rényi entropies [17], [18] in disguise, as we will clarify. In the following M b) Jump case:: In the jump case presented is a finite von Neumann algebra and τ M a in Section II-C, the generator of T has the ∶ → C L ( t)t≥0 normalized normal, faithful, tracial state (i.e. τ I 1). following form: ( H) = Let us recall the definition of the noncommutative Lp m ∗ spaces via L(x) = Q ck (x − uk x uk) , (14) k=1 (15) where the u ’s are unitary operators on . Then the k H x τ x p 1~p . S YLp(τ) ∶= [ (S S )] (16) generator Ln takes the form m Then Lp(M, τ) ≡ Lp(M) is the completion of M with ∗ ⊗n Ln(x) = Q ck (x − vk x vk) , where vk = uk . respect to this norm. Indeed, for 1 ≤ p ≤ ∞ the space is k 1 = a Banach space such that Lp(M, τ)∗ = Lpˆ(M, τ) holds If the unitary operators u generate a finite group G then 1 1 k for p + pˆ = 1 and 1 ≤ p < ∞. In this article we will focus thanks to Theorem II.4 we can find a Markov semigroup on three types of von Neumann algebras: on G and all the estimates we find on this semigroup can ● Our main example is M m, the space of m m (n) = M × be transferred to (Tt )t≥0 for all n. matrices over the field of complex numbers, and 1 Remark II.8. Unfortunately, it is not the decoherence τ(x) ≡ τm(x) ≡ m Tr(x). To keep the notations time or any other interesting quantity for L itself which at a more abstract level, we shall most of the time transfers to all the Ln, but the underlying group which refer to a finite dimensional Hilbert space H and gives the corresponding estimates. Thus, the choice of to the algebra of (bounded) linear operators B(H) the group and the classical Markov semigroup on it are and we denote by Lp(B(H)) the corresponding particularly important. non-commutative Lp space. ● If (E, F, µ) is a probability space, where F is a σ-algebra on the set E and µ a probability III.NONCOMMUTATIVE Lp SPACES AND NORM distribution, then the set of bounded complex-valued TRANSFERENCE function M = L∞(µ) is a von-Neumann algebra, In this section we introduce the main conceptual τ ∶ f ↦ Eµ(f) is a normal, faithful and tracial ideas of this article that we called group transference state and the corresponding Lp spaces are the usual techniques. These ideas and subsequent mathematical Lp(µ). results are mostly contained in [6]. All the applications ● The last key example in the transference principle we study in this article are concerned with the is the algebra of bounded Mm-valued function on properties of certain (non-commutative) functional Lp a probability space (E, F, µ), M = L∞(E, Mm), spaces. When studying primitive QMS, only the usual with trace given by (normalized Schatten) Lp spaces are required. However, in our case we are interested in non-primitive QMS with τ ∶ f ↦ τm(f(x)) dµ(x) . SE

7 1 1 For a subalgebra N ⊂ M we define the conditioned One can then show that setting x = σ− 2 ρσ− 2 and τ(x) = q Lp(N ⊂M) norm [19], [20] (see also [16], [21]) via Tr (σx) in 15, we have: p x q Y YLp N M (17) Dp ρ σ log x L τ . ( ⊂ ) ( Y ) = p 1 (Y Y p( )) ⎧ − ⎪infx ayb a L τ y L τ b L τ p q , ⎪ = Y Y 2r ( ) Y Y q ( ) Y Y 2r ( ) ≤ Moreover, the sandwiched Rényi conditional entropy ∶= ⎨sup a x b p q . ⎪ a L (τ) b L (τ) 1 Y YLq (τ) ≥ H A B ⎩⎪ Y Y 2r Y Y 2r ≤ p( S ) introduced in [17], [18] can be seen as a 1 1 1 special case of the conditional Lp norms defined in 17. Here = U − U and a, b are elements in L2r(N). For r q p They are defined for a bipartite state ρ ∈ D (HA ⊗ HB) N = M, we just find another description of Lp(M), as: q i.e. Lp(N ⊂ M) = Lp(M). Note that for a selfadjoint ∗ H A B min D ρ id σ . element x, we may assume a = a in (17). By Hölder’s p( S )ρ = − p ABSS B(HA) ⊗ B q σB ∈D(HB ) inequality, Lp(N ⊂ M) ⊂ Lp(M). In the particular case It is then easy to see that when M = Mk(N) is the algebra of k by k matrices with q k coefficients in N, the spaces L N M S Lq N ′ p( ⊂ ) ≡ p ( ( )) p log ρ p Hp A B ρ. − Y YL1 (B(HB )⊂B(HA⊗HB ) = ( S ) coincide with Pisier’s vector-valued Lp spaces [19]. We will also be concerned with norms of linear maps for p′ the Hölder conjugate. Thus, we see that between these Lp spaces. A map T ∶ Lp(M) → Lq(M) the conditional norms can also be interpreted as a is called a N-bimodule map if, for any a, b ∈ N and any generalization of conditional entropies in which we x ∈ M: condition w.r.t. to states on a subalgebra, which naturally T (a x b) = a T (x) b . includes the case of a subsystem. For two algebras N ⊂ M we will denote this conditional Rényi entropy For instance, when N Nfix is the fixed point = by Hp(MSN)ρ. Moreover, a bound on the norms defined subalgebra of a selfadjoint quantum Markov semigroup on (18) can be interpreted as a strong data processing (Tt)t≥0 acting on M, the maps Tt are N-bimodule maps inequality for conditional Rényi entropies. Indeed, given with respect to N. For N-bimodule maps and p ≤ q, the that following was proved in Lemma 3.12 of [6], generalizing T Lp N M Lq N M an earlier statement for vector valued Lp norms (see Y ∶ ∞( ⊂ ) → ∞( ⊂ )Y Lemma 1.7 of [19]): for any s 1: p q ≥ = YT ∶ L1(N ⊂M) → L1(N ⊂M)Y ≤ c , T Lp N M Lq N M (18) Y ∶ ∞( ⊂ ) → ∞( ⊂ )Y we have T Lp N M Lq N M . = Y ∶ s( ⊂ ) → s( ⊂ )Y log T ρ q log c log ρ p Y ( )YL1(N⊂M) ≤ ( ) + Y YL1 (N⊂M) We refer to [19], [20] for motivation and further holds for all states ρ. Normalizing the expressions properties. We will also use the completely bounded appropriately, we see that this implies that: version of these norms: q′ p q T L N M L N M Hq(MSN)T ρ ≥ Hp(MSN)ρ − log(c), Y ∶ s( ⊂ ) → s( ⊂ )Ycb ( ) p′ p = sup Y idm ⊗T ∶ Ls(Mm ⊗ N ⊂Mm ⊗ M) m which is stronger than a data-processing inequality for q c 1 and q p. → Ls(Mm ⊗ N ⊂Mm ⊗ M)Y . = ≥ which also does not depend on s for N-bimodule maps, as we discuss in more detail in SectionVI. B. Norm transference Noncommutative Lp norms are closely related to the sandwiched Rényi divergences introduced in [17], In [21], the authors proved the following [18]: For p 1, , these are defined for two quantum ∈ ( +∞) factorization property: given the representation α ∶ g ↦ states σ, ρ as: ∈ D (H) αg(.) = u(g)(.) u(g)∗ of a finite or compact Lie group G on the algebra of linear operators on a finite Dp (ρYσ) = B(H) 1−p 1−p p dimensional Hilbert space H, and for any t ≥ 0, define the ⎧ 2p 2p ⎪ logTrσ ρ σ co-representation π ∶ B(H) → L (G, B(H)), x ↦ (g ↦ ⎪ ker σ ker ρ or p 0, 1 ∞ ⎨ p 1 ( ) ⊆ ( ) ∈ ( ) α −1 x . Then we may transfer properties of completely ⎪ − g ( )) ⎪ , otherwise, positive maps on L G to completely positive maps on ⎩⎪+∞ ∞( ) (19) B(H). Indeed, for every positive function k on G, we define and, for p = ∞, we set ∗ 1 1 Φk ρ k g u g ρ u g dµ g . (21) ⎧ − 2 − 2 ( ) ∶= S ( ) ( ) ( ) ( ) ⎪log Yσ ρσ Y∞ ker (σ) ⊆ ker (ρ) D (ρYσ) = ⎨ ∞ ⎪ , otherwise. Here µ is the Haar measure. Therefore, the fixed-point ⎩⎪+∞ (20) algebra of the map Φk is given by the commutant of

8 u(G): translates into ′ Nfix = {σ ∈ B(H)S σu(g) = u(g)σ} = u(G) . D(Efix(ρ)Yσ) (23) ′ To see this, note for any element X ∈ u(G) we have ≤ D(Φk(ρ)SSσ) ≤ D(Efix(ρ)Yσ) + S k log k dµ , u(g)∗X u(g) = X, which shows u(G)′ ⊂ Nfix. For the and for p : other inclusion, note that for any v ∈ G and X ∈ Nfix: = ∞ ∗ ∗ D Efix ρ σ (24) u(v) Xu(v) = u(v) Φk(X)u(v) ∞( ( )Y ) D Φk ρ σ D Efix ρ σ log k . ∗ ≤ ∞( ( )SS ) ≤ ∞( ( )Y ) + Y Y∞ = S k(g) u(gv) ρ u(gv) dµ(g) Proof. Proof of (23) for the case of a finite group and ∗ = S k(g) u(g) ρ u(g) dµ(g) , p = 1 is simple and gives some intuition for the idea of the proof. The first observation is that in the case of a where in the last step we used the fact that the kernel is finite group G, we can always dilate the channel Φk as right-invariant. Thus, X ∈ u(G)′ Note that the following natural bimodule property holds Φ ρ U ρ τ U † , k( ) = Tr`2(G) G ( ⊗ k) G (25)

Φk(σ1ρσ2) = σ1Φk(ρ)σ2 . where ′ for σ1, σ2 ∈ u(G) . We then have UG = Q Ug ⊗ Sg⟩⟨gS g∈G π ○ Φk = (ϕk ⊗ id) ○ π , and τk ∈ D (`2(G)) defined as where ϕk ∶ L∞(G) → L∞(G) is defined by τk = SGS Q k(g)Sg⟩⟨gS . −1 g G ϕk(f)(g) = S k(gh ) f(h) dµ(h) . ∈ Interchanging the roles of ρ and τk, we find a channel We will denote by ∗ ∗ Ψ ∶ L1(G) → B(H) , Ψ(f) = ∑g f(g)ugρug. Therefore E ρ E ρ u g ∗ ρ u g dµ g the data processing inequality shows that fix( ) ≡ Nfix ( ) = S ( ) ( ) ( ) k 1 the conditional expectation onto the fixed-point algebra. D T ρ E ρ D ( k( )Y fix( )) ≤ G Y G The following commuting square, already mentioned in S S S S Equation (4) in the specific case of a Markov semigroup 1 = Q k(g) ln k(g) . transference, was recently found in [6]: SGS g 1 We apply this to k g kt g− , the density of this B(H) L∞(G, B(H)) ( ) = ( ) EB(H) instance of the semigroup. We combine this with the π π following decomposition which holds for the relative

Nfix entropy and a conditional expectation (see SectionV): Efix B(H) D T ρ σ D E ρ σ D T ρ E ρ , where E simply denotes the usual expectation over ( t( )Y ) = ( fix( )Y ) + ( t( )Y fix( )) B(H) (26) G, that is, for any f ∈ L∞(G, B(H)), and deduce D(Tt(ρ)Yσ) ≤ D(Efix(ρ)Yσ)+D(ktµYµ), 1 EB(H)(f) = f(g) dµ(g). where µ g is the Haar measure. For p 1 we lack SG ( ) = SGS > the Pythagorean identity in Eq. (26) and, thus, must use This in particular implies that the natural inclusion interpolation to prove the inequality. The proof for finite q q groups is a simplification of Theorem 2.5 from [21]. Let Lp(Nfix ⊂ B(H)) ⊂ Lp(B(H) ⊂ L∞(G, B(H))) η be defined such that η∗η = ρ. Here we use the fact that is completely isometric (see [20] for more details). for any ρ, The next theorem constitutes the basis of the estimates 2 that we provide in SectionV. We provide the proof in 1 » Ψ f f g g ηU , full generality in SectionA for sake of clarity and only Y ( )Yp = ] Q ( )S ⟩ ⊗ g] SGS g 2p present a simplified version for some cases. where Ψ(1) = Efix. Let pˆ be such that 1 = 1~p + 1~pˆ. + Then Theorem III.1. Let σ ∈ D(Nfix) and k ∶ G → R a σ−1~2ˆpΨ ρ σ−1~2ˆp Ψ ρ˜ measurable function such that ∫ k dµ = 1. Then for any Y k( ) Yp = Y k( )Yp ρ and σ N , and any p 1, : 1 2ˆp 1 2ˆp 1 2ˆp ∈ D(H) ∈ D( fix) ∈ ( ∞) for ρ˜ = σ− ~ ρσ− ~ , because σ− ~ is an element of the fixed point algebra and therefore commutes with Dp(Efix(ρ)Yσ) (22) U g ρ˜ D Φ ρ σ D E ρ σ D kµ µ , conjugation by g for all . While is not assured ≤ p( k( )Y ) ≤ p( fix( )Y ) + p( Y ) to be normalized, this will not pose a problem for 1 p where Dp kµ µ log k dµ. For p 1, this our calculation. For Ψ1 ρ˜ p 1, let η˜ be defined ( Y ) ∶= p−1 ∫ = Y ( )Y <

9 such that ρ˜ = η˜∗η˜. There exists an analytic function entropy of k. This is for instance the case when one ξ ∶ S = {zS0 ≤ Re(z) ≤ 1} → B(H) such that has access to the so-called modified logarithmic Sobolev constant for the underlying group. 1 ]» Q Sg⟩ ⊗ ξ(it)ug] < 1, G g S S ∞ C. Application to quantum Markov semigroups 1 ]» Q Sg⟩ ⊗ ξ(1 + it)ug] < 1 for all t ∈ R, In this subsection, we show how the machinery SGS g 2 developed in Section III-B provides estimates on the norms and entropies at the output of a quantum Markov and ξ(1~p) = η˜. Assume that YfYp < 1 and f ≥ 0, and let convolution semigroup of SectionII in terms of the 1 pz~2 VG,f (z) = » Q f (g)Sg⟩ ⊗ ξ(z)ug. kernel of their associated classical semigroup. We G g S S start by recalling the notations of SectionII: (St)t≥0, tL We then have that St = e− is a Markov semigroup on the compact group G (either Lie or finite), with right-invariant kernel 1 ipt~2 k . The QMS T , T e−tL is the transferred YVG,f (it)Y∞ = ]» Q f (g)Sg⟩ ⊗ ξ(it)ug] < 1, ( t)t≥0 ( t)t≥0 t = SGS g QMS on B(H) defined by Equation (3) through the and projective representation g ↦ u(g) of G on the finite dimensional Hilbert space H. YVG,f (1 + it)Y2 1 p(1+it)~2 The next theorem regroups all the transference = ]» Q f (g)Sg⟩ ⊗ ξ(1 + it)ug] SGS g techniques which will be frequently used in this paper 1 (see [6]). We recall that the spectral gap of the symmetric f p~2 g g ξ 1 it = ]» Q ( )S ⟩ ⊗ ( + )] Lindblad generator L, denoted by λmin(L) (resp. of L, SGS g 2 denoted by λmin(L)) is the smallest non-zero eigenvalue 1 p~2 of L (resp. of L) and is a quantity that palys a central = ]» Q f (g)Sg⟩] ]ξ(1 + it)] SGS g 2 2 role in the theory of decoherence times . 1 1 −tL −tL p~2 Theorem III.2 (Transference). Let Tt = e , St = e = ]» Q f (g)Sg⟩] ]» Q ξ(1 + it)ug] SGS g 2 SGS g 2 as above. Then < 1. i) The spectral gap for L is bigger than the spectral gap for L: λmin(L) ≥ λmin(L); By Stein’s interpolation theorem, YVG,f (1~p)Y2p ≤ 1, p ii) For any 1 ≤ p, q ≤ ∞, we have YTt ∶ Ls(Nfix ⊂ completing the proof by homogeneity. q B(H)) → Ls(Nfix ⊂B(H))Ycb ≤ YSt ∶ Lp(µG) → L µ ; It is straightforward to extend the p 1 proof q( G)Ycb = iii) For any 1 p, q , we have T E to the case of Lie groups, but p 1 requires a ≤ ≤ ∞ Y t − fix ∶ > Lp N Lq N more technical use of interpolation theory. We refer s( fix ⊂ B(H)) → s( fix ⊂ B(H))Ycb ≤ S L µ L µ ; to SectionA for the details. We note that in the context Y t − EµG ∶ p( G) → q( G)Ycb of a QMS with kernel k ≡ kt, a variation of the inequality We conclude this section by briefly explaining above is straightforward to prove given a bound on how this theorem can be applied to get estimates for the QMS. First we recall a result on the cb norm in t inf t k 1 . ( ) = { SY t − Y∞ ≤ } commutative C∗-algebra (see e.g. Theorem 3.9 of [22]). Indeed, it is easy to see that it follows from the definition Lemma III.3. Let S be an operator on the commutative of t that: ( ) Lp(µ) space. Assume that either p = 1 or q = +∞. Then † † kt g UgXU 1 UgXU ( ) g ≤ ( + ) g YS ∶ Lp(µG) → Lq(µG)Ycb = YS ∶ Lp(µG) → Lq(µG)Y . holds almost everywhere. By integrating the inequality Let us furthermore mention that for a classical above and conjugating with σ−1~2ˆp we have Markov semigroup (St)t≥0 acting on a compact group −1~2ˆp −1~2ˆp −1~2ˆp −1~2ˆp G with kernel k , a convexity argument yields σ Tt(ρ)σ ≤ (1 + )σ Efix(ρ)σ . ( t)t≥0 S L µ L µ sup k g (27) It is then easy to see that this gives for any p > 1 Y t ∶ 1( G) → ∞( G)Y = S t( )S g∈G Dp(Tt(ρ)SSσ) ≤ log(1 + ) + Dp(Efix(ρ)SSσ) and similarly and the statement for p 1 immediately follows by = YSt − EµG ∶ L1(µG) → L2(µG)Y (28) taking the appropriate limit. Thus, we see that a variation 1~2 of the transference statement is easy to obtain if we 2 = S Skt(g) − 1S dµG start from a t bound and the bound above is only ( ) 1 k = Ykt − Y2 . advantageous if we can control Y YLp(µG) or the relative

10 This gives us “for free” the following estimates on of the R.H.S. of (32). That is, they quantify how fast the norm of the transferred QMS between certain the semigroup "spreads out" all over the group. non-commutative Lp spaces. We will see in the next sections how to apply these estimates to concrete situations arising in quantum information theory. IV. EXAMPLES

Corollary III.4. Let Tt and St be as above, and Here, we illustrate the method developed in the λmin(L) ≥ λmin(L) be their respective spectral gaps. Then for all t ≥ 0 previous sections by listing examples of known QMS, that can be seen as transferred from classical semigroups. YTt − Efix ∶ L1(B(H)) → L1(B(H))Y We focus on the decoherence-time of the QMS, a ≤ YTt − Efix ∶ L1(B(H)) → L1(B(H))Ycb (29) notion that generalizes the mixing-time to non-primitive evolutions. We recall that for a general QMS (not ≤ YSt − EµG ∶ L1(µG) → L1(µG)Y necessarily selfadjoint), the decoherence time is defined and for all s, t ≥ 0 for any ε > 0 as 2 YTt s − Efix ∶ L1(B(H)) → L1(Nfix ⊂B(H))Y + tdeco(ε) (33) −λmin(L)s 1 ≤ e Ykt − Y2 (30) † † ∶= inf{t ≥ 0; YTt (ρ) − Efix(ρ)Y1 ≤ ε ∀ρ ∈ D(H)} . −λmin(L)s e kt 1 . (31) ≤ Y − Y2 In all the examples below however, the QMS is Proof. Equation (29) is a direct consequence of the selfadjoint. We also recall the definition of the mixing definition of the cb norm and of Lemma III.3. In order time of a classical primitive Markov process (St)t≥0 to prove Equations (30) and (31), we just note that we tmix(ε) = inf{t ≥ 0 SYSt(f) − EµG (f)YL1 µG have Tt Efix = Efix Tt. This implies ( ) ≤ ε ∀f ≥ 0, EµG [f] = 1} . (34) Tt+s − Efix = (Ts − Efix)(Tt − Efix) . Remark the difference in the normalization of the norms and (Ts − Efix)Tt = Tt+s − Efix. Therefore, using in both definitions. In the quantum case, density matrices Theorem III.2, are normalized with respect to the unnormalized trace 2 whereas in the classical case, we look at the evolution YTt+s − Efix ∶ L1(B(H)) → L1(Nfix ⊂B(H))Y 2 of states normalized with respect to the probability Tt Efix L1 L1 Nfix ≤ Y − ∶ (B(H)) → ( ⊂B(H))Y distribution µG. 2 2 × YTs − Efix ∶ L1(Nfix ⊂B(H)) → L1(Nfix ⊂B(H))Y Then, Equation (29) in Corollary III.4 implies that

≤ YSt − EµG ∶ L1(µG) → L1(µG)Y for a transferred QMS (Tt)t≥0 with associated classical semigroup S , we have for any ε 0: × YTs − Efix ∶ L2(B(H)) → L2(B(H))Y . ( t)t≥0 > The result follows by the deﬁnition of the spectral gap tdeco(ε) ≤ tmix(ε) . (35) as well as Equation (28). This shows that the decoherence time of a QMS is controlled by the mixing time of any classical Markov Note that the norms in the statement above are semigroup from which it can be transferred. In the just the trace and diamond distance between the channels. examples below, the classical mixing time is estimated For Hörmander systems, the following kernel estimates thanks to known results on functional inequalities that go back to the seminal work of Stein and Rothshield [7], we mainly introduce in SectionVI and SectionC, apart see also [23]. for the last example in Section IV-D which is directly computed in SectionC. Theorem III.5. Let V = {V1, ..., Vm} be a Hörmander system such that K iterated commutators span a Lie algebra of dimension d. Then LV has a spectral gap and there exists a constant CV > 0 such that, for all A. The depolarizing QMS 0 < t ≤ 1: Perhaps the simplest QMS that one can think of sup k g C t−Kd~2 (32) S t( )S ≤ V is the depolarizing semigroup on n : g∈G B(C ) I n I n Remark that for Markov kernel on graph, dep C dep −t −t C L (ρ) = ρ − ,Tt (ρ) = e ρ + (1 − e ) . estimates of the form sup k g C t−α~2 with n n g S t( )S ≤ L (36) α, c > 0 also hold in general. We shall discuss several examples of such estimates in SectionVI. From a This QMS can be seen to be transferred from the quantum information perspective, such bounds can be uniform walk on the group Zn × Zn, via the projective interpreted as saying that the maximum output min representation given by the discrete Weyl matrices entropy of these semigroups is bounded by the logarithm {Ui,j}i,j∈[n] (see e.g. [10]). Indeed, using Equation (11)

11 and denoting by L the transferred QMS given by this Now, it results from the logarithmic Sobolev d representation, we find that for all ρ ∈ D(C ), constant for the uniform walk on Zn, computed in [3], 1 n that ρ ρ U ρ U ∗ (37) L( ) = 2 Q − i,j i,j 1 − ln ε n ln(n − 1) n i,j 1 t ε n ln ln n = deco( ) ≤ n 1 + 2 n 2 n − ( − ) I C = ρ − Tr(ρ) (38) ln(n) ln ln(n) n ∼ . dep n→∞ 2 = L (ρ) , (39) b) Dephasing from the n-dimensional 1 where we took ci,j ∶= n2 for all i, j. This choice implies torus:: Take the representation of the n-dimensional that the uniform random walk on the complete graph with torus that consists of diagonal unitary matrices: 2 dep n vertices transfers to (Tt )t≥0. Using the logarithmic ⎡ 2it1π ⎤ Sobolev constant for the complete graph given in [24], ⎢e 0 ⎥ n ⎢ ⎥ we find the following upper bound on the mixing time T ∋ (t1, ..., tn) ↦ ⎢ ⋱ ⎥ . ⎢ 2it π⎥ dep ⎢ 0 e n ⎥ of (Tt )t≥0: ⎣ ⎦ The QMS associated to the heat semigroup and the above tdeco(ε) ≤ tmix(ε) deco 2 2 representation corresponds to (Tt )t≥0. This simply 2 1 − ln ε n ln(n − 1) 2 ≤ n + ln ln n follows from Equation (8) by taking the generators n2 1 2 n2 2 n − ( − ) Aj ∶= Sj⟩⟨jS of T , so that the generator of the transferred ∼ ln(n) ln ln(n) . QMS is equal to n→∞ 1 n This can be compared with the tighter bound x j j x x j j 2 j j x j j L( ) = 2 Q S ⟩⟨ S + S ⟩⟨ S − S ⟩⟨ S S ⟩⟨ S that one can get from the modified logarithmic Sobolev j=1 dep constant α1 (see Section V-D), from which we (L ) = x − Ediag[x] . can obtain [25], [26]: 1 √ t Then the estimation (47) leads to the following bound − n − 2 Yρt − n IC Y1 ≤ 2 ln n e , on the decoherence time of this QMS: so that 1 1 ln (n) √ tdeco(ε) ≤ ln n ln n + 6 − ln ε ∼ . 2 ln n 2 2 n→∞ 2 tdeco(ε) ≤ 2 ln ∼ 2 ln ln n . ε n→∞ Hence the estimate found on the decoherence time from the continuous torus turns out to be sharper B. The dephasing QMS than the one found from the discrete torus. Moreover, We recall that the dephasing quantum Markov these two bounds can be compared with the one found semigroup (also called decoherent QMS in [27]) on via decoherence-free modified logarithmic Sobolev n B(C ) with n ≥ 3, is given by inequality in [27], which implies that deph deph † √ t L (ρ) = ρ − Ediag[ρ] , − 2 YTt (ρ) − Efix(ρ)Y1 ≤ 2 ln n e , T deph ρ e−t ρ 1 e−t E ρ , t ( ) = + ( − ) diag[ ] so that √ where Ediag denotes the projection on the space of 2 ln n matrices that are diagonal in some prefixed eigenbasis. tdeco(ε) ≤ 2 ln ∼ ln ln n . ε n Here, we show how simple representations of the discrete →∞ and continuous torus both lead to the dephasing quantum Once again, we see that the transfer method does not Markov semigroup. immediately lead the best decoherence-time. a) Dephasing from the discrete torus:: Choose the uniform random walk on Zn of kernel C. Collective decoherence K(j, k) = 1~n for any j, k ∈ Zn. A simple unitary n representation of Zn is given by taking H = C and The bounds provided by the transference method j for the examples studied in the last two sections, namely Uj ∶= U , j ∈ Zn, the depolarizing and the dephasing semigroups, are where U denotes the Weyl unitary operator given by worse than the already known ones derived from the 2iπ 2i(n−1)π n U = diag (1, e n , ..., e n ) on B(C ), where the modified logarithmic Sobolev inequality. In this section, diagonal is chosen to be the one corresponding to Ediag. on the other hand, we show that our method provides One can easily verify from Equation (11) that the QMS an easy way of deriving new estimates for collective deph 2 ⊗n (Tt )t≥0 coincides with the generator of the transferred decoherence on n-register systems, that is on (C ) . QMS corresponding to the uniform kernel on Zn, since The power of the method lies in the fact that the by a direct calculation E X 1 U −j XU j. constants derived are independent of the choice of the diag[ ] = n ∑j∈Zn representation. In particular, we get estimates that are

12 independent of the number of qubits by choosing tensor where product representations. n (n) ⊗k−1 n−k σ I 2 σi I 2 . We focus on two particular examples of collective i ∶= Q C ⊗ ⊗ C k 1 decoherence, namely the weak and the strong collective = The difference with Equation (40) arises from the decoherences. We first recall the definition of the Pauli n n 2 consideration of all three Pauli operators σ( ), σ( ) matrices on C : x y (n) and σz as Lindblad operator. We consider the 0 1 0 −i 1 0 σx ∶= , σy ∶= , σz ∶= . three-dimensional simple Lie group SU(2) of associated 1 0 i 0 0 −1 generators σx, σy, and σz spanning the Lie algebra a) Weak collective decoherence:: We su(2), as well as the n-fold representation SU(2) ∋ ⊗n recall the generator of the weak collective decoherence g ↦ Ug , where U denotes the defining spin 1~2 2 on n qubits: representation of SU(2): for any ψ ∈ C and g ∈ SU(2), 1 wcd x σ(n) 2x x σ(n) 2 σ(n) x σ(n) , Ug ψ = g ψ . Ln ( ) ∶= 2(( z ) + ( z ) ) − z z (40) Just like previously, an easy use of Equation (13) shows that the semigroup transferred from the heat semigroup where on SU 2 via the above tensor product representation n ( ) (n) ⊗i−1 n−i coincides with the strong collective decoherence QMS σz ∶= Q I 2 ⊗ σz ⊗ I 2 . C C (up to a rescaling of the Lindblad operators by a i=1 √ One can easily show that the completely mixed state factor 2). An easy application of Equation (35) and n wcd the estimate of Theorem IV.3 for n 3 provides − 2 ⊗n 2 ⊗n = 2 I(C ) is invariant, since Ln (I(C ) ) = 0. n the following dimension-independent bound for the Moreover, since σ( ) is self-adjoint, wcd is selfadjoint z Ln decoherence time of the strong collective decoherence with respect to that state. QMS: The spectral gap for this QMS was computed 34 3 3 tdeco(ε) ≤ − 8 ln ε + 2 ln 1 + ln . (42) wcd 3 2 4 in [16] and found to be equal to λmin(Ln ) = 2 for any n 2. From this and the universal upper bound ≥ D. Random SWAP gate on the logarithmic Sobolev constants found in the same article, the authors conclude that the weak collective We consider two QMS that informally represent decoherence QMS satisfies a random SWAP gate Fij, applied at random times to n two registers i, j on H⊗ , where H is a d dimensional tdeco(ε) = O(n) . (41) Hilbert space. Both QMS are transferred from a classical n Σ We will see that the transference method leads to a better semigroup on the permutation group on elements n. ⊗n estimate. Let us first consider the heat diffusion on the In both cases, the unitary representation on H is the 1 2 n canonical one: one dimensional torus T , which we represent on (C )⊗ as follows: uω ∶ e1 ⊗ ⋯ ⊗ en ↦ eω(1) ⊗ ⋯ ⊗ eω(n) , ω ∈ Σ . 1 iθσ n T ∋ θ ↦ (e z )⊗ . The first QMS is induced by so-called random transpositions F One can easily verify that the QMS transferred via the (RT). Here the SWAP gate ij is applied i, j above representation is the weak collective decoherence to any pair of registers H ⊗ H placed in the ( )’s semigroup (up to a rescaling of the Lindblad operators by registers: √ a factor of 2) as a direct consequence of Equation (13). RT 1 L (ρ) = Q(ρ − FijρFij) Then, by Equation (35) and the estimate of Section IV-E, n ij we find for all t 0, and any ρ : ≥ ∈ D(H) This QMS can be easily seen as being transferred from ¼ wcd,n » t † − 2 the classical semigroup with generator YTt (ρ − Efix(ρ))Y1 ≤ 2 + π~t e , RT 1 which represents a fast convergence independent of the L f ω f ω f σijω ( )( ) = n Q( ( ) − ( )) number n of qubits of the system (remark that we set the ij dimension of a single copy to be 2, but the result would acting on `∞(Σn), where σij is the transposition (ij). depend on this dimension otherwise). For our second QMS we only allow nearest neighbor b) Strong collective decoherence:: We (NN) interaction on a cyclid 1D grid: recall the generator of the strong collective decoherence NN ρ ρ F ρF , on n qubits: L ( ) = Q ( − j(j+1) j(j+1)) 1≤j≤n 1 scd x σ(n) 2x x σn 2 σ(n) x σ(n) , where the n + 1 registered is identified with the 1. It can Ln ( ) ∶= Q 2(( i ) + ( i ) ) − i i NN i∈{x,y,z} be seen that L is being transferred from the generator

13 on the permutation group permutation group, and compare the energy form n NN 1 2 E f f σabσ f σ . L (f)(ω) = f(ω) − f(σj(j+1)ω) . E ( ) = Q Q S ( ) − ( )S Q Σn j=1 S S σ∈Σn (ab)∈E 2 According to [28], the latter can be simulated with local Let E′ ⊂ {1, ..., n} another generating set such that the gates. For the random transposition model, local means graph is complete. For every a, b ∈ E, we can find a that only two registers are involved, whereas for the geodesic path γab ∶ {1, ..., m} → E′ and observe that nearest neighbor interaction, local means neighboring m gates. The different normalizations are chosen to fit with 2 2 (f(τabσ) − f(σ)) ≤ m Q(f(τjτaj bσ) − f(τaj bσ)) . existing random transposition models in the literature, in j=1 particular [29]. ab ′ Here τj = τj comes from the generating set E . In our case the longest possible path is ≤ n. This implies that Theorem IV.1. Following the notations above, we have f LRT 2 EE( ) t1, (ε) ≤ 4(1 + ln n) − ln ε (43) ∞ 1 j 2 NN n δ j f ρ τ σ f τ σ L 2 2 ≤ Q Q Q τcd,ρ Q( ( ab aj ,b ) − ( aj ,b )) t ε 2n 1 ln n ln ε , (44) ′ ab Σn 1,∞ ( ) ≤ (( + ) − ) (c,d)∈E ab∈E j S S σ 1 where the mixing time t1, ε is defined as (see also 2 ∞( ) n δ j f τ σ f σ = Q Q Q Q τcd,ρ Q( ( cd ) − ( )) SectionVI) ′ ab Σ σ (c,d)∈E ab∈E j S S σ tL ε 1 2 1,∞( ) = n (Q Q Q )(f(τcdσ) − f(σ)) . Σ ′ −tL S S σ (c,d)∈E (a,b)∈E,(c,d)∈Γ = inf{t ≥ 0 ; Ye − µG ∶ L1(µG) → L∞(µG)Y ≤ ε} . cd Consequently we obtain the following estimate on the Since, we may take geodesic, no ( ) is counted double. E′ j, j 1 1 j n E a, b a decoherence time of the quantum SWAP evolutions: Thus for = {( + )S ≤ ≤ } and = {( )S ≠ b , we deduce that RT } L 2 tdeco ε 4 1 ln n ln ε (45) 3 ( ) ≤ ( + ) − f n ′ f . NN EE( ) ≤ EE ( ) L 2 2 tdeco (ε) ≤ 4n ((1 + ln n) − ln ε) . (46) 1 RT Thanks to the normalization factor n for L is implies that Proof. Remark that by ordering of the classical Lp norm NN −1 2 RT −1 2 and by transference, it suffices to estimate t1,∞(ε) for LSI(L ) ≤ n LSI(L ) ≤ 2n ln n . random transposition models on the permutation group for LTR and LNN . Let us start with the RT model. Note that our estimate also implies that the spectral gap λ LNN 1 Indeed, according to [29] we know that ( ) ≥ 2n2 . Thanks to [24], we deduce RT NN 1 tL ε c ln n ln ε t 1 e L 1 ln ln n! 2n2 ln n 1,∞ ( ) ≤ ( − ) 1,2( ~ ) ≤ ( + 4 ) for some unknown constant c. Such an estimate is not ln2 n 2n2 1 . available for nearest neighbor interaction. Our starting ≤ ( + 2 ) point is the LSI-inequality By symmetry this implies

RT NN LSI(L ) ≤ 2 ln(n) tL 1 e2 4n2 1 ln n 2 . 1,∞ ( ~ ) ≤ ( + ( )) n 2 for ≥ . According to [24] this implies Using the spectral gap this yields

1 NN t1,2(1~e) ≤ (1 + ln ln nn!)2 ln n . tL ε 4n2 1 ln n 2 ln ε . 4 1,∞ ( ) ≤ (( + ) − ) Hence Note that we have an automatic cb-norm estimate in this ln n case, and hence transference, allows us to estimate the t 1 e2 4 ln n 1 4 1 ln n2 . 1,∞( ~ ) ≤ ( + 2 ) ≤ ( + ) decoherence time of the tensor swaps. Thus for arbitrary ε, because of the spectral gap 1, we find LRT 2 t1, (ε) ≤ 4(1 + ln n) − ln ε . RT ∞ Remark IV.2. The correct estimate MLSI(L ) ∼ 1 2 However, the factor (ln n) is too large, because for the modified (not complete) logarithmic Sobolev Diaconis-Shahshahani proved that as n goes to ∞ inequality has only recently been found [30]. The t1,2(1~e) ∼ ln n. Since the spectral gap is of order 1, standard inductive procedure appears not directly NN then the estimates for t1,2(ε) requires an additional term applicable for L , though. However, our proof shows 1 NN 2 2 − ln ε as above. For the nearest neighbour model we first that LSI− (L ) ≤ cn (1 + ln n), and hence trivially consider a graph, in our case the Caley graph of the we find a bound MLSI. We conjecture that for every

14 graph, we always have the (two-way) private, quantum, entanglement-assisted

−1 −1 classical capacity and classical capacity of a transferred CLSI (LE) ≤ c ln ln SV S LSI (LE) . semigroup (Tt)t≥0 converging to its associated 2 3 conditional expectation E E . Roughly This would yield a bound of order cn (1 + ln n) for fix ≡ Nfix the inverse of the CLSI constant. At any rate a better speaking, the capacity is the ultimate rate of transmission estimate of order cn2 would be highly desirable. of a certain resource through a quantum channel such that in the limit of infinitely many uses of the channel, E. Compact Lie groups the success probability of the transmission of this Here, we recall some well-known estimates for resource converges to 1 after encoding and decoding the heat semigroup defined on various compact Lie operations. We refer to [32, Chapter 8] for a precise groups. definition of the various capacities considered here and a) 1-dimensional torus: The following note that we will express all capacities in e−bits, as they estimate was derived in example 1 of Section 3 of [1]: are more convenient in this context. ¼ » Computing the exact value of the capacity is most −1 −t Yh ↦ kt(gh ) − 1Y2 ≤ 2 + π~2t e . often out of reach of current techniques. Instead, we are interested in upper bounding them. More particularly, b) n-dimensional torus (n 1):: The > we will mostly be interested in showing strong converse logarithmic Sobolev constant associated to the heat bounds on these capacities. A bound on a capacity semigroup on the n dimensional torus n is known to T is called a strong converse bound if we have that if achieve the bound 1 c SHeat λ SHeat 1. In Theorem ~ ( ) ≤ ( ) = we exceed the transmission rate given by the bound, 5.3 of [1], the following upper bound on its kernel (and the probability of successful transmission of a certain in fact on the kernel of any uniformly elliptic generator) resource goes to 0 exponentially fast as the number was found: of channel uses goes to infinity. Our method relies −1 sup Yh ↦ kt(gh ) − 1Y2 (47) on relating norm estimates to bounds on entropic n g∈T quantities derived from the sandwhiched Rényi entropies. 1 1 Intuitively speaking, as the QMS T converges to ≤ exp −t + log n log n + 6 . ( t)t≥0 2 2 Efix, we expect that its capacity also converges to that c) Matrix Lie groups:: In [1], precise of the conditional expectation, and wish to quantify this estimates on the kernel of diffusion semigroups on convergence. For all of the results in this chapter, we various Riemannian manifolds were obtained starting will assume that we know the decomposition of the fixed from a curvature dimension inequality CD(ρ, ν). point algebra Nfix ⊂ B(H) ∶ Applying this to the curvature dimension inequalities m N I d . (51) satisfied by semisimple Lie groups [31], Saloff-Coste fix = ? Mnk ⊗ C k derived the following straightforward corollary (stated k=1 here as a theorem for sake of completeness): It is easy to find such decompositions for transferred semigroups in terms of the decomposition into Theorem IV.3. Let G, g be a real connected ( ) irreducible representations of the representation we use semi-simple compact Lie group of dimension n endowed to define the semigroup. We recall our notation for states with the Riemannian metric induced by its Killing form. † on Nfix: we write σ ∈ D(Nfix) whenever Efix(σ) = σ Then, the heat diffusion satisfies † (recall also that Efix = Efix for unital, not necessarily −1 sup Yh ↦ kt(gh ) − 1Y2 (48) self-adjoint QMS, as we assume for transferred QMS). g∈G Moreover, in [33], [34] closed formulas are derived for 16 1 n the capacities of conditional expectations in terms of the ≤ exp 1 + λ(∆) −t + + 2 log 1 + n log , n 2 4 decomposition of the underlying fixed point algebra. As where λ(∆) is the spectral gap of (G, g). In particular, we will see soon, roughly speaking, all these capacities for ε > 0 and tn = 2(1 + ε) log n, the above bound will be at most the limiting capacity plus an additive converges to 0. Moreover, the following bounds the error at time logarithmic Sobolev and Poincaré constants hold: t() ∶= inf {t ≥ 0 SYkt − 1Y∞ ≤ } n 1 λ(∆) ≥ ≥ , (49) for transferred semigroups. The main technical tool 8(n − 1) 8 needed for this section is Theorem III.1. We will also 4(n − 1) c(∆) ≤ ≤ 4 . (50) show how to obtain capacity bounds from the modified n logarithmic Sobolev inequality [25] in Section V-D. This V. CAPACITY BOUNDS, RESOURCETHEORIESAND will in general be better than the one we obtain with ENTROPICINEQUALITIES control over t(), but it should be noted that finding an Here we will discuss how to apply the estimates estimate on the logarithmic Sobolev constant of a QMS provided in Section III-B to obtain upper bounds on is a very hard problem in general (see [6], [35], [36]).

15 Another advantage of the t(ε) control is the fact that channel quantifies at which rate quantum information we can control all the relative entropies in the parameter can be reliably transmitted with a quantum channel, range p ∈ [1, ∞], which will be of crucial importance and is denoted by Q(T ). Note that we always have for our later applications. This is due to the fact that Q(T ) ≤ P(T ) and thus, any upper bound on the many strong converse bounds are only known in terms private capacity extends to a bound on the quantum of Rényi entropies for p > 1. capacity. Moreover, we may also consider variations of these capacities in which we also allow for unlimited Before moving to the results, let us briefly explain classical communication between the sender and the our proof techniques: using the upper bound in (22), we receiver of the output of the quantum channel. These have are usually called the two-way private and quantum capacities and we will denote them by T and D T ρ σ P↔( ) p( t()( Y ) T , respectively. Clearly, we have T T . p Q↔( ) P( ) ≤ P↔( ) D E ρ σ log k . ≤ p( fix( )Y ) + (Y t()YLp(µG)) We refer to e.g. [38] for a precise definition of these p − 1 quantities. One way to bound the classical Rényi divergence on the Using the techniques above, we can derive strong right-hand side of the above inequality is by invoking converses on the two-way quantum and private capacities k the following chain of inequalities: Y t()YLp(µG) ≤ based on the results of [38] and mixing time estimates. p−1 p More specifically, in [38] the authors show that for a Ykt()Y∞ , by Riesz-Thorin theorem and the fact that k ε 1 quantum channel T the quantity Y t( )YL1(µG) = , so that ∶ B (H) → B (H)

Emax T (53) Ykt()Y = sup Skt()(g)S ≤ 1 + sup Skt()(g) − 1S ≤ 1 + . ( ) g∈G g∈G = sup inf D∞ (T ⊗ id (ρ) SSσ) (52) σ ,σ SEP ρ∈D(H⊗H) ∈D(H⊗H) ∈ Although these estimates based on t() do not require is a strong converse upper bound on the two-way private the full power of transference, in the sense that it is also and quantum capacities of T . Here SEP stands for the possible to derive the outer inequalities without resorting convex subset of D(H ⊗ H) of separable states, that is, to interpolation as remarked after Theorem III.1, we still believe that these inequalities are of interest. This SEP A B A B is due to the fact that it is usually nontrivial to = {Q piρi ⊗ ρi ; pi ≥ 0 , Q pi = 1 , ρi , ρi ∈ D(H)} . derive them without noting that the underlying quantum i i channels arise as transferred semigroups. Moreover, one We then have: can directly use an entropy decay estimate instead of the bound (52) from the modified logarithmic Sobolev Theorem V.1 (Bounding the two-way quantum and Let T be a transferred QMS. inequality for the underlying classical Markov kernel private capacity). ( t)t≥0 Then: (kt)t≥0. All of our bounds will be based on comparing max log(nk) ≤ the capacity of the semigroup at time t to that of its k t Tt , Tt max log nk limit as → ∞ and using transference techniques to P↔( ( )) Q↔( ( )) ≤ k ( ) + estimate how close the channel is to its limit. Note, Moreover, the upper bound is a strong converse bound. however, that our estimates will be tighter than working directly with continuity bounds for capacities. To the best Proof. We have the following chain of inequalities: of our knowledge, known continuity bounds for various capacities such as the ones of [37], allow to conclude Emax(Tt) that the capacity of two channels with output dimension = sup inf D∞ (Tt ⊗ id (ρ)Yσ) σ ,σ SEP d differ by a factor of order log(d) if they are -close ρ∈D(H⊗H) ∈D(H⊗H) ∈ in diamond norm. However, in the case of transference ≤ sup inf D∞ (Tt ⊗ id (ρ)Yσ) σ N ρ∈D(H⊗H) ∈D( fix)⊗D(H) we will be able to obtain that the capacities are close σ∈ SEP under similar assumptions. ≤ sup inf D∞ (Efix ⊗ id (ρ)Yσ) + σ N ρ∈D(H⊗H) ∈D( fix)⊗D(H) σ SEP A. (Two way) Private and Quantum Capacity and ∈ max log nk , Entanglement Breaking Times = k ( ) + The private capacity quantifies the rate at which where in the first inequality we used the fact that classical information that is secret to the environment restricting the infimum over separable states such that can be reliably transmitted by a given quantum channel one half lies in the fixed point algebra can only in the asymptotic limit of many channel uses. For a increase the quantity. In the second inequality we quantum channel T , the private capacity is denoted by applied Theorem III.1 to the transferred semigroup

P(T ). Similarly, the quantum capacity of a quantum corresponding to the representation Ug ⊗ IH. Finally, it

16 remains to show the last equality. The first term on the breaking if and only if its Choi matrix is separable [43]. r.h.s, maxk log(nk), corresponds to the capacity of the It follows from the transference principle that conditional expectation, as computed in [34]. It remains T id E id (54) to show that, for conditional expectations, the infimum Y t() ⊗ − fix ⊗ Y1→1 ≤ in Equation (53) is attained at points with one half of Note that, as we have an erdogic semigroup, Efix(X) = 1 the state in the fixed point algebra of this conditional Tr X I d E id d ( ) C and, thus, fix⊗ will yield the maximally expectation. To see that this is indeed the case, note that mixed state when applied to the maximally entangled for any ρ ∈ D (H ⊗ H) we have state. Choosing the maximally entangled state as our input to the channel, it follows from (54) that the Choi inf D E id ρ σ ∞ ( fix ⊗ ( )Y ) −1 σ∈D(H⊗H),σ∈ SEP matrix of Tt is separable for t(d ) and, therefore, the map becomes entanglement breaking for this time. ≥ inf D∞ (Efix ⊗ id (ρ)YEfix ⊗ id (σ)) σ∈D(H⊗H),σ∈ SEP It is then straightforward to adapt the various = inf D∞ (Efix ⊗ id (ρ)Yσ) σ∈D(Nfix)⊗D(H),σ∈ SEP convergence results we have to obtain estimates on by the data processing inequality and the fact that the time the transferred QMS becomes entanglement conditional expectations are projections. The lower breaking. Note that the situation is a bit more subtle if bound on the capacities follows from the lower bound in the semigroup is not assumed to be primitive. Consider the example of T deph as defined in Equation (36). Theorem III.1 and the expression for the quantum and ( t )t≥0 private capacities of conditional expectations. We can see that in this case the semigroup is not entanglement breaking for any finite t ≥ 0 but is entanglement breaking in the limit t . This shows In a similar fashion, one can use that the relative → ∞ that the entanglement breaking time might be infinite if entropy of entanglement [39] of a channel T we drop the assumption of primitivity. As a matter of ER T sup inf D T id ρ σ fact, in [41], the authors show that this is always the ( ) = σ SEP ( ⊗ ( )Y ) ρ∈D(H⊗H) ∈ case for non-primitive semigroups. is a strong converse bound on the private and quantum capacities of T [40] in order to derive the following B. Classical capacity Theorem V.2 (Bounding the one-way quantum and The classical capacity of a quantum channel is the highest rate at which classical information can be private capacity). Let (Tt)t≥0 be a transferred QMS. Then: transmitted through a quantum channel with vanishing error probability. We will denote the classical capacity max log nk Tt , Tt max log nk k ( ) ≤ P( ( )) Q( ( )) ≤ k ( ) + of a quantum channel by C(T ). One can show that [18]: 1 Moreover, the upper bound is a strong converse bound. ⊗n T lim inf sup Dp T ρ σ C( ) ≤ n ⊗n ( ) SS →∞ σ∈DH ρ ⊗n n Although the last theorems provide bounds for A ∈DHA all times, in case of primitive QMS we expect that the (55) semigroup becomes entanglement breaking at some point for any p > 1. Moreover, this is a strong converse and, thus, all the aforementioned capacities become 0. bound. Taking p = 1 in the bound above also gives an We recall that an entanglement breaking channel is one upper bound on the classical capacity [44], albeit not a whose action on one part of a bipartite entangled state strong converse one. Bounding the classical capacity is a always yields a separable state. Using our methods we notoriously difficult problem because of the nonadditivity can also estimate these times (see also [41]). To this end, of the output entropy. However, for transferred groups we we define have:

Deﬁnition V.3 (Entanglement breaking time). Let Theorem V.5. Let (Tt)t≥0 be a transferred QMS. Then: (Tt)t≥0 be a primitive QMS. We deﬁne its entanglement m m breaking time, t , to be given by EB log Q ni ≤ C(Tt()) ≤ log Q ni + . i=1 i=1 t T inf t 0 T is entanglement breaking . EB(( t)t≥0) = { ≥ S t } Moreover, the upper bound is a strong converse bound.

Theorem V.4. Let (Tt)t≥0 be a primitive transferred ⊗n d Proof. First, note that the semigroup Tt corresponds to semigroup on B C . Then the channel we obtain by transfering the Markov kernel −1 n tEB((Tt)t≥0) ≤ t(d ) . kn,t = A kt Proof. In [42], the authors show that all states in the ball i=1 1 n with radius d in the Hilbert Schmidt norm around the on G . Moreover, the conditional expectation related ⊗n bipartite maximally mixed state are separable. Moreover, to the semigroup is clearly Efix. We may obtain an it is well-known that a quantum channel is entanglement upper bound to Equation (55) by restricting the inﬁmum

17 ⊗n ⊗n to states that are on Nfix. Let σ ∈ D(Nfix). By was obtained in [34]. There, they show that for a ⊗n conditional expectation E we have Theorem III.1, for any ρ ∈ D HA , fix m ⊗n ⊗n 2 Dp T ρ σ Dp E ρ σ n . t() ( ) SS ≤ ( fix ( ) SS ) + CEA(Efix) = log Q ni . i=1 where the last inequality follows from III.1, the fact that Using similar ideas as before we can estimate the k is a product and the elementary inequality log 1 n,t() ( + entanglement assisted classical capacity using relative x x. ) ≤ entropy transference. ⊗n sup Dp Tt (ρ)Yσ ⊗n ( ) ρ∈DHA Theorem V.6 (Bounding the entanglement assisted ⊗n classical capacity). Let T be a transferred QMS. ≤ sup Dp(Efix (ρ)Yσ) + n . ( t)t≥0 ⊗n ρ∈DHA Then: m m Moreover, we have 2 2 log Q ni ≤ CEA(Tt()) ≤ log Q ni + . inf sup D E⊗n ρ σ i=1 i=1 ⊗ p( fix ( )Y ) n ⊗n σ∈D(Nfix) ρ ∈DHA Furthermore, if (St)t≥0 satisfies an α-MLSI and G is ⊗n finite, then: = inf sup Dp(Efix (ρ)Yσ) , σ ⊗n ⊗n ∈DHA ρ∈DHA m 2 −2αt which follows from an application of the data processing CEA(Tt) ≤ log Q ni + e log(SGS) i=1 inequality and the fact that Efix is a projection, as in the proof of Theorem V.1. The upper bound then follows Moreover, the upper bounds are in the strong converse ⊗n sense. from taking the infimum over all σ ∈ D(Nfix), dividing the expression by n, taking the limit n → ∞ and the expression for the classical capacity of a conditional Proof. The proof is similar to the ones of Theorems V.1 expectation given in [34]. The lower bound follows by and V.5 and Proposition V.13, and hence is omitted. an argument similar to that given before for the other capacities.

C. Entanglement-assisted classical capacity D. Capacities from a modified logarithmic Sobolev The entanglement-assisted classical capacity of a inequality quantum channel is the highest rate at which classical information can be transmitted through a quantum channel with vanishing error probability given that the Similarly to the derivation of decoherence times, sender and receiver share and potentially consume an one expects to get tighter bounds on the various unlimited amount of entanglement. We refer to [32, capacities by directly applying a quantum functional Section 8.1.3] for a precise definition. We will denote the inequality, when the latter is known. As was the case entanglement-assisted capacity of a quantum channel T with decoherence times, the decay of the capacities we obtain does not depend on particular properties of by CEA(T ) and note that it is an upper bound on the classical capacity of a quantum channel. In [45], the the representation at hand and will in general not be authors show that the following quantity is an upper tight. For capacities, the right functional inequality to bound on the entanglement assisted classical capacity consider is the modified logarithmic Sobolev inequality (EAC) of a quantum channel in the strong converse (MLSI). To the best of our knowledge, this connection sense1: between a MLSI and capacity bounds cannot be found in the literature beyond primitive semigroups [46], so χEA(T ) = inf sup (56) we establish it here for more general semigroups. Here, σ A∈D(HA) Sψ⟩⟨ψS∈D(HA⊗HB) we still assume that (Tt)t 0 is a quantum Markov D T id ψ ψ σ ρ , ≥ ( ⊗ (S ⟩⟨ S)Y A ⊗ B) semigroup on B(H) that is symmetric with respect to the Hilbert Schmidt inner product. Then, instead of using where ρB is the reduced density matrix of T ⊗id(Sψ⟩⟨ψS) the entropy comparison theorem (Theorem III.1), one can on system B and where Sψ⟩⟨ψS stands for the rank-one simply decompose the relative entropy between T ρ , orthogonal projection on the norm-one vector ψ ∈ H. t( ) A closed formula for the entanglement assisted capacity ρ ∈ D(H), and any state σ ∈ D(Nfix) as follows:

Lemma V.7. For any ρ ∈ D(H), and any σ ∈ D(Nfix), 1note that we are just rewriting the mutual information in terms of the relative entropy in (56). D(Tt(ρ)Yσ) = D(Efix(ρ)Yσ) + D(Tt(ρ)YEfix(ρ)) .

18 Proof. We remark that the proof also generalizes to

compact Lie groups, provided a bound on D(µt0 YµG) D(Tt(ρ)Yσ) for some t0 > 0 to obtain an analogue of (58) and then = Tr(Tt(ρ)(ln(Tt(ρ)) − ln σ)) apply the relative entropy decay estimate. We are not = Tr(Tt(ρ)(ln(Tt(ρ)) − ln(Efix(ρ)))) aware of techniques for bounding this entropy in the Lie group case except once again invoking inequalities + Tr(Tt(ρ)(ln(Efix(ρ)) − ln σ)) like (32) and then bounding the relative entropy by the D T ρ E ρ Tr E ρ ln E ρ ln σ = ( t( )Y fix( )) + ( fix( )( ( fix( )) − )) max-relative entropy. Thus, it would be interesting to D Tt ρ Efix ρ D Efix ρ σ , = ( ( )Y ( )) + ( ( )Y ) obtain more fine-tuned bounds on D(µt0 YµG). where in the third line we used that Efix is a conditional expectation with respect to the completely mixed state, With this tool at hand, the following result can be † proved in a very similar fashion as Theorems V.1, V.2, so that Efix = E and for any σ ∈ D(Nfix) ∩ D(H) , fix + V.5 and V.6: ln(σ) = Efix(ln(σ)). −tL Theorem V.9. Let (Tt = e )t≥0 be a quantum Markov d We recall that, given a faithful quantum Markov semigroup on B(C ) that is symmetric with respect to −tL semigroup (Tt = e )t≥0, its decoherence-free modified the Hilbert Schmidt inner product. Then, for any t ≥ 0: logarithmic Sobolev constant α1 has been defined (L) −α1(L⊗id)t Tt , Tt max log nk 2 e log d in [27] as follows: Q( ) P( ) ≤ k + ( )

Tr(L(ρ)(ln ρ − ln Efix(ρ))) 2 −α1(L⊗id)t α1(L) ∶= inf CEA(Tt) ≤ log Q nk + 2 e log(d) . ρ∈D+(H) D(ρYEfix(ρ)) k (we recall our convention that L is a positive Proof. The proof proceeds completely analogously to the semi-deﬁnite operator). It is the largest constant α > 0 one of Theorem V.1. The only difference lies in the use of such that the following decay in relative entropy occurs2: Lemma V.7 instead of Theorem III.1 to bound the relative

−α t entropy. In this case, we will obtain a remaining term of D(Tt(ρ)YEfix(ρ)) ≤ e D(ρYEfix(ρ)) . the form D(ρYEfix ⊗ id(ρ)). But another application of The classical logarithmic Sobolev inequality is then Lemma V.7 shows that this term is bounded by 2 log(d). defined in an analogous way. An extension of Theorem III.1 then gives the Note that we cannot apply an estimate on α Theorem V.8. Let (Tt)t 0 be a transferred QMS from a 1 (L) ≥ directly to obtain capacity bounds for the classical finite group G on B(H), σ ∈ D(Nfix) and suppose that capacity due to the need of regularization. In order to do St has an MLS constant α1(L) > 0. Then, for any state so, we need to show that the following complete version ρ ∈ D(H) and t ≥ 0: of the modified logarithmic Sobolev inequality holds D(Efix(ρ)Yσ) ≤ D(Tt(ρ)Yσ) (57) (see [6], [27]): define the complete modified logarithmic −α1(L)t Sobolev (cMLS) constant ≤ D(Efix(ρ)Yσ) + e log (SGS)

αc inf inf Proof. It follows from Theorem III.1 that (L) ∶= k k∈N ρ∈D+(H⊗C ) Tr id k ρ ln ρ ln Efix id k ρ D(Tt(ρ)Yσ) ≤ D(Efix(ρ)Yσ) + kt log(kt)dµG. ((L ⊗ B(C ))( )( − (( ⊗ B(C ))( )))) SG . D(ρY(Efix ⊗ id k )(ρ)) Now note that B(C ) The cMLS constant αc is known to tensorize [6], [27]: t t kt log(kt)dµG = D(µtYµG), for any two QMS Tt e−L t 0 and Qt e−K t 0 SG ( = ) ≥ ( = ) ≥ where g is an arbitrary element of the group and where αc(L ⊗ id + id ⊗K) ≥ min{αc(L), αc(K)} . dµ k δ dµ . Remark also that µ S δ . Thus, t = t ∗ g G t = t( g) By a simple look at the proof of the tensorization of as we assumed that S satisfies a MLSI, we conclude t α for the generalized depolarizing semigroup [35], that 1(L) [36] one can derive the positivity of its cMLS constant. −α1(L)t D(µtYµG) ≤ e log(SGS), (58) This can be readily extended to the case of a simple semigroup of generator of the form L = id −Efix: where in the last step we used the fact that D(µtYµG) = d log(SGS) − S(µt) ≤ log(SGS) for any µt. This yields the Lemma V.10. For any subalgebra N of B(C ) with claim. conditional expectation EN associated to the completely N mixed state, the simple semigroup (Tt )t≥0 of generator N 2the theory of functional inequalities for the exponential decay of L = id −EN of fix point algebra N satisfies Rényi entropies is not well-established beyond the primitive case [47], N [48] and for p = +∞. We leave this to future work. αc(L ) ≥ 1 .

19 Proof. For sake of clarity, denote by idk the identity map as compared to the quantum MLSI method, due to the k k on B(C ). Then, for any ρ ∈ D+(H ⊗ C ), relative lack of maturity of the latter field. −tL Tr((id −EN ) ⊗ idk(ρ)(ln ρ − ln(EN ⊗ idk)(ρ))) Proposition V.13. Let (Tt = e )t≥0 be a transferred QMS from a finite group G such that S with MLS = D(ρY(EN ⊗ idk)(ρ)) + D((EN ⊗ idk)(ρ)Yρ) ( t)t≥0 constant α1(L) > 0. Then: ≥ D(ρY(EN ⊗ idk)(ρ)) . m m −α1(L)t log Q ni ≤ C(Tt) ≤ log Q ni + e log(SGS) . i=1 i=1 The link to the classical capacity of the QMS is Proof. The proof is essentially the same as the one made in the following theorem. of Theorem V.5, but instead of bounding the relative −tL Theorem V.11. Let (Tt = e )t≥0 be a symmetric entropy of kt using t(), we apply Theorem V.8 instead. d quantum Markov semigroup on B(C ) with positive Moreover, note that the MLSI of classical semigroups cMLS constant αc(L). Then, for any t ≥ 0: tensorizes [50]. m −αc(L)t Remark V.14. The method of Proposition V.13 can be C(Tt) ≤ e log d + log Q ni . i=1 extended to other capacities in a similar way as what we did before from the existence of a MLSI constant directly Proof. We use Equation (55) for p = 1, which is equal to the classical capacity of a quantum channel (see [18]): for the quantum Markov semigroup (Tt)t≥0. Since the method is identical to those for the case of the classical C(Tt) capacity, we do not pursue this path here. 1 ⊗n ≤ lim inf sup D(Tt (ρ)Yσ) n ⊗n ⊗ →∞ σ∈D(H ) ρ n n A ∈D(HA ) E. Resource theories and entropic inequalities 1 ⊗n ≤ lim inf sup D(Tt (ρ)Yσ) In a related setting, it is also possible to apply n (n) ⊗ →∞ σ∈D(Nfix(L )) ρ n n ∈D(HA ) our techniques to obtain estimates for different relative 1 −αc(L) t ⊗n entropies of a resource [51]. In the framework of resource ≤ e lim sup D(ρYEfix(ρ)) n ⊗ →∞ n ρ n theories, one is usually given a sequence of sets of free ∈D(HA ) n 1 states Fn ⊂ D (H⊗ ), which are supposed to model ⊗n + lim inf sup D(Efix(ρ)Yσ) , those states that do not provide any resources. We will n (n) ⊗ →∞ n σ∈D(Nfix(L )) ρ n ∈D(HA ) usually denote F1 by F. One is also given a set of free where we used Lemma V.7 as well as the definition of operations, which are those quantum channels that cannot the cMLS constant in the last line. The rest of the proof convert free states into resource states. The relative follows the same lines as for the one of Theorem V.5 entropy of a resource DF is then defined as and the one of Theorem V.9. DF (ρ) = inf D (ρYσ) . σ∈F Remark V.12. Note that, although we expect that this One can then show that the regularized version of D ρ , i.e. lim n−1D ρ⊗n , quantifies the optimal method will yield tighter bounds for a given semigroup, it F ( ) Fn ( ) does not yield strong converse bounds for the capacities, conversion rate of one resource to another whenever the neither does it provide bounds on the private two-way set of free states and operations satisfies some natural capacities. In order to get a bound on the two-ways properties. We refer to [Theorem 1] [51] for more details. private capacities, or a strong converse bound on the The connection to our techniques arises from the classical capacity of symmetric channels, we would fact that for many prominent resource theories, like the need to extend the theory of complete logarithmic resource theory of coherence [52] or the resource theory Sobolev inequalities to sandwiched Rényi divergences. of asymmetry [53], the set of free states has the additional This falls outside the scope of this paper and will be property of being naturally related to an algebra N. For done elsewhere. One exception is the two-way quantum instance, in the case of coherence, the free states are capacity, as the results of [49] show that the relative described as those states that are diagonal in a given entropy of entanglement gives a strong converse for this basis and, thus, are naturally contained in the algebra capacity. of diagonal operators in a certain basis. More precisely, whenever E we can use our techniques As mentioned at the beginning of the section, fix(D (H)) ⊂ F to bound the relative entropy of a resource and its another way to upper bound capacities is to combine the regularized version for outputs of a given semigroup: upper bounds found in Theorem III.1 with a modified logarithmic Sobolev constant for the classical semigroup Proposition V.15. Let F be a set of free states for of kernel (kt)t≥0. This method first has the advantage a resource theory and (Tt)t≥0 be a transferred QMS of generally providing better bounds than the estimates from a finite or Lie group G on B(H) such that ⊗n ⊗ based on t(ε). Moreover, it is easier to use in practice Efix(D (H )) ⊂ Fn for all n, where Fn denotes the

20 ⊗n set of free states corresponding to H . Then for any group G on B(H), with corresponding kernel (kt)t≥0. state ρ ∈ D (H⊗) we have Then for all t ≥ 0, the following holds 1 n A ⊗ D Tt Efix kt ln kt dµG . (60) lim DFn (Tt(ρ)) ≤ kt log(kt)dµG. ( Y ) ≤ S n→∞ n SG Proof. Note that it follows from Theorem III.1 that for Moreover, for any p ≥ 1 and any reference system R ⊗n ⊗n the following entropic chain rule holds: for all ρ, σ ∈ any n and σ ∈ Efix(D (H )) ⊂ Fn: D(H ⊗ HR), ⊗n D((Tt(ρ)) Yσ) Dp((idR ⊗Tt)(ρ)Y(idR ⊗Efix)(σ)) ⊗n ⊗n ⊗n ⊗n ≤ D(Efix(ρ) Yσ) + kt log(kt )dµG. SG ≤ Dp(ρYσ) + Dp(νtYµG) (61) ⊗n ⊗n Thus, by picking σ = Efix(ρ ) ∈ Fn and by the where dνt = ktdµG. additivity of the relative entropy we conclude that Proof. For each n ∈ N, we use (23) for the transferred n ⊗n semigroup T ⊗ id and its corresponding inf D(Tt(ρ) Yσ) ≤ n kt log(kt)dµG, ( t ⊗ Rn )t≥0 σ∈Fn SG ⊗n conditional expectation Efix ⊗ idRn , where Rn is an n from which the claim follows. arbitrary reference system, so that for any ρ ∈ D(H⊗ ⊗ idRn ): ⊗n ⊗n Thus, we see that, as in the case of the capacities, D(Tt ⊗ idRn )(ρ)Y(Efix ⊗ idRn )(ρ)) it is possible to upper bound the regularized maximal ⊗n ⊗n ≤ kt ln kt dµG = n kt ln kt dµG . relative entropy of any output of a transferred semigroup S S by computing or estimating the purely classical relative Inequality (60) follows from Equation (59). Inequality entropy between the kernel and the Haar measure on the (61) is a simple consequence of (22) and the data group. processing inequality for Efix. The statement for other The upper bounds obtained in Theorem III.1 can values of p can be proved in a similar way. also be similarly used to upper bound the amortized Remark V.17. In [55], the following chain rule for channel relative entropy [54] defined for any two relative entropies was derived: given two quantum channels E, F ∶ D(HA) → D(HB) by channels E, F, and for any ρ, σ ∈ D(HR ⊗ H) A D (EYF) A D((idR ⊗E)(ρ)Y(idR ⊗F)(σ)) ≤ D(ρYσ) + D (EYF) , = sup D((idR ⊗E)(φRA)Y(idR ⊗F)(ψRA)) (62) φRA,ψRA D φ ψ , leaving as an open problem whether such an inequality − ( RAY RA) A still holds for p-divergences, where Dp (EYF) ≡ where R denotes a reference system of arbitrary large 1 ⊗n ⊗n limn→∞ n Dp(E YF ). In view of the above dimension. It was shown in [55] that Theorem, we found that a similar chain rule holds for A 1 n n p-divergences in the restricted case when E ≡ Tt and D (EYF) = lim D(E⊗ YF ⊗ ) , (59) n→∞ n F ≡ Efix, up to a weakening of the bound, since where the channel relative entropy is defined as A Dp (TtYEfix) ≤ Dp(νtYµG). D(EYF) ∶= sup D((idR ⊗E)(φRA)Y(idR ⊗F)(φRA)) . Similar bounds can be derived for the TRO channels φ RA defined in [34]. Upper bounding the amortized channel relative entropy in terms of a single-letter entropic expression is not an easy F. Examples task in general and it has many applications in the context of channel discrimination [54]. The particular case of It is straightforward to translate the estimates transferred semigroups is tractable, as they fall under the in the last sections to obtain estimates for various category of environment-parametrized channels of [54, capacities of transferred semigroups. We will now Proposition 33]. There, the authors show a single-letter illustrate these bounds for three noise models of practical expression bound the amortized channel relative entropy. relevance: collective decoherence, depolarizing noise and In the next corollary, we reprove their upper bound in dephasing noise. As far as we know, these are the first estimates available for capacities of collectively the case when E ∶= Tt is the semigroup Tt at time t, and decohering quantum channels. On the other hand, the F ∶= Efix is the conditional expectation onto the fixed capacities of depolarizing and dephasing channels are point algebra of (Tt)t≥0, while also obtaining a stronger bound for all Rényi entropies: widely studied and we use these examples to benchmark the quality of our bounds. We observe that our bounds Corollary V.16 (Chain rule for p-Rényi divergences). show the right exponential decay of the capacities for

Let (Tt)t≥0 be a transferred QMS from a ﬁnite or Lie large time, but are not able to capture it for small

21 times. The depolarizing and dephasing semigroups are of the simple form discussed before, that is, a difference of a conditional expectation and the identity. Thus, as expected, we may obtain better bounds by a direct application of a modiﬁed logarithmic Sobolev inequality. We will also discuss quantum channels stemming from representations of ﬁnite groups that are not of simple form, such as random transposition channels. As mentioned before, it is straightforward to turn any mixing time for a reversible chain on a group into a capacity bound and, thus, this list of examples is by no means exhaustive.

a) Collective decoherence:: It follows from the results in sectionIV that both weak collective decoherent and strong collective decoherent semigroups c) Dephasing noise:: Again, as discussed will reach their limiting capacity up to an additive error deph in sectionIV, the dephasing semigroup T t 0 −1 ( t ) ≥ > 0 in time O log , showing that encoding corresponds to the transferred channel we obtain by into the decoherence-free subspaces of these channels transfering the uniform random walk on Zn. It then is essentially optimal. follows from the estimates in section C-A that the following upper bound holds for the two way quantum capacity: b) Depolarizing noise:: As discussed in sectionIV, the depolarizing channel deph Q↔ Tt I n dep −t C −t n − 1 n − 1 Tt (ρ) = (1 − e ) + e ρ. ≤ log(n)exp −t − log(n − 1) log log(n) . n n 2n − 4 corresponds to transferred channel we obtain by This capacity was computed in [39], where they show transfering the uniform random walk on Zn × Zn that via the projective representation given by the discrete deph Weyl matrices. It then follows from the estimates in Q↔ Tt section C-A that the following upper bound holds for the 1 e−t 1 e−t log n − e−t log − e−t two way quantum capacity of the depolarizing channel = ( ) + n + n + (similar estimates would also follow from transfering the −t t 1 − e heat semigroup on the unitary group with the natural + 1 − e− log . representation): n

dep Q↔ Tt 2 2 n −1 n −1 2 2 exp − n2 t + 2n2 4 log(n − 1) log log(n ) + 1 − . ≤ 2 Moreover, we may estimate when the depolarizing channel becomes entanglement breaking and, thus, when the quantum capacity becomes 0. The estimate we obtain is dep tEB Tt 2 2 n n 2 ≤ (1 + log(2n)) + log log(n ) . n2 − 1 2(n2 − 2) To the best of our knowledge, the best available bound on the two way quantum capacity of these channels is the one in [39], where they show that We see that the bounds we obtain using our methods are dep Q↔ Tt ≤ log(n) − H2(pn,t) − pn,t log(n − 1) close to the state of the art, specialized bounds, at large times. Moreover,the slope of the decay is close to the 2 n −1 −t with pn,t = n2 (1 − e ) and H2 the binary entropy for correct one, except at small times. t log 1 n and 0 else. Again using a MLSI inequality argument, we ≤ − n+1

22 obtain that the quantum capacity of the dephasing We also recall the deﬁnition of the mixing time3 of a channel is bounded by classical primitive Markov process (St)t≥0 deph −t Q Tt ≤ 2e log(n). tmix(ε) (64)

= inf{t ≥ 0 SYSt(f) − EµG (f)YL1 µG ≤ ε d) Random SWAP gate:: Consider the ( ) ∀f ≥ 0, EµG [f] = 1} . natural representation of the permutation semigroup Sn d ⊗n acting on C with and the semigroup with on Sn In this section, we study some aspects of RT 1 ij with generator L (f)(ω) = n ∑ij(f(ω) − f(σ ω)). the theory of decoherence time that goes beyond d ⊗n The transferred semigroup acting on C is then transference. In this matter, it appears that the most given by the random transposition semigroup relevant quantity is a regularized form of the decoherence time that we call the complete decoherence time RT 1 L (ρ) = Q(ρ − FijρFij) , (cb-decoherence time) and defined as n ij tcb ε (65) F e f f e deco( ) where ij( ⊗ ) = ⊗ acts between the different † † registers i and j of the tensor product. It then follows ∶= inf{t ≥ 0; YTt − Efix ∶ L1(Tr) → L1(Tr)Ycb ≤ ε} . from the results listed in Section IV-D and our results Here L1 Tr denotes the L1 space defines with respect to 1−ln ε ( ) on capacities that for times t ≥ log n + 2 various the unnormalized trace. Remark that the only difference capacities of this quantum channel are at most ε away with Equation (63) is the choice of the completely from their limiting capacity. Similarly, we can estimate bounded norm instead of the usual one. We thus have decoherence for the nearest neighbour interaction the trivial bound: n NN cb tdeco ε t ε , L (ρ) = Q(ρ − Fj(j+1)ρFj(j+1)) . ( ) ≤ deco( ) j=1 which shows that the cb-decoherence time controls It again follows from the results in Section IV-D that for NN the usual decoherence time. Remark that for classical 5 −tL t ≥ 32π2 (2 + ln n − ε) the quantum channel e is semigroups, both definitions coincide thanks to ε away from its various limiting capacities. Lemma III.3 and therefore one can overlook this notion It is also possible to bound the capacities through of decoherence time for transferred QMS. the MLSI constant given in Equation (89) together with Proposition V.13. This way we get that, for instance, the In the application to quantum capacities , t classical capacity considered here is at most e− log n! ∼ it will yet be an other possible definition of the e−t n log n away from the limiting capacity. However, in decoherence time that will be useful. We thus define RT this case we know [56] that αc(L ) ≥ 1 and hence the ultracontractive (complete) decoherence time (UC-cb t n t Theorem V.11 shows that for ε = e− log d = e− n log d, decoherence time for short) as the channel is ε away from its limiting classical capacity. tcb ε inf t 0; (66) This is better than both bounds. 1,∞( ) ∶= { ≥ † † ∞ Similarly, for the case of nearest-neighbor swaps, YTt − Efix ∶ L1(Tr) → L1 (Nfix ⊂ B(H)) Ycb ≤ ε} . the results listed in Section IV-D and for instance The term “ultracontractive” comes from the associated 2 Theorem V.5 show that for times t ∼ cn (log(n) − ln ε), functional inequality that we define and study in the capacities considered here quantum channel are at Section VI-B. It is also closely related to an other most ε away from their limiting capacity. functional inequality, namely hypercontractivity, that we study in SectionB. Finally it also provides an upper bound on the decoherence time by the ordering of the Lp norms. VI.FUNCTIONALINEQUALITIESANDESTIMATES One can generalize these definitions to include FORDECOHERENCETIMES: BEYONDTRANSFERENCE general amalgamated Lp spaces for 1 ≤ p ≤ q ≤ ∞: cb tp,q(ε) ∶= inf{t ≥ 0; (67) † † p q The different applications of transference so far YTt − Efix ∶ Ls(Nfix ⊂ B(H)) → Ls(Nfix ⊂ B(H)) Ycb ≤ ε} . have been based on the decoherence time of the The relationship between all these definitions will be transferred QMS. We recall that for a general QMS (not studied in Section VI-A, following the original approach necessarily selfadjoint), the decoherence time is defined for any ε > 0 as 3Remark the difference in the normalization of the norms in both definitions. In the quantum case, density matrices are normalized with tdeco(ε) (63) respect to the unnormalized trace whereas in the classical case, we look † † at the evolution of states normalized with respect to the probability inf t 0; Tt ρ E ρ 1 ε ρ . ∶= { ≥ Y ( ) − fix( )Y ≤ ∀ ∈ D(H)} distribution µG.

23 of Saloff-Coste in the classical case [1]. We then focus and hence by interpolation for 1 1−θ θ we have q = ∞ + 1 on the application to quantum capacities in Section VI-C q YTt − Efix ∶ L1(B(H)) → L1(Nfix ⊂ B(H))Y 21~qvL t 1−1~q . ≤ 1,∞( ) A. Interpolation method For the next step we interpolate this inequality with The choice of the trace norm as the measure T E L Lq N 2 of distance to equilibrium in the definition of Y t − fix ∶ q(B(H)) → q( fix ⊂ B(H))Y ≤ the decoherence time is justified by its operational 1 1−η η and get ( p = 1 + q ) that interpretation as a measure of distinguishability between q two density matrices. One could also ask how YTt − Efix ∶ Lp(B(H)) → Lp(Nfix ⊂ B(H))Y this compares with other choices, such as other 2η2(1−η)~qvL t (1−η)(1−1~q) 2 vL t 1~p−1~q . ≤ 1,∞( ) ≤ 1,∞( ) Lp norms. This makes even more sense in the quantum situation, where different non-commutative The proof for the cb-norm is identical. norms appear: conditioned and completely bounded. In the classical case, this was discussed by Saloff-Coste in [1] using interpolation theory. We first briefly sketch the main message in the classical setting (see [1]): given a classical primitive Markov semigroup (St)t≥0 on a One can get a finer control of the decoherence group G, with generator L and Markov kernel (kt)t≥0, time from the simple remark that, for a selfadjoint QMS Saloff-Coste proposed to study for all 1 p the ≤ ≤ +∞ (Tt)t 0 (not necessarily transferred), as Tt ○Efix = Efix quantities ≥ for all t ≥ 0: L v1,1(t + s + r) L 1 v1,p(t) ∶= sup Ykt(x, ⋅) − Yp L x∈G ≤ v1,2(t + s + r) 1 q St µ L1 µG Lp µG . T L N L N = Y − E G ∶ ( ) → ( )Y ≤ Z t ∶ ∞( fix ⊂ B(H)) → ∞( fix ⊂ B(H))Z T Lq N L2 vL r . Remark that the mixing-time corresponds to the study of Y s ∶ ∞( fix ⊂ B(H)) → ∞(B(H))Y 2,2( ) L v1,1(t). This definition is justified by the ordering of the (69) Lp norms, which implies that for any 1 ≤ p ≤ q and any Now in the last term, each individual term can be t ≥ 0, L L L estimated using particular functional inequalities (resp. v1,1(t) ≤ v1,p(t) ≤ v1,q(t) . ultracontractivity (UC), hypercontractivity (HC) and More generally, we define Poincaré inequality (PI)) of the classical semigroup using transference. In practice, since these estimates L vp,q(t) ∶= YSt − EµG ∶ Lp(µG) → Lq(µG)Y . are independent of the representation of the transferred t QMS, we expect that they provide bounds that are less In the quantum case, let Tt e− L t 0 be a QMS on ( = ) ≥ tight than the ones one would get if one had access to B(H). We do not assume that (Tt)t≥0 is a transferred QMS or that it is selfadjoint. Mimicking the classical the exact UC/HC/PI constants of the QMS. case, we define:

L We conclude this section with a property which vp,q(t) ∶= (68) p q allows to connect hypercontractive estimates with YTt − Efix ∶ Ls(Nfix ⊂ B(H)) → Ls(Nfix ⊂ B(H))Y . ultracontractive ones (see SectionB). L,cb We denoted by vp,q the same quantity but defined with respect to the cb norm. We recall that in both cases, the Proposition VI.2. Let (Tt)t≥0 be a selfadjoint QMS. definition of the norm is independent of the choice of the Then vL t 2 vL 2t (70) subscript s which is thus arbitrarily taken to be equal to 1,2( ) = 1,∞( ) ∞ or 1. The following proposition remains true when replacing Proof. In the selfadjoint setting, we exploit that L L,cb vp,q by vp,q . 1 2 YTt − Efix ∶ Lu(Nfix ⊂ B(H)) → Lu(Nfix ⊂ B(H))Y With the above notations, we have for 2 ∞ Proposition VI.1. = YTt − Efix ∶ Lu(Nfix ⊂ B(H)) → Lu (Nfix ⊂ B(H))Y any 1 ≤ p ≤ q ≤ +∞ together with the fact that T E T E 2, in 1 1 2t − fix = ( t − fix) vL t 2 vL t p − q . p,q( ) ≤ 1,∞( ) order to get 1 ∞ Proof. We first observe that YT2t − Efix ∶ L2(Nfix ⊂ B(H)) → L2 (Nfix ⊂ B(H))Y 1 2 Tt Efix L L2 . YTt − Efix ∶ L1(B(H)) → L1(B(H))Y ≤ 2 ≤ Y − ∶ 2(B(H)) → (B(H)))Y

24 To prove the other inequality, we use that by definition YxY1 = 1. We find that 1 2 β(1~q−1) Tt Efix L Nfix L2 Nfix t Tt x q Y − ∶ 2( ⊂ B(H)) → ( ⊂ B(H))Y Y ( )YL1(Nfix⊂B(H)) 2 β(1~q−1) β(1~p−1~q) sup Tt Efix x t rp,q t 2 T x p = Y( − )( )YL2(τ) ≤ ( ~ ) Y t~2( )YL1 (Nfix⊂B(H)) YxYL1(N ⊂B(H))≤1 2 fix β(1−1~q) β(1~p−1) ≤ rp,q 2 (t~2) sup T E x , T E x ≤ ⟨( t − fix)( ) ( t − fix)( )⟩ 1−θ θ T x q Tt x YxYL1(N ⊂B(H))≤1 Y t~2( )YL Nfix Y ( )YL1 2 fix 1( ⊂B(H)) (B(H)) β 1 1 q β 1 1 q 1 θ ( − ~ ) ( − ~ ) q − sup T2t Efix x ∞ ≤ rp,q 2 (sup t YTtxYL Nfix ) . ≤ Y( − )( )YL Nfix 1( ⊂B(H)) 2 ( ⊂B(H)) t≤t0 YxYL1(N ⊂B(H))≤1 2 fix 1 1 θ θ 1 ∞ Here − . For x the supremum is finite, ≤ YT2t − Efix ∶ L2(Nfix ⊂ B(H)) → L2 (Nfix ⊂ B(H))Y , p = q + 1 ∈ B(H)) and hence where in the third line we use again that Tt is selfadjoint β(1−1~q) β(1−1~q) 1−θ 1~1−θ 2 sup t Tt x Lq N 2 rp,q . together with T2t − Efix = (Tt − Efix) and the Hölder Y ( )Y 1( fix⊂B(H)) ≤ t

In certain situations, it may happen that B. Complete ultracontractivity Rp,q(α, t0) holds only for a short time t0 < 1. However in this article we will only consider example where we can take t 1. For sake of clarity, we thus present our In this section we introduce (complete) 0 = result only in this case. The general case t 0 would ultracontractivity (or, more generally, the Varapoulos 0 > follow similarly by considering for instance Tˆ e−tT . dimension) for general selfadjoint QMS, not nessecarily t = t transferred ones. These provide better estimates for small times. Finally, using ultracontractivity (or more generally the notion of Varapoulos dimension), we obtain control Definition VI.3. We say that a (not necessarily primitive) on different decoherence times of a QMS. QMS has p, q -Varapoulos dimension α if there exists ( ) Theorem VI.5. Let L be the generator of a transferred cp,q > 0 and t0 > 0 such that for all t ≤ t0 QMS with spectral gap λmin(L) which satisfies q UC C, α and recall the definition: YTt ∶ Lp(B(H)) → Lp(Nfix ⊂ B(H))Ycb ( ) α − 2 (1~p−1~q) cb cb ≤ cp,q(α) t . (Rp,q(α, t0)) tp,q(ε) ∶= inf{t ≥ 0 ; vp,q(t) ≤ ε} .

For p = 1, q = +∞ and t0 = 1, we say that (Tt)t≥0 is Then ultracontractive, and we denote it by UC(Cα, α). 1 ln 1 ε tcb ε 1 ln C ( ~ ) . p,q( ) ≤ + λ α + 1 p 1 q The reason for the factor 1~2 comes from the min(L) ~ − ~ n n behaviour of the heat kernel on R or T . A beautiful Proof. By transference, we have extrapolation argument by Raynaud shows that heat 2 kernel estimates for small times essentially do not depend YT1+t − E ∶ L1(B(H)) → L1(Nfix ⊂ B(H))Ycb 2 −λmin(L)t on p, q. Again, we give a prove for general selfadjoint ≤ YT1 ∶ L1(B(H)) → L1(Nfix ⊂ B(H))Ycb e QMS, independently of the transference construction. 1~2 −λmin(L)t ≤ Cα e . Lemma VI.4. For all 1 ≤ p ≤ q, Rp,q(α, t0) and Thanks to Proposition VI.1 this implies R1, α, t0 are equivalent (i.e. they hold equivalently ∞( ) cb −λmin(L)t v 1 t Cα e up to the constant cp,q). 1,∞( + ) ≤

α and Proof. Let us deﬁne β = 2 and vcb 1 t C1~p−1~q e−λmin(L)(1~p−1~q)t . β(1~q−1~p) p,q( + ) ≤ α rp,q = sup t 0

25 (St)t≥0 is primitive with spectral gap λmin(L) and that This in turn is equivalent to it is ultracontractive with constants C, α > 0: (1 − ε)EN ≤cp T ≤cp (1 + ε)EN , −α~2 YSt ∶ L1(µG) → L∞(µG)Y = sup Skt(g)S ≤ C t . g∈G where ≤cp means that the inequality remains true for all ampliations E id and T id for all positive Then N ⊗ Mm ⊗ Mm integers m ≥ 0. In particular, cb 1 ln(1~ε) tp,q(ε) ≤ 1 + ln C + . T T 1 ε E 1 ε E εΦ 1 ε E. λmin(L) 1~p − 1~q = − ( − ) + ( − ) = + ( − ) T −(1−ε)E † We already discussed in Section III-C how Here Φ = ε is normalized so that Φ (1) = 1, such ultracontractivity holds in general for Hörmander i.e. Φ is a channel. Obviously, it is also an N bimodule systems and finite groups. map. A classical result from Varapoulos says that ultracontractivity is equivalent to the following Sobolev Proof of Proposition VI.7. Since we assume α to be inequality (see [57]): convex, we find 2 2 f C f , Lf µ f , α Tt α 1 ε Efix εΦ Y YLθ (µG) ≤ −⟨ ⟩ G + Y YL2(µG) ( ) = (( − ) + ) ≤ (1 − ε)α(Efix) + εα(Φ) where θ = 2α~(α−2). This inequality itself implies Nash inequality: ≤ (1 − ε)α(Efix) + ε sup α(S) . S 2(1+2~α) 2 4~α f C f , Lf µ f f . The lower bound follows from convexity, because Efix Y YL2(µG) ≤ −⟨ ⟩ G + Y YL2(µG) Y YL1(µG) is obtained as an average. In the quantum case, this last inequality was studied in [58] in the case of primitive selfadjoint QMS. In [56] it Remark VI.9. Many capacities are either convex was shown that Varopoulos’ theorem remains true in the functions of the channel, for example the entanglement cb-category. assisted capacity, or admit a controllable convex roof, for example the side-channel assisted capacity Q ≤ Qss from ∗ C. Application to quantum capacities [59]. Assuming that the maximal value α = supS α(S) is of order log n, we see that for t tcb ε , the leading The value tcb ε is a complete decoherence ≥ 1,∞( ) 1,∞( ) term of α T is α E . In that sense, tcb ε is a time, usually bigger than t and t . It provides a ( t) ( fix) 1,∞( ) mix deco “universal” coherence time. In the transference situation universal bound for convex functions on channels. we can get a hold of this constant via commutative t Proposition VI.7. Let Tt = e− L be a selfadjoint methods. semigroup with fixpoint algebra Nfix and conditional expectation E , let ε 0 and t tcb ε . Let α be APPENDIX A fix > ≥ 1,∞( ) convex function on channels. Then ENTROPY COMPARISON THEOREM Let G a compact group with Haar measure µG α(Efix) ≤ α(Tt) ≤ (1 − ε)α(Efix) + ε sup α(S) S and let g ↦ u(g) be a projective representation of G on some finite dimensional Hilbert space . For a bounded where the supremum is take over all channels S with the H measurable function k G +, we define the operator same input and ouput dimension. ∶ → R Φk on B(H) as: We need the following observation due to Li Gao: ∗ Φk(ρ) ∶= S k(g) u(g) ρ u(g) dµG(g) .

Lemma VI.8. (Li Gao) Let N ⊂ Mn and T ∶ Mn → Mn We recall that the ﬁxed-point algebra of the map Φk is be a completely positive N-bimodule map and 0 < ε < 1 given by the commutant of u(G): such that ′ Nfix = {σ ∈ B(H)S σu(g) = u(g)σ} = u(G) T E L L∞ N ε . Y − ∶ 1(Mn) → 1 ( ⊂Mn)Ycb ≤ and that the following bimodule property holds: for any Then there exists a completely positive N-bimodule map σ1, σ2 ∈ Nfix, Φ such that Φk(σ1 ρ σ2) = σ1 Φk(ρ) σ2 . T = (1 − ε)E + εΦ . Proof of Lemma VI.8. Indeed, we refer to [6] for the fact + + that an N bimodule map admits a modiﬁed Choi matrix Theorem A.1. Let x ∈ Nfix and k ∶ G → R a bounded χT , and that YχT −1Y ≤ ε holds iff YT −EN ∶ L1(Mn) → measurable function such that ∫ k dµG = 1. Then, for any L1(N ⊂ Mn)Y ≤ ε. Then we deduce that positive y ∈ B(H), and any p ≥ 1 of Hölder conjugate pˆ: 1 1 1 1 − − − − (1 − ε)1 ≤ χT ≤ (1 + ε)1 . Yx 2p ˆ Φk(y)x 2p ˆ Yp ≤ YkYp Yx 2p ˆ Efix(y)x 2p ˆ Yp .

26 Moreover, for any states ρ, σ ∈ D+(H) such that σ ∈ u(g)ψ), so that D(Nfix): ′ ∗ Efix(ρ) = (idB(H ) ⊗EµG )(V ρV ) D(Φk(ρ)SSσ) ≤ D(Efix(ρ)SSσ) + k log k dµG . (71) ∗ SG = u(g) ρ u(g) dµG(g) . SG Next, identify the space V (H) with a subspace of r c L2(µG) ⊗h H′ ≅ B(L2(µG), H′), so that V (ψ) = ηψ. In order to prove Theorem A.1, we need to introduce a few functional analytical notions: given two Lemma A.2. V (H) is a TRO. (possibly infinite dimensional) Hilbert spaces and , ˜ H K Proof. Let ψ, ϕ, ψ, ϕ˜ ∈ H. We compute we denote by Tp(H, K) the Banach space of linear ˜ ∗ ˜ ∗ operators x with norm ⟨ψ, ηψηϕ(ϕ˜)⟩ = ⟨ψ, u(g) ψ⟩⟨ϕ, u(g)ϕ˜⟩ dµG(g) ∶ H → K H SG ∗ 1~2 ∗ 1~2 ˜ x , xx x x , (72) Tr ϕ˜ ψ Efix ψ ϕ Y YTp(H K) ∶= Y YTp~2(K) = Y YTp~2(H) = (S ⟩⟨ S (S ⟩⟨ S)) ˜ 1 ψ, Efix ψ ϕ ϕ˜ q q = ⟨ (S ⟩⟨ S) ⟩H where YaY q ∶= Tr(SaS ) . With a slight abuse of T (H) ψ,˜ σϕ˜ , notations, we will also denote this norm by YxYp. Next, ≡ ⟨ ⟩H (74) given two Hilbert spaces H and K and a positive invertible element σ ∈ B(H) we recall that the Kosaki ˜ 1 where σ ∶= Efix(Sψ⟩⟨ϕS). Since this holds for all ψ ∈ H, norms x σ p x , form an interpolation ∗ ′ Y YL̂p(σ) ∶= Y YTp(H K) ηψηϕ(ϕ˜) = σϕ˜. We also have that, since σ ∈ u(G) , for family (see. Theorem 4 of [34] for more details). We any ψ′ ∈ H will also use the column and row spaces. For a Hilbert ∗ ′ ∗ ′ c η ′ σϕ˜ g ψ , u g σϕ˜ σ ψ , u g ϕ˜ space H: we denote the column space H ∶= B(C, H) ψ ( )( ) = ⟨ ( ) ⟩H = ⟨ ( ) ⟩H r ∗ and row space H ∶= B(H, C). From this one may = ησ∗ψ′ (ϕ˜)(g) . cp c r construct the interpolation spaces , 1 p and H = [H H ] ~ ∗ ∗ ∗ ∗ ′ rp r c Hence η ′ ηψηϕ η ∗ ′ . Since Efix ψ ϕ ψ , , 1 p (see [60] for an introduction to these ψ = σ ψ (S ⟩⟨ S) ∈ H H = [H H ] ~ ∗ ∗ spaces and their properties). Importantly, we also recall ησ∗ψ′ ∈ V (H) , so that V (H) is a TRO. that if a pair of spaces X0 and X1 form an interpolation scale [X0,X1]θ, and if Y0 ⊆ X0 and Y1 ⊆ X1 are Next, for any Ω ∈ T2(H), define the following given by the same projector, then [Y0,Y1]θ is also an dual operators: interpolation scale. The Haagerup tensor product will L µ ′ also play an important role in the proof of Theorem A.1. ηˆ H ⊗ 2( G) → H , Ω ∶ χ u g ∗ Ω χ g dµ g It is defined as follows: given two operator spaces ↦ ∫G ( ) ( ) G( ) ′ X ⊂ B(H) and Y ⊂ B(K), the Haagerup tensor norm ∗ H → H ⊗ L2(µG) ηˆΩ ∶ is defined on X ⊗ Y as ψ ↦ (g ↦ Ω∗ u(g)ψ) . ηˆ V YzYX⊗hY ∶= By definition, the operators Ω span the space (H) ⊗h r 1 2 1 2 . Since V is a TRO, as shown in Lemma A.2, inf x x∗ ~ y∗ y ~ . H (H) Y Q k k YB(H) Y Q k k YB(K) V r is also a TRO. z=∑k xk⊗yk k k (H) ⊗h H Finally, we recall that a ternary ring of operators (TRO) Lemma A.3. Let 1 ≤ p ≤ ∞ of Hölder conjugate pˆ, and is a closed operator subspace X of B(H, K) with the Lk ∶ L2(µG) → L2(µG) denote left multiplication by property k for k ∶ G → R+ bounded measurable. Then, for any ∗ xy z ∈ X for all x, y, z ∈ X. (73) Ω ∈ T2(H) and positive operator σ ∈ L(V (H)) ≡ Nfix,

1 1 1 1 If X is a TRO and if σ is in the left algebra L(X) ∶= σ− 2p ˆ ηˆ 1 L 2 k 2 σ− 2p ˆ ηˆ . Y Ω( H ⊗ k )Y2p ≤ Y YL2p(µG) Y ΩY2p span{xy∗S x, y ∈ X} ⊂ B(K), we may construct an (75) interpolation scale Xp,σ = [X∞,X1,σ]1~p as the space 1 p Proof. By Theorem 5.2 of [34], for any positive operator X equipped with the norm YxYσ,p = Yσ xYp. For more r information, see Theorem 5.2 in the appendix of [34]. Let σ ∈ L(V (H) ⊗h H ) ≡ L(V (H)) ≡ Nfix, the spaces X˜ V r equipped with the Kosaki norm H′ be a copy of H that corresponds to the output space of p,σ ∶= (H) ⊗h H x form an interpolation scale. In particular, the channel Φ . Now, given ψ , let η L µ ′ Y YL̂p(σ) k ∈ H ψ ∶ 2( G) → H 1 X˜2p,σ X˜ , X˜2,σ 1 . Next, observe that σ− 2 ηˆΩ and its adjoint be given by = [ ∞ ] p ∈ r 1 V . Therefore, assuming that σ− 2 ηˆ ∗ (H) ⊗h H Y ΩYL̂p σ < ηψ(χ) = χ(g) u(g) ψ dµG(g), ( ) SG 1, then there exists an analytic function ξ ∶ {z ∈ C ∶ 0 ≤ ∗ ˜ ˜ ηψ ϕ g ψ, u g ϕ . Re z 1 X X2,σ, of ﬁnite dimensional range ( )( ) = ⟨ ( ) ⟩H ( ) ≤ } → ∞ + 1 such that ξ(1~p) = σ− 2 ηˆΩ [61], Denote by V ∶ H → H′ ⊗L2(µG) the Stinespring dilation of the conditional expectation that is given by V ψ g ξ it ξ it 1, and ξ 1 it 1 . = ( ↦ Y ( )Y∞ = Y ( )YL̂∞(σ)< Y ( + )YL̂2(σ)<

27 Let us deﬁne the analytic function x z where p is the Hölder conjugate of 1 q 2 and where pz ( ) ∶= ≤ ≤ z 2 ~ 1 2 we assume that Tt t 0 is selfadjoint. We insist that the σ ξ(z)( H ⊗ Lk ). We have that ( ) ≥ situation is more tricky than in the UC case, because as it itp 2 1 2 Yx(it)Y∞ = Yσ ξ(it)( H ⊗ Lk )Y∞= Yξ(it)Y∞ < 1 . mentioned in Lemma III.3, the only operator norms that r transfer to the cb case are when the image is L∞. Next, since ξ(1+it) ∈ V (H)⊗h H , there is Ω′ ∈ T2(H) such that ξ 1 it ηˆ ′ . Therefore, ( + ) ≡ Ω Hypercontractivity has been originaly studied in p 2 1 2 2 1 2 the quantum case in [62] in the context of spin system, Yx(1 + it)Y2 = Yσ ξ(1 + it)( H ⊗ Lk )Y2 1 p than in [25] for decoherence time. Up to now, very few 2 2 2 ′ 1 = Yσ ηˆΩ ( H ⊗ Lk )Y2 examples are known where it is possible to estimate 1 p 2 ∗ ′ 2 the hypercontractive constant. Transference provides a = Sk(g)S Yσ u(g) Ω Y2 dµG(g) SG new way to do so. It must however be noted that when p 1 2 ∗ 2 ′ concerned with applications as the ones presented in = S Sk(g)S Yu(g) σ Ω Y2 dµG(g) G this article, a direct application of transference lead to 1 2p 1 2 k 2 σ 2 Ω′ = Y YL2p(µG) Y Y2 better result than ﬁrst proving HC via transference and 1 2p 1 2 k 2 σ 2 ξ 1 it then applying transference. = Y YL2p(µG) Y ( + )Y2 1 2p k 2 , < Y YL2p(µG) Thus, our motivation in this section is simply to show that using transference, one can obtain bounds where in the fourth line, we used that σ ∈ Nfix. Then, using Stein’s interpolation theorem: on the hypercontractive (and thus also log-Sobolev) constants for a large class of QMS: the transferred QMS. 1 1 1 σ− 2p ˆ ηˆ 1 L 2 x 1 p k 2 , Only few such bounds existed previously in the literature Y Ω( H ⊗ k )Y2p = Y ( ~ )Y2p ≤ Y YL2p(µG) and only in the primitive case (see [16], [63]) so we and the result follows after rescaling. thought it meaningful to write this technique explicitily in this article. Our main result is the following one.

Proof of Theorem A.1. We use the notation of the theorem. The following holds for ρ Ω 2: = S S A. Main result 1 − 2p ˆ 1 2 Yσ ηˆΩ ( ⊗ L 1 )Y L µ , ′ H k 2 T2p(H⊗ 2( G) H ) Definition B.1. We say that the quantum Markov 1 1 σ− 2p ˆ ηˆ 1 L ηˆ∗ σ− 2p ˆ ′ = Y Ω ( H ⊗ k) Ω YTp(H ) semigroup (Tt)t≥0 on B(H) is (weakly) hypercontractive 1 1 if there exist two non-negative constants c 0, d 1 − ∗ − > ≥ = \ k(g) σ 2p ˆ u(g) ρ u(g) dµG(g)σ 2p ˆ \ c SG ′ such that for all t ln p 1 : Tp(H ) ≥ 2 ( − ) 1 1 − − p 1 1 σ 2p ˆ Φ ρ σ 2p ˆ ′ , 2 − p = Y k( ) YTp(H ) YTt ∶ L2(B(H)) → L2 (Nfix ⊂ B(H))Y ≤ d (HC (c, d)) where Lk ∶ L2(µG) → L2(µG) denotes the operator of multiplication by k. Therefore, for the claim to hold, it The study of hypercontractivity and its application suffices to show that for 1 ≤ p ≤ ∞, to the estimation of the decoherence time was the subject 1 of the recent article [16]. We can slightly improve − 2p ˆ 1 1 ′ Yσ ηˆΩ ( ⊗ L )Y 2p L2 µG , H k 2 T (H⊗ ( ) H ) their results (Proposition 7.1) using Equation (69) and 1 1 k 2 σ− 2p ˆ ηˆ ′ . ≤ Y YL2p(µG) Y ΩYT2p(H⊗L2(µG),H ) transference. This follows from Lemma A.3. Differentiation at p = 1 Proposition B.2. Assume that the transferred QMS gives the entropic inequality (71). (Tt)t≥0 on B(H) is HC(c, d). Assume furthermore that the group G is finite with SGS ≥ e. Then for all ρ ∈ D(H) and all κ 0, APPENDIX B > √ RANSFERENCEOFHYPERCONTRACTIVITY † † 1−κ T Tt ρ E ρ d e , [ ( ) − fix( )[1 ≤ In the classical theory and compared to c κ t = ln ln SGS + . ultracontractivity (UC), it is wellknown that an even 2 λmin(L) finer control of the decoherence time can be realized Proof. Remark that by transference, using the hypercontractivity property (HC) of the id L1 N Lq N semigroup at intermediate times, using Equation (69). Z ∶ ∞( fix ⊂ B(H)) → ∞( fix ⊂ B(H))Z Hypercontractivity is concerned with controlling the term id L1 N Lq N ≤ Z ∶ ∞( fix ⊂ B(H)) → ∞( fix ⊂ B(H))Zcb q 2 T L N L id L1 µG Lq µG Y t ∶ ∞( fix ⊂ B(H)) → ∞(B(H))Y ≤ Y ∶ ( ) → ( )Y p 1~qˆ = YTt ∶ L2(Nfix ⊂ B(H)) → L2(B(H))Y , ≤ SGS ,

28 where qˆ is the Hölder conjugate of q. We now apply Remark B.5. A similar statement holds if c Equation (69) with t = 0, s = 2 ln ln SGS and qˆ = 1+ln SGS we replace YSt0 − Eµ ∶ L2(µG) → L∞(G)Y by to get: St Eµ L2 µG Lp µG for some p0 2. Z p0 − ∶ ( ) → 0 ( )Zcb > 1 1 However, one can apply Lemma III.3 only for p0 L 1~qˆ 2 − qˆ −r λmin(L) = +∞ v1,1(s + r) ≤ SGS d e so we choose to stick to this case. √ 1 r λ ≤ d e − min(L) . We have used that, as SGS ≥ e, then (ln SGS)~(1+ln SGS) ≤ Proof. By a similar argument as in the proof of Theorem 1 qˆ 1, which gives SGS ~ ≤ e. This concludes the proof by 4.7 in [16] (see also [63] in the case of the usual Schatten taking r = κ~λmin(L) and t ≥ s + r. norms, and Theorem 3.9 and 3.10 of [24] in the classical setting), we get from Equation (78) that a complete In practice, even if it is hard to find the tightest logarithmic Sobolev inequality cLSI2 (t0,M) holds (see constants for which the above functional inequality is Section B-B for the definition). By a simple adaptation of satisfied, it is still possible to obtain good bounds based √ Theorem 4.5 in [16], we get cLSI2 (c, 2) with c given on ultracontractivity. As we can use transference on by Equation (79). Gross’ integration Lemma for the cb ultracontractivity, we can obtain in this way bound norms allows us to conclude (cf. [64] in the case of a on the hypercontractive constants. However the bound finite group, and Theorem B.8 in the case of a compact we would obtain on the decoherence time from such Lie group). an estimate would not be better from using only ultracontractivity. Theorem B.3. The proof starts by a simple application of Theorem B.3. Let S be a reversible Markov ( t)t≥0 the transference method: first, by (iii) of Theorem III.2: semigroup on a compact Lie or finite group G, with p right-invariant kernel and assume there exists t0 ≥ 0 and YTt ∶ L2(B(H)) → L2(Nfix ⊂ B(H))Y (80) M 0 such that p > ≤ YTt ∶ L2(B(H)) → L2(Nfix ⊂ B(H))Ycb S L µ L µ . YSt − EµG ∶ L1(µG) → L2(µG)Y (76) ≤ Y t ∶ 2( G) → p( G)Ycb = sup Yh ↦ kt(g) − 1Y2 ≤ M. Then, notice that g∈G S E L µ L G Let g ↦ u(g) be a unitary representation of G on Y t − µ ∶ 2( G) → ∞( )Ycb some finite dimensional Hilbert space and let (Tt)t≥0 = YSt − Eµ ∶ L2(µG) → L∞(G)Y be the corresponding transferred QMS defined as in √ = Yh ↦ kt(h) − 1Y2 , Equation (3). Then (Tt)t≥0 satisfies HC(c, 2) with 1 c λ t ln M 1 . (77) ≤ λ ( min(L) 0 + + ) where we used Lemma III.3 in the first line. The result min(L) follows from a direct application of Lemma B.4. The proof of Theorem B.3 proceeds by consecutive uses of the transference method of Theorem III.2 as well as the following interpolation result: In Theorem 3.7 of [3], the authors showed, conversely to the above theorem, how to obtain bounds Lemma B.4. Let S be a reversible Markov ( t)t≥0 of the form of (76) from estimates on the log-Sobolev semigroup on a compact Lie or finite group G, with constant. Similar bounds were obtained from the Bakry right-invariant kernel. Assume that Emery condition via Poincaré, logarithmic Sobolev and Nash inequalities. We refer to SectionC for YSt0 − Eµ ∶ L2(µG) → L∞(G)Y ≤ M (78) a more detailed discussion. This allows us to get for some t0 0 and M 0. Then, the semigroup ≥ > hypercontractivity estimates for a QMS (Tt)t≥0 from (St)t≥0 is hypercontractive with respect to the completely hypercontractivity of any classical Markov semigroup bounded norm: for all 2 ≤ p, there exist c > 0 such that S from which T can be transferred: for c ( t)t≥0 ( t)t≥0 for all t ≥ 2 ln (p − 1): example, the following corollary is a direct consequence √ 1 1 of (87) and Theorem B.3: S L µ L µ 2 2 − p , Y t ∶ 2( G) → p( G)Ycb ≤ √ (cHC2(c, 2)) Corollary B.6 (From classical to quantum hypercontractivity). Let S be a reversible Markov with ( t)t≥0 semigroup on a finite group G, with right-invariant 1 c λ L t ln M 1 , (79) kernel satisfying HC(c, 0). Then the QMS (Tt)t 0 ≤ λ L ( min( ) 0 + + ) √ ≥ min( ) defined in Equation (3) satisfies HC(c′, 2), with where λmin(L) is the spectral gap of the generator L of 2 c c′ ln ln G . the semigroup St t 0. ≤ + S S ( ) ≥ λmin(S) 2 29 B. Completely bounded Gross lemma for classical of positive matrix valued functions with coefficients in A, diffusions with spectrum uniformly bounded away from 0. Then, we define the Lq-entropy and the Dirichlet form as follows: In this section, we briefly describe the proof given elements f, g ∈ A(Mm)++, of Gross’ integration lemma relating the complete q q q q logarithmic Sobolev inequality to the hypercontractivity Entq(f) = τ(f log f ) − τ(f ) log τ(f ) , with respect to the completely bounded normes defined f τ f q−1 id L f . Eq, id m⊗L ( ) ∶= ( ( Mm ⊗ )( )) in Section III-A for classical diffusions. The proof is M similar to the ones of [64] for quantum (and hence Next, we define the notions of completely bounded classical) Markov semigroups in finite dimensions (see hypercontractivity and of complete logarithmic Sobolev also [65] for the case of the modified logarithmic Sobolev inequality: Given q ≥ 1, the semigroup (St)t≥0 is said 4 inequality). to

Let (St)t≥0 be a semigroup on the algebra - be q-completely hypercontractive if there exist c > L E, , µ of real bounded measurable functions on 0, d 1 such that for all p q and all t c log p−1 : ∞( F ) ≥ ≥ ≥ 2 q−1 the probability space (E, F, µ) that is reversible with 1 1 p q − p respect to µ. The semigroup is described by its kernel YSt ∶ Lq(C IM ⊂ M) → Lq (C IM ⊂ M)Ycb ≤ d (cHC c, d ) (kt)t≥0 via Equation (2) that we recall here: q( ) - satisfy a q-complete logarithmic Sobolev inequality St(f)(x) = kt(x, y) f(y) dµ(y) . (81) SE if there exist c ≥ 0, d ≥ 1 such that for all m ∈ N, and all f ++: When extended to its action on the space L2(µ) ∈ A(Mm) of square integrable real valued functions on E, the q Entq f c q, id f log d f . ( ) ≤ E Mm ⊗L( ) + ( )Y YLq (Mm) semigroup (St)t≥0 is strongly continuous, and we denote (cLSIq(c, d)) by (L, dom(L)) its associated generator. Since the domain of L is not usually known in practice, we The equivalence between cLSIq(c, d) and cHCq(c, d) will work on a dense subspace of it. In fact, it will was proved in [64] in the case of a Markov chain defined be convenient to assume for technical reasons that the on a finite sample space (and even for quantum Markov following hypothesis, already used in [66], [67], holds: semigroups in finite dimensions). Here, we extend this equivalence to the present abstract setting, which in Hypothesis B.7. There exists an algebra A of bounded particular incorporates the case of classical diffusions. measurable functions, containing all the constants, dense in all the spaces Lp(µ), p ≥ 1 as well as in dom(L), that is stable under composition with multivariate smooth Theorem B.8. Let (St)t≥0 be a classical Markov functions. We also assume that for any sequence {fn} semigroup defined on the algebra L∞(E, F, µ) of of A that converges to a function f in L2(µ), and every bounded measurable functions on some measure space smooth bounded function Φ ∶ R → R with bounded (E, F, µ), and assume that µ is an invariant measure derivatives, there exists a subsequence {Φ(fnk )} of of (St)t≥0 for which (St)t≥0 is reversible. Further {Φ(fn)} converging towards Φ(f) in L1(µ) and such assume the existence of a subalgebra A satisfying that LΦ(fnk ) converges to LΦ(f) in L1(µ). Hypothesis B.7. Then, (i) If cHC c, d holds, then cLSI c, d holds. For any m ∈ N, the algebra Mm(L∞(µ)) q( ) q( ) coincides with the algebra L∞(E, Mm) of bounded (ii) If cLSI2(c, d) holds, then cHCq(c, d) holds for any measurable functions with values in Mm, with norm q ≥ 2.

YfYL∞ m defined as supx E Yf(x)Y m . For sake of (M ) ∈ M We now briefly sketch a proof of Theorem B.8: As simplicity, we denote this norm by Y.Y . Next, for any ∞ usual, the first step towards establishing a Gross lemma f ∈ L∞(Mm), define the following trace on L∞(Mm): is to provide a formula for the differential at p = q of 1 τ f Tr f x µ dx . q p ( ) ∶= Mm ( ( )) ( ) p S L I L L I E SE m ↦ Y t(p) ∶ ∞(C E ⊂ ∞) → ∞(C E ⊂ L∞( ))Ycb q We denote the completions of the Lp norms associated = sup Y idMm ⊗ St(p) ∶ Lq(Mm ⊂ L∞(Mm)) → to that state L , p 1, and denote the norms m p(Mm) ≥ Lp L , . q (Mm ⊂ ∞(Mm))Y associated to it by Y YLp(τ). To simplify the notations, we will get rid of the indices identifying the underlying for some increasing, twice differentiable function t ∶ tr 1 spaces and introduce the normalized traces ∶= m Tr . [1, ∞) → [0, ∞). The proof of the differentiability The main difference to [64] in our setting arises from follows closely the one of Lemma 9 of [64]: given the possible unboundedness of the generator L of the classical semigroup St t 0, which will not be an issue ( ) ≥ 4These inequalities in particular imply the primitivity of the as long as we carry out our differentiations in the algebras semigroup. One could easily extend these inequalities to non-primitive A(Mm)++ ∶= {g = (gij), gij ∈ A ∀ij ∈ {1, ..., m}, g > 0} classical semigroups, which however play no role in this article.

30 ++ p f ∈ A(Mm) , the Lq (Mm ⊂ L∞(Mm)) norms take is twice continuously differentiable. Moreover, the following useful form: d 1 1 1 − 2r(p) − 2r(p) p p τ((X ft(p) X ) ) f p (82) dp Y YLq (Mm⊂L∞(Mm)) = 1 1 1 1 ⎧ − 2r − 2r p p ⎪ inf τ((X f X ) ) p ≥ q = p 1 ⎪ X ++, tr X 1 2 1− p ⎪ ∈Mm ( )= p τ(gp) ⎨ 1 1 1 2r 2r p p p p p p p ⎪ sup τ((X f X ) ) p ≤ q . τ(log(gp)gp) + tr Eµ[gp] log X − τ(gp) log τ(gp) ⎪ ++ ⎪ X∈Mm , tr(X)=1 1 1 ⎩ 2 ′ p−1 − 2r(p) − 2r(p) +p t (p)τ gp (X (idMm ⊗L)(ft(p))X ) . where 1 1 1 . Therefore, we can restrict our analysis r = q − p (83) to the one of the following function 1 Moreover, the function X ↦ p p 1 1 F ∶ [1, ∞) × L (Mm) ∋ (p, g) ↦ τ(g ) , − 2r(p) − 2r(p) p ∞ τ((X ft(p) X ) ) is Fréchet differentiable on ++ for all p 1, . by considering the differentiation of F ○ G, where Mm ∈ [ ∞) 1 1 − − Proof. The only point that remains to be G ∶ [1, ∞) ∋ p ↦ (p, gp) , gp ∶ x ↦ X 2r ft p (x) X 2r ( ) proven is the Fréchet differentiability of ++ 1 1 where ft p St p f , with f M and for some − 2r(p) − 2r(p) p ( ) ∶= ( )( ) ∈ A( ) X ↦ τ((X ft(p) X ) ), which follows ++ fixed X ∈ Mm . Both functions p ↦ gp (see Lemma from a general argument on the Fréchet differentiability p τ gp g 8 of [68]) and ↦ ( ) at fixed are differentiable of noncommutative Lp spaces, see [69]. with continuous derivatives. The latter holds by means of bounded convergence. Therefore, the function p ↦ Next, we define the marginal state on m as F p, gp itself is differentiable and the chain rule holds: M ( ) follows: d p ∂ p ∂ q 1 µ f F ○ G(p) = τ(gp )Sp1=p + τ(p, τp2 )Sp2=p . E [ ] dp ∂p1 ∂p2 γ ∶= , (84) τ(f q) The first term above simply follows from a bounded ˜ convergence theorem: Defining the function X ↦ G(X) that associates the term in between parentheses on the right hand side of ∂ p p ++ 1 Equation (83) to any operator X m , the following τ(gp )Sp1=p = τ( log(gp) gp ) . ∈ M ∂p1 lemma is a straightforward extension of Lemma 10 of The second term arises from the differentiability in [64]: L1(Mm) of the map p ↦ gp, with: Lemma B.10. There exists κ > 0 and K < ∞ such that ∂ r′ p 1 1 ( 2) − ( ) − ( ) for all p q and X m, X γ L τ κ, g X 2r p2 log X , f X 2r p2 ≤ ∈ M Y − Y 1( ) ≤ p2 = 2 { ( ) t(p2)} ∂p2 2r(p2) R R 1 1 R G˜ X R − ( ) − ( ) R ( ) R t′ p X 2r p2 id L f X 2r p2 . R gp L τ gq L τ t p t q R + ( 2) ( Mm ⊗ )( ) RY Y p( ) − Y Y q ( ) − ( ( ) − ( )) 2 q−1 R R q YgqYL τ R xp yp R q ( ) R Since the function 0, 0, x, y − , 2 [ ∞) × [ ∞) ∋ ( ) ↦ x−y ≤ K (t(p) − t(q)) . p ≥ 1 admits a double integral form as in [68], it follows p1 from Lemma 8 of that same paper that p2 gp is Proof. The proof consists in a simple Taylor expansion ↦ 2 1 p p differentiable in L1(Mm), and therefore so is p2 ↦ of the function p ↦ τ(gp) , and we refer to the proof τ gp1 ( p2 ), with derivative: of Lemma 10 of [64] for more details. ∂ τ gp ( p2 )Sp2=p ˜ ∂p2 From the very definition of the function G, one 1 1 1 easily derives the following formula: p−1 − 2r(p) − 2r(p) = p τ gp X {log(X), ft p }X 2p2 ( ) q G X G γ gq D γ X , 1 1 ( ) − ( ) = Y YLq (τ) ( Y ) ′ − 2r(p) − 2r(p) +t (p) X (idMm ⊗L)(ft(p)) X . where D(ρYσ) denotes the (normalized) relative entropy Since we restrict the differentiation to operator-valued between two densities ρ, σ: functions in the algebra Mm ++, the same argument A( ) D ρ σ τ ρ log ρ log σ . would further provide that F ○ G is twice continuously ( Y ) ∶= ( ( − )) differentiable, as long as the function t is. The following q The link to the Lp(Mm ⊂ L∞(Mm)) norms is made in lemma hence extends Lemma 9 of [64] to the case of the next lemma which establishes the proximity to the general classical semigroups: density γ of the optimizer X in the definition of the norms, for p close to q. Lemma B.9. Let t ∶ [1, ∞) → [0, ∞) be an increasing, twice continuously differentiable function. Then, for any Lemma B.11. For any 0 ε κ, there exists δ 0 such 1 1 < ≤ > ++ − 2r(p) − 2r(p) p X ∈ Mm , the function p ↦ τ((X ft(p) X ) ) that for all p, q ∈ (1, ∞), such that Sp−qS < δ, there exists

31 ++ ++ X ∈ Mm , τ(X) = 1, such that Lemma B.14. For any f ∈ A(Mm) , and any q > 1: 2 f p g , q YPt(p)( )YLq (Mm⊂L∞(Mm)) = Y pYLp(τ) f f . E2, id m ⊗L( ) ≤ Eq, id m ⊗L( ) γ X ε . M 4(q − 1) M Y − YL1(τ) ≤ Proof. Such an inequality was derived under various Proof. The proof is identical to the one of Lemma 11 of conditions in the classical and quantum case (see [64] and for this reason is omitted. e.g. Proposition 3.1 of [70], or [27]) and readily extends In the case when p > q, the optimizer of to the finite von Neumann algebraic case. Equation (82) can actually be further characterized: The above lemma allows us to show that cLSI2 Lemma B.12. There exists η 0 such that for any implies cLSIq, so that the proof of Theorem B.8 becomes > 1 p p a simple consequence of Lemma B.13. q < p < q + η, the function X ↦ τ(gp) is strictly ˜ ++ convex, and there exists a unique X ∈ Mm such that it is identically equal to f p . The optimizer APPENDIX C Y t(p)YLq (M⊂L∞(Mm)) X˜ satisfies: CLASSICAL MARKOVSEMIGROUPS 1 1 In this appendix we will briefly review the − 2r(p) − 2r(p) Eµ X ft(p) X X˜ . definitions of some classical Markov semigroups we = 1 1 p − 2r(p) − 2r(p) discussed in the main text. We also list some of the YX ft(p) X YL τ p( ) functional inequalities known for these semigroups. Proof. Once again, the proof is identical to the one of Lemma 12 of [64]. In particular, it relies on the uniform continuity of Schatten p-norms that is known to hold in A. Finite groups a general von Neumann algebraic context. Given a finite group and a discrete time Markov 1 chain of kernel k(g, h) = k(gh− ), consider the kernel Combining the last two lemmas, we conclude that of the associated continuous time chain St t 0 defined p ( ) ≥ there exists a unique positive definite optimizer of the Lq by norms, for q < p close enough, and that this minimizer is close to the operator γ defined in Equation (84). One kt(x, y) = S G S exp(−t ( id −k))(x, y) . (86) can also use these results to show that the function p ↦ By construction, this kernel is right-invariant, and the f p is continuous (see Lemma 13 of Y t(p)YLq (Mm⊂L∞(Mm)) theory developed in Section III applies. In Theorem [64] for a proof). The above tools can also be use to p 3.7 of [3], the authors showed how to obtain bounds prove the following differentiation of the Lq norm, the of the form of (76) from estimates on the log-Sobolev proof of which we also omit since it is identical to the constant. Adapting their result to our setting, they showed one of Theorem 7 of [64]: that for a reversible Markov semigroup (St)t≥0 with d 1 associated right-invariant kernel (kt)t≥0 on a finite group ft p p (85) dp Y ( )YLq (Mm⊂L∞(Mm))V = 2 q−1 of cardinality G 35, p=q q Yft(q)Yq S S > q q q q −1 1−γ τ f log f tr µ f log µ f sup G h kt gh 1 2 e , (87) t(q) t(q) − E [ t(q)] (E [ t(q)]) Y ∋ ↦ ( ) − Y ≤ g∈G 2 ′ q−1 q t q τ f L ft q . + ( ) t(q) ( ( )) for This differentiation is the key tool to prove Theorem B.8: c γ t = ln ln SGS + , γ > 0, we first assume the following two results hold: 2 λ(∆) where λ ∆ denotes the spectral gap of the generator Lemma B.13. (i) If cHCq(c, d) holds, then ( ) of the chain and c its log-Sobolev constant. Thus, a cLSIq(c, d) holds. bound on the spectral gap and log-Sobolev constant are (ii) If cLSIp(t)(c, d) holds for all t ≥ 0, then cHCq(c, d) holds. sufficient for our purposes. We now list some of the constants for some Markov semigroups which are of Proof. The proof of these implications uses interest in quantum information theory. Equation (85) and is identical to the one of Theorem 4 n a) The hypercube:: Let G = Z2 and define of [64]. The only difference resides in ii where one ( ) the following classical Markov chain: for i = 1, ..., n, invokes the density of in all the Lp spaces in order n A let ei be the vector in Z2 with all coordinates 0 but in to show that hypercontractivity holds for any initial the ith coordinate, which is set to be 1. Next, define a bounded operator valued function f f0. n = probability mass function Q on Z2 by setting k(0) = k(ei) = 1~(n+1) for i = 1, .., n, and k(x) = 0 otherwise. The reduction to q = 2 follows from a standard Stroock-Varopoulos inequality relating the Dirichlet form 5The conventions in [24] are slightly different from the ones that we q 1 q q id f, f − to id f 2 , f 2 . use in this article: in particular, their constant α is related to our weak E Mm⊗L ( ) E Mm⊗L ( ) log-Sobolev constant c as follows: 2αc = 1.

32 In words, at each time, the discrete-time chain jumps for some constant c. In [71], it was shown that from one vertex to a neighboring one with probability the spectral gap of the corresponding continuous time RT 1~(n + 1), and stays where it was with same probability. semigroup (St )t≥0 is The strong logarithmic Sobolev constant and the RT 2 spectral gap for this chain are known [24], [71]: λmin S . ( ) = n2 1 Hyp 1 More recently, it was also shown that the MLSI constant = λ(S ) = . c (SHyp) n + 1 for this transposition model satisfies the following A direct application of Corollary B.6 shows that any bounds [30]: Sym Hyp associated QMS T t 0 obtained from S t 0 via 2 4 ( t ) ≥ √ ( t ) ≥ α SRT . (89) Equation (3) satisfies HC(c, 2), with n2 ≤ 1( ) ≤ n2 n 1 log n log 2 c 2 n 1 ( + ) ( ). ≤ ( + ) + 2 ACKNOWLEDGMENT b) The finite circle:: We now consider I.B. is partially supported by French A.N.R. the simple random walk on G = Zm with m ≥ grant: ANR-14-CE25-0003 "StoQ". M.J and N.L. 4, of associated kernel k(x, x ± 1) = 1~2 and are partially supported by DMS NSF 1800872. uniform stationary measure. The spectral gap of the D.S.F. acknowledges financial support from VILLUM Cir corresponding continuous time Markov chain (St )t≥0 FONDEN via the QMATH Centre of Excellence (Grant is given by the formula [24] no. 10059), the graduate program TopMath of the Elite 2π Network of Bavaria, the TopMath Graduate Center λ SCir 1 cos ( ) = − m of TUM Graduate School at Technische Universität Cir München and by the Technische Universität München It was shown in [24] that (St )t≥0 satisfies the following bound: – Institute for Advanced Study, funded by the German Excellence Initiative and the European Union Cir 2 YSt − µG ∶ L2(Zm) → L∞(Zm)Ycb Seventh Framework Programme under grant agreement Cir 2 no. 291763. Moreover, D.S.F. acknowledges support = YSt − µG ∶ L2(Zm) → L∞(Zm)Y √ from the QuantERA ERA-NET Cofund in Quantum 5m 16π2 t m 1 2 1 exp + e−2t . Technologies implemented within the European Union’s ≤ + √ − 2 + 8 πt 5m 2 Horizon 2020 Programme (QuantAlgo project) via the In particular, in the case m ≥ 5, the above expression Innovation Fund Denmark. C.R. acknowledge financial 2 2 support from the TUM university Foundation Fellowship. yields the following simpler bound for t∞ = 5m ~16π : CR acknowledges support by the DFG cluster of SCir µ L L 2 e . Y t∞ − G ∶ 2(Zm) → ∞(Zm)Ycb ≤ excellence 2111 (Munich Center for Quantum Science Now, chose the uniform random walk of kernel and Technology). k(x, y) = 1~m for any x, y ∈ Zm. This is a special case of the Markov chain studied in Theorem A.1 of [24], for REFERENCES which the strong log-Sobolev constant c(K) was shown to be equal to6 [1] L. Saloff-Coste, “Precise estimates on the rate at which certain diffusions tend to equilibrium,” Mathematische Zeitschrift, vol. log m 1 1 217, no. 1, pp. 641–677, 1994. c K ( − ), λ S 1 . (88) ( ) = 2 4 m ( ) = − m [2] P. Diaconis, L. Saloff-Coste et al., “Comparison techniques for − ~ random walk on finite groups,” The Annals of Probability, vol. 21, c) Random Transpositions: In SectionIV no. 4, pp. 2131–2156, 1993. we considered two random transposition models on the [3] P. Diaconis and L. Saloff-Coste, “Logarithmic Sobolev inequalities for finite Markov chains,” Annals of Applied permutation group Σn: Probability, vol. 6, no. 3, pp. 695–750, 1996. [4] M. Ledoux, “On Talagrand’s deviation inequalities for product RT 1 ij L (f)(ω) = (Q f(ω) − f(σ ω) , measures,” ESAIM Probab. Statist., vol. 1, pp. 63–87, 1995/97. n ij [5] D. A. Lidar and T. A. Brun, Quantum error correction. Cambridge University Press, 2013. where σij is a swap of the ith and jth subsystems, and [6] L. Gao, M. Junge, and N. LaRacuente, “Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions,” n NN Annales Henri Poincaré, vol. 21, no. 11, pp. 3409–3478, Nov. L (f)(ω) = Q f(ω) − f(σj(j+1)ω) . 2020. j=1 [7] L. P. Rothschild and E. M. Stein, “Hypoelliptic differential We refer to [29] for operators and nilpotent groups,” Acta Math., vol. 137, no. 3-4, pp. 247–320, 1976. RT tL ε c ln n ln ε [8] R. Alicki and K. Lendi, Quantum dynamical semigroups and 1,∞ ( ) ≤ ( − ) applications. Springer, 2007, vol. 717. [9] A. Frigerio, “Stationary states of quantum dynamical 6We recall once again that our definition of the strong log-Sobolev semigroups,” Communications in Mathematical Physics, constant c(K) corresponds to 1~2α in [24]. vol. 63, no. 3, pp. 269–276, 1978.

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