SSPECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS
LI GAO AND CAMBYSE ROUZ´E
Abstract.
We propose a conceptually simple method for proving various entropic inequalitiesby means of spectral estimates. Our approach relies on a two-sided estimate of the relativeentropy between two quantum states in terms of certain variance-type quantities. As an ap-plication of this estimate, we answer three important problems in quantum information theoryand dissipative quantum systems that were left open until now: first, we prove the existence ofstrictly contractive constants for the strong data processing inequality for quantum channels thatsatisfy a notion of detailed balance. Second, we prove the existence of the complete modifiedlogarithmic Sobolev constant which controls the exponential entropic convergence of a quantumMarkov semigroup towards its equilibrium. Third, we prove a new generalization of the strongsubadditivity of the entropy for the relative entropy distance to a von Neumann algebra un-der a specific gap condition. All the three results obtained are independent of the size of theenvironment. Introduction
The quantum relative entropy as a fundamental measure is widely used in quantum infor-mation theory. For two quantum states with density matrices ρ and σ , with supp( ρ ) ⊆ supp( σ ),the relative entropy of ρ with respect to σ is defined as D ( ρ (cid:107) σ ) = tr( ρ ln ρ − ρ ln σ ) , where tr is the matrix trace. In this paper, we study several important inequalities satisfied bythe relative entropy using spectral methods.One key property of the quantum relative entropy is its monotonicity under the action of aquantum channel Φ (i.e. completely positive trace preserving map) [35, 45, 36]: for all states ρ and σ D (Φ( ρ ) (cid:107) Φ( σ )) ≤ D ( ρ (cid:107) σ ) . (1)Hence, it is natural to ask whether the decrease observed in (1) is strict. This question has beenintensively studied for classical channels and more general entropies (see e.g. [1, 18, 19, 15, 34, 42]and the references therein), and considered in [31, 37, 26] for quantum channels. In the generalsetting, one way of formulating this question is to seek for a constant c ≡ c Φ < ρ : D (Φ( ρ ) (cid:107) Φ ◦ E ∗ ( ρ )) ≤ c D ( ρ (cid:107) E ∗ ( ρ )) , (2)where E ∗ can be viewed as the projection onto the invariant states of Φ. More importantly, onemay ask whether the above inequality can be made stable against the addition of a noiseless Zentrum Mathematik, Technische Universit¨at M¨unchen, 85748 Garching, Germany
E-mail addresses : [email protected], [email protected] . a r X i v : . [ qu a n t - ph ] F e b SPECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS auxiliary system. In other words, does there exist a constant c (cid:48) ≡ c (cid:48) Φ such that, for any n ∈ N and all bipartite states ρ : D ((Φ ⊗ id n )( ρ ) (cid:107) (Φ ◦ E ∗ ⊗ id n )( ρ )) ≤ c (cid:48) D ( ρ (cid:107) ( E ∗ ⊗ id n )( ρ )) , (3)where id n denotes the identity channel on the algebra M n of n × n complex-valued matrices. Inthis paper, we will refer to (2) as the strong data processing inequality (SDPI) and to (3) as the complete strong data processing inequality (CSDPI). These are important improvements over thedata processing inequality that have applications to quantum state preparation and quantumchannel capacities [4, 9].In the context of continuous time Markovian evolutions [27], the SDPI corresponds to the so-called modified logarithmic Sobolev inequality. Let Φ t = e t L ∗ be a quantum Markov semigroupgenerated by a Lindbladian L ∗ , i.e. a continuous semigroup of quantum channels. The semigroupΦ t is said to satisfy the α - modified logarithmic Sobolev inequality ( α -MLSI) for some α > ρ , D (Φ t ( ρ ) (cid:107) E ∗ ( ρ )) ≤ e − αt D ( ρ (cid:107) E ∗ ( ρ )) . Quantum Markov semigroups model the time evolution of an important class of open quantumsystems, whereas MLSIs characterize a strong entropic convergence property for these systems. Inthe last decade, the MLSI and its connection to other functional inequalities have been intensivelystudied for quantum Markov semigroups (see e.g. [30, 37, 43, 17, 5, 11, 12]). More recently, asan analog of CSDPI, the complete version of MLSI was introduced in [22] and further studiedin [32, 7, 8]. We say that a quantum Markov semigroup Φ t satisfies the α - complete modifiedlogarithmic Sobolev inequality ( α -CMLSI) for some α > n ∈ N and all bipartite states ρ , D (Φ t ⊗ id n ( ρ ) (cid:107) E ∗ ⊗ id n ( ρ )) ≤ e − αt D ( ρ (cid:107) E ∗ ⊗ id n ( ρ )) . The motivation for introducing CMLSIs mainly goes back to the tensorization property of MLSI.In the classical case, if two semigroups S t , T t both satisfy α -MLSI, then S t ⊗ T t also satisfies α -MLSI [6]. Tensorization is a useful property that can be used to obtain MLSI for compos-ite systems in terms of the dynamics on smaller subsystems. Nevertheless, tensor stability ofMLSI fails for general non-primitive quantum Markov semigroups [7], which naturally requirethe stronger definition of CMLSI. CMLSIs are used as a key condition to study the modifiedlogarithmic Sobolev constant α of quantum Gibbs samplers on lattice spin systems (see [9]).The data processing inequality is closely related to another celebrated inequality in quantuminformation theory, namely strong subadditivity (SSA) [33]. SSA can be equivalently stated interms of relative entropies as follows: for any tripartite state ρ ABC , D (cid:16) ρ ABC (cid:13)(cid:13)(cid:13) AB d AB ⊗ ρ C (cid:17) ≤ D (cid:16) ρ ABC (cid:13)(cid:13)(cid:13) A d A ⊗ ρ BC (cid:17) + D (cid:16) ρ ABC (cid:13)(cid:13)(cid:13) B d B ⊗ ρ AC (cid:17) . Here AB d AB is the completely mixed state on AB and ρ C is the reduced density on C (and similarlyfor the other terms). In its more general form introduced by Petz [39], SSA states that givenany four matrix algebras N ⊂ N , N ⊂ M , and corresponding projections E , E , E N from M onto N , N and N , the following inequality holds: for all states ρ on M , D ( ρ (cid:107) E N ∗ ( ρ )) ≤ D ( ρ (cid:107) E ∗ ( ρ )) + D ( ρ (cid:107) E ∗ ( ρ )) . (4) PECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS 3 as long as E ◦ E = E ◦ E = E N . This last commutation property is usually referred to as a“ commuting square ” condition and was introduced by Popa [41]. It is natural to consider SSA-type inequalities when the “commuting square” condition is not fully satisfied. For instance, inthe context of classical lattice spin systems, where the projections are conditional expectationsonto different regions of the lattice with respect to a given Gibbs measure, the commuting squarecondition corresponds to the infinite temperature regime [3]. For the finite temperature regime,(4) has to be modified in the following way [14, 16]: there exists a constant c (cid:48)(cid:48) > ρ , D ( ρ (cid:107) E N ∗ ( ρ )) ≤ c (cid:48)(cid:48) (cid:0) D ( ρ (cid:107) E ∗ ( ρ )) + D ( ρ (cid:107) E ∗ ( ρ )) (cid:1) , (5)where the constant c (cid:48)(cid:48) is a measure of the violation of the commutation relation (cid:107) E ◦ E − E N (cid:107) insome appropriate norm. This inequality, called approximate tensorization of the relative entropy ,was recently considered in the quantum setting by one of the authors [3]. There, a bound similarto (5) was derived with a further additive error term vanishing on classical states. However, thegeneral question of finding bounds like (5) without additive error term was left unresolved.In this paper, we study the entropy inequalities mentioned above and answer the followingopen problems:(i) the existence of a complete strong data processing inequality on matrix algebras (seeSection 3);(ii) the existence of the complete modified logarithmic Sobolev inequality on matrix algebras(see Section 4);(iii) the approximate tensorization of the relative entropy (see Section 5).All our proofs rely on a conceptually simple tool, namely a two-sided estimate of the relativeentropy functional via a so-called Bogoliubov-Kubo-Mori Fisher information (see Lemma 2 in Sec-tion 2 for more details). This control allows us to approach each of the three constants describedabove by means of certain spectral gap conditions for the maps considered. Given the simplicityof our approach, we believe it will also prove useful in other areas of quantum information theory.
Note:
The existence of CMLSI constant for self-adjoint semigroups on matrix algebras were alsoindependently obtained by the first author, Marius Junge and Haojian Li [24] using a completelydifferent method.
Aknowledgements.
CR is supported by a Junior Researcher START Fellowship from the MC-QST. CR is grateful to Daniel Stilck Fran¸ca, Angela Capel and Ivan Bardet for stimulatingdiscussions. They particularly thank Daniel Stilck Fran¸ca for very useful comments on a prelim-inary version of the paper. Li Gao thanks Marius Junge and Haojian Li for helpful discussions.2.
Preliminaries
Relative Entropy and Conditional expectation.
Let H be a finite dimensional Hilbertspace and B ( H ) (resp. T ( H )) be the bounded operators (resp. trace class operators) on H . Wedenote tr for the standard matrix trace, (cid:104)· , ·(cid:105) HS for the trace inner product and (cid:107) · (cid:107) for theHilbert-Schmidt norm. The corresponding Hilbert-Schmidt space is denoted by T ( H ). Wewrite a † for the adjoint of an operator a ∈ B ( H ), and Φ ∗ (or Φ ∗ ) for the adjoint of a map T : B ( H ) → B ( H ). The identity operator on H is denoted as H and the identity map on B ( H )is id H . SPECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS
We say that an operator ρ is a state (or density operator) if ρ ≥ ρ ) = 1. We denoteby D ( H ) the set of states on H . For two states ρ and σ , their relative entropy is defined as D ( ρ (cid:107) σ ) = (cid:40) tr( ρ ln ρ − ρ ln σ ) , if supp( ρ ) ⊆ supp( σ )+ ∞ , otherwise , where supp( ρ ) (resp. supp( σ )) is the support projection of ρ (resp. σ ). Let N ⊂ B ( H ) be avon Neumann subalgebra. Recall that a conditional expectation onto N is a completely positiveunital map E N : B ( H ) → N onto N satisfyingi) for all X ∈ N , E N ( X ) = X ii) for all a, b ∈ N , X ∈ B ( H ), E N ( aXb ) = aE N ( X ) b .We denote by E N ∗ its adjoint map with respect to the trace inner product, i.e.tr( E N ∗ ( X ) Y ) = tr( XE N ( Y )) . For a state ρ , the relative entropy with respect to N is defined as follows D ( ρ (cid:107)N ) := D ( ρ (cid:107) E N ∗ ( ρ )) = inf E N∗ ( σ )= σ D ( ρ (cid:107) σ ) , where the infimum is always attained by E N ∗ ( ρ ). Indeed, for any σ satisfying E N ∗ ( σ ) = σ , wehave the identity (see [28, Lemma 3.4]) D ( ρ (cid:107) σ ) = D ( ρ (cid:107) E N ∗ ( ρ )) + D ( E N ∗ ( ρ ) (cid:107) σ ) . Hence the infimum is attained if and only if D ( E N ∗ ( ρ ) (cid:107) σ ) = 0. More explicitly, a finite di-mensional von Neumann (sub)algebra is always given by a direct sum of matrix algebras withmultiplicity, i.e. N = n (cid:77) i =1 B ( H i ) ⊗ C K i , H = n (cid:77) i =1 H i ⊗ K i . Denote P i as the projection onto H i ⊗ K i . There exists a family of density operators τ i ∈ D ( K i )such that E N ( X ) = n (cid:77) i =1 tr K i ( P i XP i ( K i ⊗ τ i )) ⊗ K i , E N ∗ ( ρ ) = n (cid:77) i =1 tr K i ( P i ρP i ) ⊗ τ i , (6)where tr K i is the partial trace with respect to K i . Write d H = dim( H ). A state σ satisfies E N ∗ ( σ ) = σ if and only if σ = n (cid:77) i =1 p i σ i ⊗ τ i for some density operator σ i ∈ D ( H i ) and a probability distribution { p i } ni =1 . Denote D ( E N ) := { σ ∈ D ( H ) | E N ∗ ( σ ) } as the subset of states that are invariant under E N ∗ . For any σ ∈ D ( E N ), E N ∗ ( σ Xσ ) = σ E N ( X ) σ . PECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS 5
Subalgebra index and Max-relative entropy.
Let
M ⊂ B ( H ) be a finite dimensionalvon Neumann algebra and N ⊂ M be a subalgebra. The trace preserving conditional expectation E N , tr : M → N is defined so that for any X ∈ M and Y ∈ N ,tr( XY ) = tr( E N , tr ( X ) Y ) .E N , tr is self-adjoint and corresponds to taking τ i = d − K i K i in (6). We recall the definition of theindex associated to the algebra inclusion N ⊂ M , C ( M : N ) = inf { c > | ρ ≤ c E N , tr ( ρ ) for all states ρ ∈ M} ,C cb ( M : N ) = sup n ∈ N C ( M ⊗ M n : N ⊗ M n ) , where the supremum in C cb ( M : N ) is taken over all finite dimensional matrix algebras M n . Theindex C ( M : N ) was first introduced by Popa and Pimsner in [40] for the connection to subfactorindex and Connes entropy, and the completely bounded version C cb ( M : N ) was studied in [23].These indices are closely related to the notion of maximal relative entropy. Recall that for twostates ρ, ω , their maximal relative entropy is D max ( ρ (cid:107) ω ) = ln inf { c > | ρ ≤ c ω } . Indeed, ln C ( M : N ) = sup ρ ∈D ( E M , tr ) D max ( ρ (cid:107) E N , tr ( ρ )) . For all finite dimensional inclusion
N ⊂ M , the index C ( M : N ) is explicitly calculated in [40,Theorem 6.1] (hence also for C cb ( M : N )). In particular, for M = B ( H ) and N = (cid:76) ni =1 B ( H i ) ⊗ C K i , C ( B ( H ) : N ) = n (cid:88) i =1 min { d H i , d K i } d K i , C cb ( B ( H ) : N ) = n (cid:88) i =1 d K i . (7)For example, if we take D ⊂ B ( H ) to be the subalgebra of diagonal matrices and C as themultiple of identity C ( B ( H ) : D ) = C cb ( B ( H ) : D ) = d H ,C ( B ( H ) : C ) = d H , C cb ( B ( H ) : C ) = d H . In this paper, we consider a generalization of these indices for a general conditional expectation E N : M → N . We recall that here M n is the n -dimensional matrix algebra and E N ⊗ id M n is aconditional expectation from M ⊗ M n → N ⊗ M n . Denote τ = n (cid:77) i =1 H i ⊗ τ i . (8)Note that E N and E N ∗ are uniquely defined by τ as follows, E N ( X ) = E N , tr ( τ Xτ ) , E N ∗ ( ρ ) = τ E N , tr ( ρ ) τ . (9)In particular, E N is faithful if and only if τ is. Next, we define C τ ( M : N ) := inf c { c > | ρ ≤ c E N ∗ ( ρ ) for all states ρ ∈ M} C τ, cb ( M : N ) := sup n ∈ N C τ ⊗ n ( M ⊗ M n : N ⊗ M n ) . (10) SPECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS
Since τ commutes with N , C τ ( M : N ) ≤ µ min ( τ ) − C ( M : N ) , C τ, cb ( M : N ) ≤ µ min ( τ ) − C cb ( M : N ) (11)where µ min ( τ ) = min i µ min ( τ i ) is the minimal eigenvalue of τ . Combined with (7), this implies C τ ( M : N ) and C τ, cb ( M : N ) are finite iff τ is faithful. Moreover, for any invariant state σ ∈ D ( E N ), by the obvious bound σ ≤ τ , we also have C τ ( M : N ) ≤ µ min ( σ ) − C ( M : N ) , C τ, cb ( M : N ) ≤ µ min ( σ ) − C cb ( M : N ) . (12)2.3. A key lemma.
We shall now discuss the key lemma that will be used in the later sections.Given a density operator ρ ∈ D ( H ), we define the multiplication operatorΓ ρ ( X ) := (cid:90) ρ s X ρ − s ds . Γ ρ is a positive operator on the Hilbert-Schmidt space T ( H ) := L ( B ( H ) , tr) and hence inducesa weighted L -norm (semi-norm if ρ is not invertible) defined for X ∈ B ( H ) as (cid:107) X (cid:107) ρ := (cid:104) X, Γ ρ ( X ) (cid:105) HS = (cid:90) tr( X † ρ s Xρ − s ) ds . We denote by L ( ρ ) the corresponding L -space. The inverse operator of Γ ρ is given byΓ − ρ ( X ) := (cid:90) ∞ ( ρ + r ) − X ( ρ + r ) − dr , which is the double operator integral for the function f ( t ) = ln t and operator ρ (see e.g. [11]).For an invertible density ρ , we denote by slight abuse of notations the corresponding weighted L -norm as (cid:107) X (cid:107) ρ − := (cid:104) X, Γ − ρ ( X ) (cid:105) HS = (cid:90) ∞ tr( X † ( ρ + r ) − X ( ρ + r ) − ) dr . and the corresponding L space as L ( ρ − ). This is a special case of quantum χ divergenceintroduced in [44, Defnition 1] for the logarithmic function. It is easy to see that (cid:107) Γ ρ ( X ) (cid:107) ρ − = (cid:107) X (cid:107) ρ , (cid:107) Γ − ρ ( X ) (cid:107) ρ = (cid:107) X (cid:107) ρ − . Lemma 1. If ρ ≤ c σ for any two states ρ, σ and some c > , then for any X ∈ M and all µ , µ > , (cid:90) ∞ tr( X † ( µ σ + r ) − X ( µ σ + r ) − ) dr ≤ c (cid:90) ∞ tr( X † ( µ ρ + r ) − X ( µ ρ + r ) − ) dr . In particular, (cid:107) X (cid:107) σ − ≤ c (cid:107) X (cid:107) ρ − . PECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS 7
Proof.
This is a standard comparison. Using cyclicity of the trace and the fact that t (cid:55)→ t − isoperator anti-monotone, (cid:90) ∞ tr( X † ( µ ρ + r ) − X ( µ ρ + r ) − ) dr ≥ (cid:90) ∞ tr( X † ( cµ σ + r ) − X ( µ ρ + r ) − ) dr ≥ (cid:90) ∞ tr( X † ( cµ σ + r ) − X ( cµ σ + r ) − ) dr = (cid:90) ∞ c tr( X † ( µ σ + rc ) − X ( µ σ + rc ) − ) dr = 1 c (cid:90) ∞ tr( X † ( µ σ + r ) − X ( µ σ + r ) − ) dr . In the last equality, we used the change of variable r → rc . (cid:3) Our key lemma is a two-sided estimate of D ( ρ (cid:107) σ ) via the inverse weighted norm. Lemma 2.
Let ρ and σ be two invertible density operators and suppose ρ ≤ c σ for some c > .Then k ( c ) (cid:107) ρ − σ (cid:107) σ − ≤ D ( ρ (cid:107) σ ) ≤(cid:107) ρ − σ (cid:107) σ − (13) where k ( c ) = c ln c − c + 1( c − . Note that k ( c ) ≤ / for c ≥ .Proof. For the lower bound, we consider ρ t = (1 − t ) σ + tρ, t ∈ [0 ,
1] and the function f ( t ) = D ( ρ t (cid:107) σ ). We have f (0) = 0, f (1) = D ( ρ (cid:107) σ ) and [31] f (cid:48) ( t ) = tr(( ρ − σ ) ln ρ t − ( ρ − σ ) ln σ ) ,f (cid:48)(cid:48) ( t ) = (cid:90) ∞ tr (cid:16) ( ρ − σ ) 1 ρ t + r ( ρ − σ ) 1 ρ t + r (cid:17) dr = (cid:107) ρ − σ (cid:107) ρ − t . Note that f (cid:48) (0) = 0 and ρ t ≤ ( ct + (1 − t )) σ . We have for the lower bound D ( ρ (cid:107) σ ) = (cid:90) (cid:16) (cid:90) s f (cid:48)(cid:48) ( t ) dt (cid:17) ds = (cid:90) (cid:90) s (cid:107) ρ − σ (cid:107) ρ − t dtds ≥ (cid:90) (cid:90) s
11 + ( c − t dtds (cid:107) ρ − σ (cid:107) σ − ≥ k ( c ) (cid:107) ρ − σ (cid:107) σ − , where we used Lemma 1 and k ( c ) = (cid:90) (cid:90) s
11 + ( c − t dtds = c ln c − c + 1( c − . SPECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS
The upper bound is special case of [44, Proposition 6]. Here we present a different proof usingsimilar idea from lower bound. Note that ρ t = (1 − t ) σ + tρ ≥ (1 − t ) σ . Then, D ( ρ (cid:107) σ ) = (cid:90) (cid:90) s (cid:107) ρ − σ (cid:107) ρ − t dtds ≤ (cid:90) (cid:90) s − t (cid:107) ρ − σ (cid:107) σ − dtds = (cid:90) (cid:90) s − t dtds (cid:107) ρ − σ (cid:107) σ − . Note that the upper bound does not need the assumption ρ ≤ c σ . (cid:3) Now given a conditional expectation E N : B ( H ) → N , it follows immediately from the abovethat for any state ρ and ρ N = E N ∗ ( ρ ), k ( C τ ( M : N )) (cid:107) ρ − ρ N (cid:107) ρ − N ≤ D ( ρ (cid:107) ρ N ) ≤(cid:107) ρ − ρ N (cid:107) ρ − N , (14)where C τ ( M : N ) is the index defined in (10).3. Complete strong data processing inequalities
In this section, we provide universal bounds on the complete strong data processing inequalityconstant in the case of a quantum channel satisfying the detailed balance condition. We firstdiscuss the notion of detailed balance and its connection to the spectral gap. Given a faithfulstate σ and 0 ≤ s ≤
1, we define the multiplication operatorΓ σ,s ( X ) = σ − s Xσ s . Γ σ,s is a positive operator on the Hilbert-Schmidt space and induces the following weighted innerproduct (cid:104) X, Y (cid:105) σ,s := tr( X † σ − s Y σ s ) , (cid:107) X (cid:107) σ,s = (cid:104) X, X (cid:105) σ,s . We denote by L ( σ, s ) the corresponding L space. A map Φ ∗ : B ( H ) → B ( H ) is self-adjointwith respect to (cid:104)· , ·(cid:105) σ,s if Φ ◦ Γ σ,s = Γ σ,s ◦ Φ ∗ , where Φ is the adjoint of Φ ∗ for the trace inner product. Denote H = − ln σ, ∆ σ ( X ) = σXσ − , α t ( X ) = e itH Xe − itH , t ∈ C as the modular generator, modular operator, and modular automorphism group of σ respectively.It was proved in [11, Theorem 2.9] that under the assumption Φ ∗ ( a † ) = (Φ ∗ ( a )) † , Φ ∗ is self-adjointwith respect to (cid:104)· , ·(cid:105) σ,s for some s (cid:54) = 1 / ∗ commutes with ∆ σ and hence Φ ∗ isself-adjoint with respect to (cid:104)· , ·(cid:105) σ,s for all s ∈ [0 , ∗ satisfies σ -DBC (detailedbalance condition) if Φ ∗ is self-adjoint with respect to (cid:104)· , ·(cid:105) σ, . Note thatΓ σ = (cid:90) Γ σ,s ds . Thus, we have Γ σ ◦ Φ ∗ = Φ ◦ Γ σ and hence Γ − σ ◦ Φ = Φ ∗ ◦ Γ − σ if Φ ∗ satisfies the σ -DBC. PECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS 9
Let E N : M → N be a conditional expectation. It can be readily seen that E N satisfies the σ -DBC condition for all σ ∈ D ( E N ) (satisfying σ = E N ∗ ( σ )). Hence ∀ s ∈ [0 , , Γ σ,s ◦ E N = E N ∗ ◦ Γ σ,s ⇒ Γ σ ◦ E N = E N ∗ ◦ Γ σ . In particular, E N is the projection onto N for the L -norms (cid:107) · (cid:107) σ,s for any s ∈ [0 ,
1] and (cid:107) · (cid:107) σ ,for all σ ∈ D ( E N ). Indeed, for any X ∈ M , (cid:104) E N ( X ) , X − E N ( X ) (cid:105) σ,s = (cid:104) Γ σ,s ◦ E N ( X ) , X − E N ( X ) (cid:105) HS = (cid:104) E N ∗ ◦ Γ σ,s ( X ) , X − E N ( X ) (cid:105) HS = (cid:104) Γ σ,s ( X ) , E N ( X − E N ( X )) (cid:105) HS = 0 . Now, let Φ : M ∗ → M ∗ be a quantum channel, i.e. a completely positive trace preservingmap. Its adjoint Φ ∗ is a completely positive unital map. Let N be the multiplicative domain ofΦ ∗ , i.e. N := { a ∈ M | Φ ∗ ( aa † ) = Φ ∗ ( a )Φ ∗ ( a † ) , Φ ∗ ( a † a ) = Φ ∗ ( a † )Φ ∗ ( a ) } . Then Φ ∗ restricted on N is a ∗ -isomorphism (see [47, Theorem 5.7]). Lemma 3.
Let
Φ : M ∗ → M ∗ be a quantum channel and let N be the multiplicative domain of Φ ∗ . Then,i) There exists an invariant state σ such that Φ( σ ) = σ If in addition σ is full-rank and Φ ∗ satisfies σ - DBC ,ii) Φ ∗ is a contraction on L ( σ, s ) for any s ∈ [0 , and L ( σ ) . Φ ∗ restricted on N is a ∗ -isomorphism and an L -isometry on L ( σ, s ) for all s ∈ [0 , , as well as on L ( σ ) .iii) Let E N : M → N be the conditional expectation such that E N ∗ ( σ ) = σ . Then Φ ∗ ◦ E N = E N ◦ Φ ∗ , (Φ ∗ ) ◦ E N = E N ◦ (Φ ∗ ) = E N . Proof. i) Viewing Φ as a linear map, Φ has eigenvalue 1 because Φ ∗ ( ) = . Since Φ preservesself-adjointness, we have an operator a = a † such that Φ( a ) = a . Let a + (resp. a − ) be thepositive (resp. negative) part of a . We have Φ( a ) = Φ( a + ) − Φ( a − ) = a . Because Φ is positiveand trace preserving, Φ( a + ) and Φ( a − ) are positive andtr(Φ( a + )) + tr(Φ( a − )) = tr( a + ) + tr( a − ) = (cid:107) a (cid:107) . This implies Φ( a + ) = a + and Φ( a − ) = a − , which proves i). For any X , (cid:107) Φ ∗ ( X ) (cid:107) σ,s = tr (cid:16) Φ ∗ ( X † ) σ − s Φ ∗ ( X ) σ s (cid:17) = tr (cid:16) Φ ∗ ( α i − s ( X ) † )Φ ∗ ( α i − s ( X )) σ (cid:17) ≤ tr (cid:16) Φ ∗ ( α i − s ( X ) † α i − s ( X )) σ (cid:17) = tr (cid:16) α i − s ( X ) † α i − s ( X ) σ (cid:17) = (cid:107) X (cid:107) σ,s Here, in the inequality we used Kadison-Schwarz inequality and the second to last equalityfollows from Φ( σ ) = σ . Note that α s ( N ) = N for any s ∈ C . Then for any X ∈ N ,Φ ∗ ( α i − s ( X ) † )Φ ∗ ( α i − s ( X )) = Φ ∗ ( α i − s ( X ) † α i − s ( X )) and the above inequality becomes an equality. This proves ii) for L ( σ, s ) for all s ∈ [0 , L ( σ ) follows by integra-tion. For iii), we first note that for any X ∈ N , Φ ∗ Φ ∗ X = X . Indeed, (cid:104) Φ ∗ Φ ∗ X, X (cid:105) σ,s = (cid:104) Φ ∗ X, Φ ∗ X (cid:105) σ,s = (cid:107) X (cid:107) σ,s . This further implies Φ ∗ ( X ) ∈ N is in the multiplicative domain becauseΦ ∗ (Φ ∗ ( X † )Φ ∗ ( X )) = Φ ∗ (Φ ∗ ( X † X )) = X † X = (Φ ∗ ) ( X † )(Φ ∗ ) ( X † ) . Also, Φ ∗ is invariant on the orthogonal complement of N because (cid:104) X, Φ ∗ ◦ (id − E N )( Y ) (cid:105) σ,s = (cid:104) Φ ∗ ( X ) , (id − E N )( Y ) (cid:105) σ,s = 0 . That completes the proof. (cid:3)
We see from the above lemma that under σ -DBC, Φ ∗ is a self-adjoint contraction on L ( σ, s )(also L ( σ )), and N is the union of the eigenspace of Φ ∗ for eigenvalue 1 and −
1. The eigenspacefor eigenvalue 1 is the fixed point space of Φ ∗ , which is a subalgebra F ⊂ N . For each invariantstate σ = Φ( σ ), we have σ = E F∗ ( σ ). In finite dimensions, there always exists 0 < (cid:15) < (cid:107) Φ ∗ (id − E N ) : L ( σ, s ) → L ( σ, s ) (cid:107)≤ (1 − (cid:15) ) , which is a spectral gap condition. The next lemma shows that this spectral gap condition isindependent of s ∈ [0 ,
1] and of the choice of invariant state σ . Lemma 4.
Let
Φ : M ∗ → M ∗ be a quantum channel and Φ ∗ be its adjoint. Suppose Φ ∗ satisfy σ - DBC for some full-rank invariant state σ satisfying Φ( σ ) = σ . Then,i) (Φ ∗ ) satisfies ρ - DBC for all states ρ ∈ D ( E N ) and Φ ∗ satisfies ρ - DBC for all invariantstates ρ .ii) For each ρ ∈ D ( E N ) , denote λ ( ρ, s ) = (cid:107) Φ ∗ (id − E N ) : L ( ρ, s ) → L (Φ( ρ ) , s ) (cid:107) . Then forall s ∈ [0 , λ ( ρ, s ) = λ ( σ, . iii) For each ρ ∈ D ( E N ) , denote λ ( ρ ) = (cid:107) Φ ∗ (id − E N ) : L ( ρ ) → L (Φ( ρ )) (cid:107) . Then λ ( ρ ) ≤ λ ( σ,
1) = λ ( σ ) . Proof.
By Lemma 3, (Φ ∗ ) | N is the identity map and has the module property(Φ ∗ ) ( aXb ) = a (Φ ∗ ) ( X ) b , ∀ a, b ∈ N . Note that for any two states ρ, σ ∈ D ( E N ), ρ − s σ s ∈ N for any real s . Therefore, we have for all s ∈ [0 , ρ,s ◦ (Φ ∗ ) ◦ Γ − ρ,s = Γ σ,s ◦ (Φ ∗ ) ◦ Γ − σ,s = Φ . This shows (Φ ∗ ) satisfies ρ -DBC. Now consider ρ satisfies Φ( ρ ) = ρ . Because both ρ, σ ∈ D ( E F ),we have ρ − s σ s ∈ F for any real s . Then it follows from the same argument above that Φ ∗ satisfies ρ -DBC. For ii), we denote ι = Φ ∗ | N to be the involution Φ ∗ restricted to N . Note that for anyreal r , (this can be verified directly by finite dimensional direct sum structure) ι ( ρ − r σ r ) = Φ( ρ ) − r σ r (15) PECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS 11 where ρ ◦ ι = Φ( ρ ). For a mean zero element Y = X − E N ( X ), (cid:107) Y (cid:107) ρ,s = (cid:107) Γ / ρ,s ( Y ) (cid:107) = (cid:107) Γ / σ,s Γ − / σ,s Γ / ρ,s ( Y ) (cid:107) = (cid:107) Γ / σ,s ( Y ) (cid:107) = (cid:107) Y (cid:107) σ,s where Y = Γ − / σ,s Γ / ρ,s ( Y ) is also a mean zero element in N ⊥ . Moreover, (cid:107) Φ ∗ ( Y ) (cid:107) σ,s = (cid:107) Γ / σ,s Γ − / σ,s Γ / ρ ) ,s Φ ∗ ( Y ) (cid:107) = (cid:107) Γ / ρ ) ,s Φ ∗ ( Y ) (cid:107) = (cid:107) Φ ∗ ( Y ) (cid:107) ρ ) ,s where we used (15) in the first line. This proves λ ( ρ, s ) = λ ( σ, s ) for each s . For the independenceof s , we have for r ∈ [0 , (cid:107) Φ ∗ ( Y ) (cid:107) σ,s = tr (cid:2) Φ ∗ ( Y ) † σ − s Φ ∗ ( Y ) σ s (cid:3) = tr (cid:2) Φ ∗ ( α i r − s ( Y )) † σ − r Φ ∗ ( α i r − s ( Y )) σ r (cid:3) = (cid:107) Φ ∗ (cid:0) α i r − s ( Y ) (cid:1) (cid:107) σ,r where α i r − s ( Y ) = α i r − s ( X − E N ( X )) = α i r − s ( X ) − E N ( α i r − s ( X )) is also in N ⊥ . Moreover, (cid:107) Y (cid:107) σ,s = (cid:107) α i r − s ( Y ) (cid:107) σ,r . For iii), the inequality λ ( ρ ) ≤ λ ( σ,
1) follows from integrating the (cid:104)· , ·(cid:105) ρ,s inner product to obtain (cid:104)· , ·(cid:105) ρ . The equality λ ( σ,
1) = λ ( σ ) follows from the fact the map Φ ∗ (id − E N ) is self-adjoint withrespect to both (cid:104)· , ·(cid:105) σ (cid:104)· , ·(cid:105) σ,s for any s ∈ [0 , (cid:107) Φ ∗ (id − E N ) (cid:107) as the maximaleigenvalue is independent of choice of Hilbert space norm chosen. (cid:3) Let Φ : M ∗ → M ∗ be a quantum channel and N be its multiplicative domain. We sayΦ satisfies c -strong data processing inequality ( c -SDPI) for some 0 < c < ρ ∈ D ( H ), D (Φ( ρ ) (cid:107) Φ ◦ E N ∗ ( ρ )) ≤ c D ( ρ (cid:107) E N ∗ ( ρ )) . (16)We say Φ satisfies the c (cid:48) -complete strong data processing inequality ( c (cid:48) -CSDPI) for some 0 Theorem 5. Let Φ : M ∗ → M ∗ be a quantum channel and N be the multiplicative domain of Φ ∗ . Assume that Φ ∗ satisfies the σ - DBC for some faithful invariant state σ = Φ( σ ) . Denote the spectral gap λ (Φ) := (cid:107) Φ ∗ (id − E N ) : L ( σ ) → L ( σ ) (cid:107) < . There exists an explicit constant c ( C τ, cb ( M : N ) , λ ) < such that λ (Φ) ≤ c CSDPI (Φ) ≤ c ( C τ, cb ( M : N ) , λ (Φ)) . (18) The same estimate holds for c SDPI (Φ) simply replacing C τ, cb ( M : N ) by C τ ( M : N ) .Proof. We first show the lower bound. Write λ ≡ λ (Φ). Taking a self-adjoint mean zero element X = Y − E N ( Y ), we have that Γ σ ( X ) is a self-adjoint element satisfying E N ∗ ( X ) = 0. Denote σ t = (1 − t ) σ + t Γ σ ( X ). Because σ is full rank, there exists (cid:15) > t ∈ [0 , (cid:15) ], σ t isa state. Now assume Φ satisfy SDPI for c = c SDPI (Φ). We have D (Φ( σ t ) (cid:107) σ ) ≤ cD ( σ t (cid:107) σ ) . Consider the function f ( t ) = cD ( σ t (cid:107) σ ) − D (Φ( σ t ) (cid:107) σ ). Taking derivatives, we have f (0) = f (cid:48) (0) =0, and [31] f (cid:48)(cid:48) (0) = c (cid:107) Γ σ X (cid:107) σ − − (cid:107) Φ(Γ σ X ) (cid:107) σ − . Note that f (cid:48)(cid:48) (0) ≥ f ( t ) ≥ t ∈ [0 , (cid:15) ]. Therefore, (cid:107) Φ ∗ ( X ) (cid:107) σ = (cid:107) Φ(Γ σ X ) (cid:107) σ − ≤ c (cid:107) Γ σ X (cid:107) σ − = c (cid:107) X (cid:107) σ − for arbitrary self-adjoint mean zero element X . This proves the lower bound λ (Φ) ≤ c SDPI (Φ) ≤ c CSDPI (Φ) . For the upper bound, denote ρ N = E N ∗ ( ρ ), ρ t = tρ + (1 − t ) ρ N and g ( t ) = D ( ρ t (cid:107) ρ N ) − D (Φ( ρ t ) (cid:107) Φ( ρ N )). We have g (0) = g (cid:48) (0) = 0, and g (cid:48)(cid:48) ( t ) = (cid:107) ρ − ρ N (cid:107) ρ − t − (cid:107) Φ( ρ ) − Φ( ρ N ) (cid:107) ρ ) − t . It follows from [31, Example 2] that g (cid:48)(cid:48) ( t ) ≥ 0. Using the spectral gap condition, Lemmas 4 and1, we also have g (cid:48)(cid:48) ( t ) = (cid:107) ρ − ρ N (cid:107) ρ − t − (cid:107) Φ( ρ ) − Φ( ρ N ) (cid:107) ρ ) − t ≥ 11 + ( C − t (cid:107) ρ − ρ N (cid:107) ρ − N − − t (cid:107) Φ( ρ ) − Φ( ρ N ) (cid:107) ρ N ) − ≥ (cid:16) 11 + ( C − t − λ − t (cid:17) (cid:107) ρ − ρ N (cid:107) ρ − N . where C = C τ ( M : N ). Thus we have for t := − λ λ ( C − , g (cid:48)(cid:48) ( t ) ≥ (cid:40)(cid:16) C − t − λ − t (cid:17) (cid:107) ρ − ρ N (cid:107) ρ − N , t ≤ t , t > t Denote a ( s ) := (cid:90) s 11 + ( C − t − λ − t dt = ln(1 + ( C − s ) C − λ ln(1 − s ). Since g (cid:48) (0) = 0, wehave g (cid:48) ( s ) ≥ a ( s ) (cid:107) ρ − ρ N (cid:107) ρ − N if s ≤ t and g (cid:48) ( s ) ≥ a ( t ) if s ≥ t . Denote b ( t ) := (1 + ( C − t ) ln(1 + ( C − t ) − ( C − t ( C − − λ ((1 − t ) ln(1 − t ) + t ) = (cid:90) t a (cid:48) ( s ) ds. PECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS 13 Thus we have, D ( ρ (cid:107) ρ N ) − D (Φ( ρ ) (cid:107) Φ( ρ N )) = g (1) − g (0) = (cid:90) g (cid:48) ( s ) ds ≥ (cid:0) (1 − t ) a ( t ) + b ( t ) (cid:1) (cid:107) ρ − ρ N (cid:107) ρ − N ≥ (cid:0) (1 − t ) a ( t ) + b ( t ) (cid:1) D ( ρ (cid:107) ρ N ) , where the last inequality follows from Lemma 2. The SDPI constant is then upper bounded by c = 1 − (1 − t ) a ( t ) − b ( t ) < . The same argument holds for the complete bounded constant by replacing C = C τ ( M : N ) by C τ, cb ( M : N ). (cid:3) Remark 6. For primitive unital quantum channels, the SDPI constant obtained in [37] is gener-ically better than the bounds found in Theorem 5. Nevertheless, our results give explicit SDPIconstants for general non-egordic GNS-symmetric quantum channels, and they are moreover in-dependent of the size of the environment. In particular, it is easy to see that the CSDPI constantsatisfies the tensorization property. Remark 7. In the last step of the proof, we obtained the following improved data processinginequality (DPI) D ( ρ (cid:107) ρ N ) − D (Φ( ρ ) (cid:107) Φ( ρ N )) ≥ (cid:0) (1 − t ) a ( t ) + b ( t ) (cid:1) (cid:107) ρ − ρ N (cid:107) ρ − N . This in particular implies that D ( ρ (cid:107) ρ N ) = D (Φ( ρ ) (cid:107) Φ( ρ N )) is saturated if and only if ρ = ρ N .This gives a new form of refinement of DPI, which is different from recovery results (see e.g.[29, 13, 25] with correction term related to the recovery error).4. Complete modified logarithmic Sobolev inequalities In this section, we use our key estimates found in Subsection 2.3 in order to derive a simpleproof of the complete modified logarithmic Sobolev inequality (CMLSI) for quantum Markovsemigroups on finite dimensional matrix algebras. A quantum Markov semigroup (QMS) ( P t ) t ≥ is a continuous semigroup of completely positive, unital maps acting on a von Neumann algebra M . Such a semigroup is characterised by its generator, called the Lindbladian L , which is definedon M by L ( X ) = lim t → t ( P t ( X ) − X ) for all X ∈ M , so that P t = e t L for all t ≥ 0. TheQMS is said to be primitive if it admits a unique full-rank invariant state σ . In this section, weexclusively study QMS that satisfy the following detailed balance condition with respect to some(possibly non-unique) full-rank invariant state σ . This detailed balance condition is also referredto as GNS-symmetry: for any X, Y ∈ M and any t ≥ σ X † P t ( Y )) = tr( σ P t ( X ) † Y ) . ( σ -DBC)Under this condition, there exists a conditional expectation E F : M → F onto the fixed pointalgebra F ⊂ M of ( P t ) t ≥ such that [20] e t L → t →∞ E F . Moreover, the generator L can be written as L ( X ) = (cid:88) j (cid:16) e − ω j / A † j [ X, A j ] + e ω j / [ A j , X ] A † j (cid:17) , (19)for some real parameters ω j such that for any invariant state σ , ∆ σ ( A j ) := σA j σ − = e − ω j A j (see [11]). In this section, we are interested in the exponential convergence in relative entropyof the semigroup towards its corresponding conditional expectation. The entropy production (sometimes also refered to as Fisher information ) of ( P t = e t L ) t ≥ is defined as the opposite ofthe derivative of the relative entropy with respect to the invariant state: for any ρ ∈ D ( E M ),EP L ( ρ ) := − ddt (cid:12)(cid:12)(cid:12)(cid:12) t =0 D ( P t ∗ ( ρ ) (cid:107) E F∗ ( ρ )) = − tr( L ∗ ( ρ )(ln ρ − ln E F∗ ( ρ ))) . We are interested in the uniform exponential convergence in relative entropy of systems evolvingaccording to a QMS towards equilibrium: more precisely, we ask the question of the existence ofa positive constant α > ρ ∈ D ( E M ), D ( P t ∗ ( ρ ) (cid:107) E F∗ ( ρ )) ≤ e − αt D ( ρ (cid:107) E F∗ ( ρ )) . After differentiation at t = 0 and using the semigroup property, this inequality is equivalent tothe following modified logarithmic Sobolev inequality (MLSI): for any ρ ∈ D ( E M ), α D ( ρ (cid:107) E F∗ ( ρ )) ≤ EP L ( ρ ) . (MLSI)The best constant α achieving this bound is called the modified logarithmic Sobolev constant ofthe semigroup, and is denoted by α MLSI ( L ). We may also consider the complete version whichrequires α CMLSI ( L ) D ( ρ (cid:107) ( E F∗ ⊗ id)( ρ )) ≤ EP ( L⊗ id) ( ρ ) (CLSI)to hold for all states ρ on M ⊗ B ( H R ) for any reference system R (or even B ( H R ) replaced by afinite von Neumann algebra). In [28], it was shown that the proof of the positivity of α CMLSI forall GNS-symmetric quantum Markov semigroups can be reduced to that for symmetric quantumMarkov semigroups, that is to those for which L = L ∗ . However, the problem of the positivityof the constant for symmetric QMS has been left open despite considerable work delved on thattopic in the recent years (see e.g. [22, 7, 8, 46]). Here, we finally provide a positive answer tothe question as a simple application of our key estimates from Subsection 2.3. First, we recallthat the Dirichlet form associated to L takes the following simple form [11, Section 5]: for anyinvariant state σ = E F∗ ( σ ), E σ ( X ) := −(cid:104) X, L ( X ) (cid:105) σ = (cid:88) j (cid:90) e ( − s ) ω j (cid:104) ∂ j ( X ) , ∂ j ( X ) (cid:105) σ,s ds , (20)where ∂ j ( X ) := [ A j , X ]. We denote (cid:107) X (cid:107) σ,ω j := (cid:90) e ( − s ) ω j (cid:104) ∂ j ( X ) , ∂ j ( X ) (cid:105) σ,s ds ⇒ E σ ( X ) = (cid:88) j (cid:107) ∂ j ( X ) (cid:107) σ,ω j . (21) PECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS 15 On the other hand, the entropy production associated to L also takes a simpler form (see [28,Lemma 2.3]): for any invariant state σ = E F∗ ( σ ),EP L ( ρ ) = (cid:88) j (cid:107) Γ σ, ◦ ∂ j ◦ Γ − σ, ( ρ ) (cid:107) ρ − ,ω j , (22)where for any X ∈ M : (cid:107) X (cid:107) σ − ,ω j = (cid:90) ∞ tr (cid:104) X † (e − ωj ρ + u ) − X (e ωj ρ + u ) − (cid:105) du . We denote the kernels corresponding to the inner products (cid:107) . (cid:107) σ,ω j and (cid:107) . (cid:107) σ − ,ω j by Γ σ,ω j , resp.Γ σ − ,ω j . Lemma 8. The following relation holds for any full-rank state σ : Γ − σ,ω j = Γ σ − ,ω j . (23) Moreover, whenever σ = E F∗ ( σ ) : Γ σ, ◦ ∂ j ◦ Γ − σ, = Γ σ,ω j ◦ ∂ j ◦ Γ − σ . (24) Proof. The first identity follows from Lemma 5.8 in [11]. The proof of the second identity followsby direct computation using the commutation relation σA j = e − ω j A j σ . (cid:3) We recall that the spectral gap λ ( L ) of the Lindbladian L is characterized as λ ( L ) := inf X E σ ( X ) (cid:107) X − E F ( X ) (cid:107) σ , (25)for a given full-rank invariant state σ . Lemma 9. The infimum in (25) is independent of the full-rank invariant state σ = E F∗ ( σ ) . Proof. We assumed here that the map L is symmetric with respect to the GNS inner product( σ -DBC), which implies self-adjointness with respect to the inner products (cid:104) ., . (cid:105) σ . Moreover, self-adjointness with respect to the GNS inner product is independent of the invariant state chosen.Therefore, L is self-adjoint with respect to (cid:104) ., . (cid:105) σ for all full-rank invariant state σ . Now, the gapis the difference between the smallest eigenvalue (here, 0) and the second smallest eigenvalue of −L , which is a quantity independent of the inner product with respect to which L is chosen,which allows us to conclude. (cid:3) We are now ready to state and prove the main theorem of this section. Theorem 10. Any GNS -symmetric quantum Markov semigroup acting on a finite dimensionalvon Neumann algebra satisfies the complete modified logarithmic Sobolev inequality. More pre-cisely, given such a QMS ( P t = e t L ) t ≥ acting on M and with fixed-point algebra F , the followingbound holds true: λ ( L ) C τ, cb ( M : F ) ≤ α CMLSI ( L ) ≤ λ ( L ) . (26) Similarly, the modified logarithmic Sobolev inequality constant is controlled by λ ( L ) C τ ( M : F ) ≤ α MLSI ( L ) ≤ λ ( L ) . (27) Proof. The proof of the upper bounds is standard and can be found in [2, 30], so we focus onthe lower bounds. We first provide a bound on the MLSI constant. For this we use the upperbound in Lemma 2: given X := Γ − E F∗ ( ρ ) ( ρ ), D ( ρ (cid:107) E F∗ ( ρ )) ≤ (cid:107) X − (cid:107) E F∗ ( ρ ) ≤ λ ( L ) − E E F∗ ( ρ ) ( X ) , where λ ( L ) is the spectral gap of L . Next, we have by (21) that E E F∗ ( ρ ) ( X ) = (cid:88) j (cid:107) ∂ j ( X ) (cid:107) E F∗ ( ρ ) ,ω j (1) = (cid:88) j (cid:107) Γ E F∗ ( ρ ) ,ω j ◦ ∂ j ◦ Γ E F∗ ( ρ ) − ( ρ ) (cid:107) E F∗ ( ρ ) − ,ω j (2) = (cid:88) j (cid:107) Γ E F∗ ( ρ ) , ◦ ∂ j ◦ Γ E F∗ ( ρ ) − , ( ρ ) (cid:107) E F∗ ( ρ ) − ,ω j (3) ≤ C τ ( M : F ) (cid:88) j (cid:107) Γ E F∗ ( ρ ) , ◦ ∂ j ◦ Γ E F∗ ( ρ ) − , ( ρ ) (cid:107) ρ − ,ω j (4) = C τ ( M : F ) EP L ( ρ ) . In (1), we used the inverse relation (23); in (2) we used the relation (24); in (3) we used Lemma1 with the weights µ := e − ωj and µ := e ωj ; finally (4) follows from (22). The proof of (26)follows the exact same steps, up to replacing the constant C τ ( M : F ) by its completely boundedversion C τ,cb ( M : F ). (cid:3) Remark 11. When the semigroup is primitive and M := B ( H ), comparison to the logarithmicSobolev constant α LSI combined with standard interpolation inequalities provide the followingbounds for α MLSI [38, 30, 10]: λ ( L )ln( µ min ( σ ) − ) + 2 ≤ α LSI ( L ) ≤ α MLSI ( L )2 ≤ λ ( L ) . The lower bound can be compared with the one provided in (27) together with (11): µ min ( σ ) d H ≤ α MLSI ( L ) , µ min ( σ ) d H ≤ α CMLSI ( L ) . Clearly, these latter bound are much worse than the one above. However, they have the advantageof being the first lower bounds in the case of non-primitive QMS, and the second one works evenindependently of the size of the environment. For that same reason, the bound also tensorizes.5. Approximate tensorization and entropic uncertainty relations In this section, we consider the approximate tensorization property in a general setting. Let M be a finite dimensional von Neumann algebra equipped with faithful trace tr. Let N , N ⊂ M be subalgebras of M and N = N ∩ N . Let E N : M → N and E i : M → N i , i = 1 , 2, beconditional expectations such that E N ◦ E i = E N . If ρ is a state that satisfies E N ∗ ( ρ ) = ρ , then ρ = ρ ◦ E N = ρ ◦ E N ◦ E i = ρ ◦ E i , i = 1 , . PECTRAL METHODS FOR ENTROPY CONTRACTION COEFFICIENTS 17 Namely, every E N invariant state is both E and E invariant. Denote ρ N = E N ∗ ( ρ ) ρ i = E i ∗ ( ρ ) , i = 1 , 2. We are interested in the following approximate tensorization property: D ( ρ (cid:107) ρ N ) ≤ c ( D ( ρ (cid:107) ρ ) + D ( ρ (cid:107) ρ )) , ∀ ρ ∈ D ( E M ) . (28)It was proved in [21, Corollary 2.3] that the constant c equals to 1 if and only if E and E forma commuting square, i.e. E ◦ E = E ◦ E = E N . Using the chain rule D ( ρ (cid:107) ρ N ) = D ( ρ (cid:107) ρ i ) + D ( ρ i (cid:107) ρ N ), the inequality (28) is equivalent to the following entropic uncertainty relation D ( ρ (cid:107) ρ N ) ≥ α ( D ( ρ (cid:107) ρ N ) + D ( ρ (cid:107) ρ N )) , ∀ ρ ∈ D ( E M ) . (29)where α = c c − > / 2. Take ρ ( t ) = tρ + (1 − t ) ρ N and the function f ( t ) = D ( ρ ( t ) (cid:107) ρ N ) − α (cid:0) D ( ρ ( t ) (cid:107) ρ N ) + D ( ρ ( t ) (cid:107) ρ N ) (cid:1) . Then we have f (0) = f (cid:48) (0) = 0 and f (cid:48)(cid:48) (0) = (cid:107) ρ − ρ N (cid:107) ρ − N − α (cid:0) (cid:107) ρ − ρ N (cid:107) ρ − N + (cid:107) ρ − ρ N (cid:107) ρ − N (cid:1) . So a necessary condition for (29) and equivalently (28) is that for any state ρ , (cid:107) ρ − ρ N (cid:107) ρ − N ≥ α (cid:0) (cid:107) ρ − ρ N (cid:107) ρ − N + (cid:107) ρ − ρ N (cid:107) ρ − N (cid:1) . In particular, if we choose ρ = ρ = E ∗ ( ρ ), we have(1 − α ) α (cid:107) ρ − ρ N (cid:107) ρ − N ≥(cid:107) E ∗ ( ρ ) − ρ N (cid:107) ρ − N . Because 1 / < α < 1, for λ = (1 − α ) α this can be reformulated as the following L -clustercondition (cid:107) E ∗ ◦ E ∗ − E N ∗ : L ( ρ − N ) → L ( ρ − N ) (cid:107) = (cid:107) E ◦ E − E N : L ( ρ N ) → L ( ρ N ) (cid:107) = λ < . Since E ◦ E is identity on N and satisfies the ρ N -DBC condition, the above definition isindependent of the choice of invariant state ρ N (see Lemma 4, also [3, Theorem 2]). Note that infinite dimensions, the constant λ is always strictly less than 1. Otherwise there exists a nonzero X / ∈ N such that E ( X ) = X, E ( X ) = X and hence X ∈ N , which leads to a contradictionWe now show that the L -cluster condition is also a sufficient condition for (28): Theorem 12. Let σ ∈ D ( E N ) . Denote (cid:107) E ◦ E − E N : L ( σ ) → L ( σ ) (cid:107) = λ < as the L -clustering constant. Then for any state ρ , D ( ρ (cid:107) ρ N ) ≤ c (cid:0) D ( ρ (cid:107) ρ ) + D ( ρ (cid:107) ρ ) (cid:1) , where the constant c satisfies − λ ≤ c ≤ C τ ( M : N )(1 − λ ) (30) Similarly, for any n ∈ N and all states ρ ∈ D ( E M ⊗ id n ) , we have D ( ρ (cid:107) ( E N ⊗ id)( ρ )) ≤ c cb (cid:0) D ( ρ (cid:107) ( E ⊗ id)( ρ )) + D ( ρ (cid:107) ( E ⊗ id)( ρ )) (cid:1) where the constant c cb satisfies − λ ≤ c cb ≤ C τ, cb ( M : N )(1 − λ ) . (31) Proof. The lower bound was proven at the beginning of the section, so we focus on the upperbound. Note that E , E and E N are all projections on L ( ρ N ). By the L -cluster condition (cid:107) ρ − E ∗ ◦ E ∗ ( ρ ) (cid:107) ρ − N ≥(cid:107) ρ − ρ N (cid:107) ρ − N − (cid:107) ρ N − E ∗ ◦ E ∗ ρ (cid:107) ρ − N ≥ (1 − λ ) (cid:107) ρ − ρ N (cid:107) ρ − N . (32)Moreover, since E ∗ , E ∗ and E N ∗ are projections on L ( ρ − N ), (cid:107) ρ − ρ N (cid:107) ρ − N − (cid:107) ρ − ρ (cid:107) ρ − N − (cid:107) ρ − ρ (cid:107) ρ − N ≤ (cid:107) ρ − ρ N (cid:107) ρ − N − (cid:107) ρ − ρ (cid:107) ρ − N − (cid:107) ρ − E ∗ ( ρ ) (cid:107) ρ − N ≤ (cid:107) ρ − ρ N (cid:107) ρ − N − (cid:107) ρ − E ∗ ( ρ ) (cid:107) ρ − N = (cid:107) ρ − ρ N (cid:107) ρ − N − (cid:107) ρ − E ∗ ◦ E ∗ ( ρ ) (cid:107) ρ − N ≤ (1 − (1 − λ ) ) (cid:107) ρ − ρ N (cid:107) ρ − N , where the last line follows from (32). Namely, we have (cid:107) ρ − ρ N (cid:107) ρ − N ≤ − λ ) (cid:0) (cid:107) ρ − ρ (cid:107) ρ − N + (cid:107) ρ − ρ (cid:107) ρ − N (cid:1) . Now using Lemma 2, D ( ρ (cid:107) ρ N ) ≤ (cid:107) ρ − ρ N (cid:107) ρ − N ≤ − λ ) (cid:0) (cid:107) ρ − ρ (cid:107) ρ − N + (cid:107) ρ − ρ (cid:107) ρ − N (cid:1) ≤ C τ ( M : N )(1 − λ ) (cid:0) (cid:107) ρ − ρ (cid:107) ρ ( t ) − + (cid:107) ρ − ρ (cid:107) ρ ( t ) − (cid:1) , where ρ ( t ) = tρ + (1 − t ) ρ and ρ ( t ) = tρ + (1 − t ) ρ . As in Lemma 2, for i = 1 , D ( ρ (cid:107) ρ i ) = (cid:90) (cid:90) s (cid:107) ρ − ρ i (cid:107) ρ i ( t ) dtds . Then integrating the above inequality we have D ( ρ (cid:107) ρ N ) ≤ C τ ( M : N )(1 − λ ) (cid:0) D ( ρ (cid:107) ρ ) + D ( ρ (cid:107) ρ ) (cid:1) . That completes the proof of (30). The proof of (31) follows the exact same lines after replacing C τ ( M : N ) by C τ, cb ( M : N ) . (cid:3) Remark 13. 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