Superadditivity of quantum relative entropy for general states
aa r X i v : . [ qu a n t - ph ] O c t SUPERADDITIVITY OF QUANTUM RELATIVE ENTROPY FORGENERAL STATES ´ANGELA CAPEL, ANGELO LUCIA, AND DAVID P´EREZ-GARC´IA
Abstract.
The property of superadditivity of the quantum relative entropy statesthat, in a bipartite system H AB = H A ⊗ H B , for every density operator ρ AB one has D ( ρ AB || σ A ⊗ σ B ) ≥ D ( ρ A || σ A ) + D ( ρ B || σ B ). In this work, we provide an extensionof this inequality for arbitrary density operators σ AB . More specifically, we prove that α ( σ AB ) · D ( ρ AB || σ AB ) ≥ D ( ρ A || σ A ) + D ( ρ B || σ B ) holds for all bipartite states ρ AB and σ AB , where α ( σ AB ) = 1 + 2 (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B σ AB σ − / A ⊗ σ − / B − AB (cid:13)(cid:13)(cid:13) ∞ . Introduction and notation
The quantum relative entropy between two density operators ρ and σ in a finite dimen-sional Hilbert space, D ( ρ || σ ), is given by tr[ ρ (log ρ − log σ )] if supp( ρ ) ⊆ supp( σ ) and by+ ∞ otherwise . It constitutes a measure of distinguishability between two quantum statesand is a fundamental tool in quantum information theory [17], [26].The quantum relative entropy is the quantum analogue of the Kullback-Leibler diver-gence [12], the probabilistic relative entropy. Its origin lies in mathematical statistics,where it is used to measure how much two states differ in the sense of statistical dis-tinguishability. The larger the relative entropy of two states is, the more informationfor discriminating between the hypotheses associated to them can be obtained from anobservation.One of the main properties of quantum relative entropy is superadditivity , which statesthat in a bipartite system H AB = H A ⊗ H B one has:(1) D ( ρ AB || σ A ⊗ σ B ) ≥ D ( ρ A || σ A ) + D ( ρ B || σ B )for all ρ AB , where we use the standard notation ρ A = tr B [ ρ AB ] and tr B is the partial trace.Since (Proposition 2) D ( ρ AB || σ A ⊗ σ B ) − D ( ρ A || σ A ) − D ( ρ B || σ B ) = D ( ρ AB || ρ A ⊗ ρ B ) , (1) is equivalent to the fact that the mutual information I ρ ( A : B ) := D ( ρ AB || ρ A ⊗ ρ B ) isalways non-negative, a fact that appears ubiquitously in quantum information theory.In the form (1), superadditivity of the quantum relative entropy has found applicationsin e.g. quantum thermodynamics [7], statistical physics [17, Chapter 13] or hypothesistesting [9]. Indeed, as proven recently in [27] (building on results from [15]), the propertyof superadditivity, along with the properties of continuity with respect to the first variable, monotonicity and additivity (Proposition 1), characterizes axiomatically the quantum rel-ative entropy.The main aim of this work is to provide a quantitative extension of (1) for an arbitrarydensity operator σ AB . Note that for all ρ AB and σ AB , as a consequence of monotonicityof the quantum relative entropy for the partial trace, the following holds:(2) 2 D ( ρ AB || σ AB ) ≥ D ( ρ A || σ A ) + D ( ρ B || σ B ) . Date : October 19, 2017. It can also be defined in infinite dimensions, as well as generalized von Neumann algebras [17]. However,in this work, for simplicity we will restrict to finite dimensions.
Therefore we aim to give a constant α ( σ AB ) ∈ [1 ,
2] at the LHS of (1) that measureshow far σ AB is from σ A ⊗ σ B .Following [5] we will consider as α ( σ AB ) − to “ σ AB multipliedby the inverse of σ A ⊗ σ B ”. In the case in which σ AB and σ A ⊗ σ B commute there isa unique way to define this: σ AB ( σ − A ⊗ σ − B ). In the non-commutative case, however,there are many possible ways to define the multiplication by the inverse. The one wewill take in the result below is a symmetric analogue of the commutative case, ( σ − / A ⊗ σ − / B ) σ AB ( σ − / A ⊗ σ − / B ). Another one that will appear in the proof of this result is thederivative of the matrix logarithm on σ A ⊗ σ B evaluated on σ AB , T σ A ⊗ σ B ( σ AB ), whoseexplicit equivalent expressions shown in [13] and [21] will be presented later. Theorem 1.
For any bipartite states ρ AB , σ AB : (1 + 2 k H ( σ AB ) k ∞ ) D ( ρ AB || σ AB ) ≥ D ( ρ A || σ A ) + D ( ρ B || σ B ) , where H ( σ AB ) = σ − / A ⊗ σ − / B σ AB σ − / A ⊗ σ − / B − AB ,and AB denotes the identity operator in H AB .Note that H ( σ AB ) = 0 if σ AB = σ A ⊗ σ B . This result constitutes an improvement to (2) whenever k H ( σ AB ) k ∞ ≤ / k H ( σ AB ) k ∞ ≤ σ AB ∼ σ A ⊗ σ B . This is the case of (quantum) many body systems where suchproperty is expected to hold for spatially separated regions A, B in the Gibbs state abovethe critical temperature. Indeed, a classical version of Theorem 1 proven by Cesi [3] andDai Pra, Paganoni and Posta [5], was the key step to provide the arguably simplest proofof the seminal result of Martinelli and Olivieri [14] connecting the decay of correlations inthe Gibbs state of a classical spin model with the mixing time of the associated Glauberdynamics, via a bound on the log-Sobolev constant.1.1.
Notation.
We consider a finite dimensional Hilbert space H . We denote the setof bounded linear operators acting on H by B = B ( H ) (whose elements we denote bylowercase Latin letters: f, g ...), and its subset of Hermitian operators by A ⊆ B (whoseelements we call observables ). The set of positive semidefinite Hermitian operators isdenoted by A + . We also denote the set of density operators by S = { f ∈ A + : tr[ f ] = 1 } (whose elements we also call states and denote by lowercase Greek letters: σ, ρ ...).A linear map T : B → B is called a superoperator . We say that a superoperator T is positive if it maps positive operators to positive operators. Moreover, we denote T as completely positive if T ⊗ : B ⊗ M n → B ⊗ M n is positive for every n ∈ N , where M n is the space of complex n × n matrices. We also say that T is trace preserving iftr[ T ( f )] = tr[ f ] for all f ∈ B . Finally, if T verifies all these properties, i.e., is a completelypositive and trace preserving map, it is called a quantum channel (for more informationon this topic, see [28]).We denote by k·k the trace norm (cid:0) k f k = tr (cid:2) √ f ∗ f (cid:3)(cid:1) and by k·k ∞ the operator norm( k f k ∞ = sup {k f ( x ) k H : k x k H = 1 } ). In the following section, we will make use of this(H¨older) inequality [1]:(3) k f g k ≤ k f k k g k ∞ for every f, g ∈ B . In most of the paper, we consider a bipartite finite dimensional Hilbert space H AB = H A ⊗ H B . When this is the case, we use the previous notation placing the subindex AB (resp. A , B ) in each of the previous sets to denote that the operators consideredact on H AB (resp. H A , H B ). There is a natural inclusion of A A in A AB by identifying A A = A A ⊗ B . UPERADDITIVITY OF QUANTUM RELATIVE ENTROPY FOR GENERAL STATES 3
Relative entropy.
Let H be a finite dimensional Hilbert space, f, g ∈ A + , f verify-ing tr[ f ] = 0. The quantum relative entropy of f and g is defined by [23]:(4) D ( f || g ) = 1tr[ f ] tr [ f (log f − log g )] . Remark . In most of the paper we only consider density matrices (with trace 1). Let ρ, σ ∈ S . In this case, the quantum relative entropy is given by:(5) D ( ρ || σ ) = tr [ ρ (log ρ − log σ )] . In the following proposition, we collect some well-known properties of the relative en-tropy, which will be of use in the following section.
Proposition 1 (Properties of the relative entropy, [24]) . Let H AB be a bipartite finite dimensional Hilbert space, H AB = H A ⊗ H B . Let ρ AB , σ AB ∈S AB . The following properties hold: (1) Non-negativity. D ( ρ AB || σ AB ) ≥ and D ( ρ AB || σ AB ) = 0 ⇔ ρ AB = σ AB . (2) Finiteness. D ( ρ AB || σ AB ) < ∞ if, and only if, supp ( ρ AB ) ⊆ supp ( σ AB ) , wheresupp stands for support. (3) Monotonicity. D ( ρ AB || σ AB ) ≥ D ( T ( ρ AB ) || T ( σ AB )) for every quantum channel T . (4) Additivity. D ( ρ A ⊗ ρ B || σ A ⊗ σ B ) = D ( ρ A || σ A ) + D ( ρ B || σ B ) . These properties, especially the property of non-negativity, allow to consider the relativeentropy as a measure of separation of two states, even though, technically, it is not adistance (with its usual meaning), since it is not symmetric and lacks a triangle inequality.Let us prove now the property of superadditivity, whenever σ AB = σ A ⊗ σ B . Proposition 2.
Let H AB = H A ⊗ H B and ρ AB , σ AB ∈ S AB . If σ AB = σ A ⊗ σ B , then D ( ρ AB || σ AB ) = I ρ ( A : B ) + D ( ρ A || σ A ) + D ( ρ B || σ B ) ,where I ρ ( A : B ) = D ( ρ AB || ρ A ⊗ ρ B ) is the mutual information [20] .As a consequence, D ( ρ AB || σ A ⊗ σ B ) ≥ D ( ρ A || σ A ) + D ( ρ B || σ B ) .Proof. Since σ AB = σ A ⊗ σ B , we have D ( ρ AB || σ A ⊗ σ B ) = tr[ ρ AB (log ρ AB − log σ A ⊗ σ B )]= tr[ ρ AB (log ρ AB − log ρ A ⊗ ρ B + log ρ A ⊗ ρ B − log σ A ⊗ σ B )]= D ( ρ AB || ρ A ⊗ ρ B ) + D ( ρ A ⊗ ρ B || σ A ⊗ σ B )= I ρ ( A : B ) + D ( ρ A || σ A ) + D ( ρ B || σ B ) . Now, since I ρ ( A : B ) is a relative entropy, it is greater or equal than zero (property 1of Proposition 1), so D ( ρ AB || σ A ⊗ σ B ) ≥ D ( ρ A || σ A ) + D ( ρ B || σ B ). (cid:3) We prove now a lemma for observables (non necessarily of trace 1) which yields a relationbetween the relative entropy of two observables and the relative entropy of some dilationsof each of them. In particular, it is a useful tool to express the relative entropy of twoobservables in terms of the relative entropy of their normalizations (i.e., the quotient ofeach of them by their trace).
Lemma 2.
Let H be a finite dimensional Hilbert space and let f, g ∈ A + such that tr[ f ] = 0 . For all positive real numbers a and b , we have: (6) D ( af || bg ) = D ( f || g ) + log ab . CAPEL, LUCIA, AND P´EREZ-GARC´IA
Proof. D ( af || bg ) = 1 a tr f ( a tr [ f (log af − log bg )])= 1tr f (tr[ f log a ] + tr[ f log f ] − tr[ f log b ] − tr[ f log g ])= 1tr f (tr[ f (log f − log g )]) + log a − log b = D ( f || g ) + log ab , where, in the first and third equality, we are using the linearity of the trace, and we aredenoting log a by log a for every a ≥ (cid:3) Since the relative entropy of two density matrices is non-negative (property 1 of Propo-sition 1), we have the following corollary:
Corollary 1.
Let H be a finite dimensional Hilbert space and let f, g ∈ A + such that tr[ f ] = 0 and tr[ g ] = 0 . Then, the following inequality holds: (7) D ( f || g ) ≥ − log tr[ g ]tr[ f ] . Proof.
Since f / tr[ f ] and g/ tr[ g ] are density matrices, we have that D ( f / tr[ f ] || g/ tr[ g ]) ≥ ≤ D ( f / tr[ f ] || g/ tr[ g ]) = D ( f || g ) + log tr[ g ]tr[ f ] . (cid:3) Proof of main result
We divide the proof of Theorem 1 in four steps.In the first step, we provide a lower bound for the relative entropy of ρ AB and σ AB in terms of D ( ρ A || σ A ), D ( ρ B || σ B ) and an error term, which we will further bound in thefollowing steps. Step 1.
For density matrices ρ AB , σ AB ∈ S AB , it holds that (8) D ( ρ AB || σ AB ) ≥ D ( ρ A || σ A ) + D ( ρ B || σ B ) − log tr M, where M = exp [log σ AB − log σ A ⊗ σ B + log ρ A ⊗ ρ B ] and equality holds (both sidesbeing equal to zero) if ρ AB = σ AB .Moreover, if σ AB = σ A ⊗ σ B , then log tr M = 0 .Proof. It holds that: D ( ρ AB || σ AB ) − [ D ( ρ A || σ A ) + D ( ρ B || σ B )] == D ( ρ AB || σ AB ) − D ( ρ A ⊗ ρ B || σ A ⊗ σ B )= tr ρ AB log ρ AB − ( log σ AB − log σ A ⊗ σ B + log ρ A ⊗ ρ B ) | {z } log M = D ( ρ AB || M ) , where M is defined as in the statement of the step and in the first equality we have usedthe fourth property of Proposition 1. UPERADDITIVITY OF QUANTUM RELATIVE ENTROPY FOR GENERAL STATES 5
We can now apply Corollary 1 to obtain that D ( ρ AB || M ) = tr[ ρ AB (log ρ AB − log M )] ≥ − log tr M .It is easy to check, given the definition of M , that M = σ AB if ρ AB = σ AB , so bothsides are equal to zero in this case.Also, if σ AB = σ A ⊗ σ B , M is equal to ρ A ⊗ ρ B . In both cases we have log tr M = 0. (cid:3) Our target now is to bound the error term, log tr M , in terms of the relative entropy of ρ AB and σ AB times a constant which depends only on A , B and σ AB , and represents howfar σ AB is from being a tensor product. In the second step of the proof, we will boundthis term by the trace of the product of a term which contains this ‘distance’ between σ AB and σ A ⊗ σ B and another term which depends on ρ AB and not on σ AB . However, beforethat, we need to introduce some concepts and results.First, we recall the Golden-Thompson inequality, proven independently in [8] and [22](and extended to the infinite dimensional case in [19] and [2]), which says that for Her-mitian operators f and g ,(9) tr h e f + g i ≤ tr h e f e g i , where we denote by e f the exponential of f , given by e f := ∞ X k =0 f k k ! .The trivial generalization of the Golden-Thompson inequality to three operators insteadof two in the form tr (cid:2) e f + g + h (cid:3) ≤ tr (cid:2) e f e g e h (cid:3) is false, as Lieb mentioned in [13]. However, inthe same paper, he provides a correct generalization of this inequality for three operators.This result has recently been extended by Sutter et al. in [21] via de so-called multivariatetrace inequalities (see also the subsequent paper by Wilde [25], where similar inequalitiesare derived following the statements of [6]). Theorem 3.
Let f, g be positive semidefinite operators, and recall the definition of T g : (10) T g ( f ) = Z ∞ d t ( g + t ) − f ( g + t ) − . T g is positive semidefinite if g is. We have that (11) tr[exp( − f + g + h )] ≤ tr h e h T e f ( e g ) i . This superoperator T g provides a pseudo-inversion of the operator g with respect tothe operator where it is evaluated. In particular, if f and g commute, it is exactly thestandard inversion, as we can see in the following corollary. Corollary 2. If f and g commute, then T g ( f ) = f Z ∞ d t ( g + t ) − = f g − , and therefore tr[exp( − f + g + h )] ≤ tr h e h e − f e g i = tr h e h e − f + g i . This shows that Lieb’s theorem is really a generalization of Golden-Thompson inequality.
We use an alternative definition of this superoperator to obtain a necessary tool for theproof of Step 2. In [21, Lemma 3.4], Sutter, Berta and Tomamichel prove the followingresult:
Lemma 4.
For f a positive semidefinite operator and g a Hermitian operator the followingholds: CAPEL, LUCIA, AND P´EREZ-GARC´IA T g ( f ) = Z ∞−∞ dt β ( t ) g − − it f g − it , with β ( t ) = π πt ) + 1) − . Using this expression for T σ A ⊗ σ B ( σ AB ), we can prove the following result, which is aquantum version of a result used in [5]. Lemma 5.
For every operator O A ∈ B A and O B ∈ B B the following holds: tr[ L ( σ AB ) σ A ⊗ O B ] = tr[ L ( σ AB ) O A ⊗ σ B ] = 0 , where L ( σ AB ) = T σ A ⊗ σ B ( σ AB ) − AB .Proof. We only prove tr[ L ( σ AB ) σ A ⊗ O B ] = 0,since the other equality is completely analogous.tr[ L ( σ AB ) σ A ⊗ O B ] == tr[( T σ A ⊗ σ B ( σ AB ) − AB ) σ A ⊗ O B ]= tr[ T σ A ⊗ σ B ( σ AB ) σ A ⊗ O B ] − tr[ σ A ⊗ O B ]= tr (cid:20)Z ∞−∞ dt β ( t ) ( σ A ⊗ σ B ) − − it σ AB ( σ A ⊗ σ B ) − it σ A ⊗ O B (cid:21) − tr[ O B ]= Z ∞−∞ dt β ( t ) tr (cid:20) σ − − it A ⊗ σ − − it B σ AB σ − it A ⊗ σ − it B σ A ⊗ O B (cid:21) − tr[ O B ] , because tr[ σ A ] = 1, the integral commutes with the trace, β ( t ) is a scalar for every t ∈ R and the exponent in the power of a tensor product can be split into both terms.Now, since the trace is cyclic and using the fact that any operator in H B commuteswith every operator in H A , we have:tr[ L ( σ AB ) σ A ⊗ O B ] == Z ∞−∞ dt β ( t ) tr (cid:20) σ AB σ − it A ⊗ σ − it B σ A ⊗ O B σ − − it A ⊗ σ − − it B (cid:21) − tr[ O B ]= Z ∞−∞ dt β ( t ) tr (cid:20) σ AB (cid:18) σ − it A σ A σ − − it A (cid:19) ⊗ (cid:18) σ − it B O B σ − − it B (cid:19)(cid:21) − tr[ O B ]= Z ∞−∞ dt β ( t ) tr (cid:20) σ AB A ⊗ (cid:18) σ − it B O B σ − − it B (cid:19)(cid:21) − tr[ O B ]= Z ∞−∞ dt β ( t ) tr (cid:20) σ B σ − it B O B σ − − it B (cid:21) − tr[ O B ]= Z ∞−∞ dt β ( t ) tr (cid:20) σ − − it B σ B σ − it B O B (cid:21) − tr[ O B ]= tr[ O B ] Z ∞−∞ dt β ( t ) − tr[ O B ]= 0 , where we have used Z ∞−∞ dt β ( t ) = 1,and the fact that, for every f A ∈ B A and g AB ∈ S AB , the following holds: UPERADDITIVITY OF QUANTUM RELATIVE ENTROPY FOR GENERAL STATES 7 tr[ f A ⊗ B g AB ] = tr[ f A g A ]. (cid:3) We are now in position to develop the second step of the proof.
Step 2.
With the same notation of step 1, we have that (12) log tr M ≤ tr[ L ( σ AB ) ( ρ A − σ A ) ⊗ ( ρ B − σ B )] , where L ( σ AB ) = T σ A ⊗ σ B ( σ AB ) − AB .Proof. We apply Lieb’s theorem to the error term of inequality (8):tr M = tr exp log σ AB | {z } g − log σ A ⊗ σ B | {z } f + log ρ A ⊗ ρ B | {z } h ≤ tr[ ρ A ⊗ ρ B T σ A ⊗ σ B ( σ AB )]= tr ρ A ⊗ ρ B ( T σ A ⊗ σ B ( σ AB ) − AB ) | {z } L ( σ AB ) + tr[ ρ A ⊗ ρ B ] | {z } , where we are adding and substracting ρ A ⊗ ρ B inside the trace in the last equality.Now, using the fact log( x ) ≤ x −
1, we havelog tr M ≤ tr M − ≤ tr[ L ( σ AB ) ρ A ⊗ ρ B ].Finally, in virtue of Lemma 5, it is clear thattr[ L ( σ AB ) ρ A ⊗ ρ B ] = tr[ L ( σ AB ) ( ρ A − σ A ) ⊗ ( ρ B − σ B )].Therefore, log tr M ≤ tr[ L ( σ AB ) ( ρ A − σ A ) ⊗ ( ρ B − σ B )].Notice that if σ AB = σ A ⊗ σ B , then T σ A ⊗ σ B ( σ AB ) = ( σ A ⊗ σ B ) − σ A ⊗ σ B = AB , so L ( σ AB ) = 0. (cid:3) In the third step of the proof, we need to bound tr[ L ( σ AB ) ( ρ A − σ A ) ⊗ ( ρ B − σ B )] interms of the relative entropy of ρ AB and σ AB times a constant depending only on L ( σ AB )(since L ( σ AB ) represents how entangled σ AB is between the regions A and B ). The firstwell-known result we will use in this step is Pinsker’s inequality [4, 18]. Theorem 6.
For ρ AB and σ AB density matrices, it holds that (13) k ρ AB − σ AB k ≤ D ( ρ AB || σ AB ) . This result will be of use at the end of the proof to finally obtain the relative entropyin the right-hand side of the desired inequality. However, it is important to notice thedifferent scales of the L -norm of the difference between ρ AB and σ AB and the relativeentropy of ρ AB and σ AB in Pinsker’s inequality. Since we are interested in obtaining therelative entropy with exponent one, we will need to increase the degree of the term with thetrace we already have and from which we will construct an L -norm (since, for the moment,its degree is one). We will see later that the fact that in tr[ L ( σ AB ) ( ρ A − σ A ) ⊗ ( ρ B − σ B )]we have ( ρ A − σ A ) ⊗ ( ρ B − σ B ) split into two regions, the multiplicativity of the trace withrespect to tensor products and the monotonicity of the relative entropy play a decisiverole in the proof. CAPEL, LUCIA, AND P´EREZ-GARC´IA
Another important fact that we notice in the left-hand side of Pinsker’s inequality isthat there is a difference between two states (in fact, the ones appearing in the relativeentropy). This justifies the use of Lemma 5 at the end of Step 2, to obtain somethingsimilar to the difference between ρ AB and σ AB .We are now ready to prove the third step in the proof of Theorem 1. Step 3.
With the notation of Theorem 1, (14) tr[ L ( σ AB ) ( ρ A − σ A ) ⊗ ( ρ B − σ B )] ≤ k L ( σ AB ) k ∞ D ( ρ AB || σ AB ) . Proof.
We use the multiplicativity with respect to tensor products of the trace norm andH¨older’s inequality between the trace norm and the operator norm. Thus,tr[ L ( σ AB ) ( ρ A − σ A ) ⊗ ( ρ B − σ B )] ≤ k L ( σ AB ) k ∞ k ( ρ A − σ A ) ⊗ ( ρ B − σ B ) k = k L ( σ AB ) k ∞ k ρ A − σ A k k ρ B − σ B k . Finally, Pinsker’s inequality (Theorem 6) implies that k ρ A − σ A k ≤ p D ( ρ A || σ A ) , k ρ B − σ B k ≤ p D ( ρ B || σ B ).Therefore, k ρ A − σ A k k ρ B − σ B k ≤ p D ( ρ A || σ A ) D ( ρ B || σ B ) ≤ D ( ρ AB || σ AB ) , where in the last inequality we have used monotonicity of the relative entropy with respectto the partial trace (Proposition 1). (cid:3) If we now put together Steps 1, 2 and 3, we obtain the following expression(15) (1 + 2 k L ( σ AB ) k ∞ ) D ( ρ AB || σ AB ) ≥ D ( ρ A || σ A ) + D ( ρ B || σ B ) , with L ( σ AB ) = T σ A ⊗ σ B ( σ AB ) − AB .This inequality already constitutes a quantitative extension of (1) for arbitrary densityoperators σ AB in the sense that if σ AB is a tensor product between A and B , we recover theusual superadditivity, and in general k L ( σ AB ) k ∞ measures how far σ AB is from σ A ⊗ σ B .In the fourth and final step of the proof, we bound k L ( σ AB ) k ∞ by (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B σ AB σ − / A ⊗ σ − / B − AB (cid:13)(cid:13)(cid:13) ∞ ,a quantity from which the closeness to 0 whenever σ AB is near from being a tensor productis directly deduced. It also has some physical interpretation in quantum many bodysystems that will be discussed after proving Step 4.First, we need to introduce the setting of non-commutative L p spaces with a ρ -weightednorm [11]. The central property of these non-commutative L p spaces is that they areequipped with a weighted norm which, for a full rank state ρ ∈ S AB , is given by k f k L p ( ρ ) := tr h(cid:12)(cid:12)(cid:12) ρ / p f ρ / p (cid:12)(cid:12)(cid:12) p i /p for every f ∈ A AB .Analogously, the ρ - weighted inner product is given by h f, g i ρ := tr[ √ ρf √ ρg ] for every f, g ∈ A AB .Some fundamental properties of these spaces are collected in the following proposition. Proposition 3.
Let ρ ∈ S AB . The following properties hold for ρ -weighted norms: (1) Order. ∀ p, q ∈ [1 , ∞ ) , with p ≤ q , we have k f k L p ( ρ ) ≤ k f k L q ( ρ ) ∀ f ∈ A AB . (2) Duality. ∀ f ∈ A AB , we have k f k L p ( ρ ) = sup n h g, f i ρ , g ∈ A AB , k g k L q ( ρ ) ≤ o for /p + 1 /q = 1 . (3) Operator norm. ∀ f ∈ A AB , we have k f k L ∞ ( ρ ) = k f k ∞ , the usual operator norm. UPERADDITIVITY OF QUANTUM RELATIVE ENTROPY FOR GENERAL STATES 9
Another tool we will use in the proof of Step 4 is the following result.
Lemma 7.
Consider ρ ∈ S AB and let T be a quantum channel verifying T ∗ ( ρ ) = ρ ,where T ∗ denotes the dual of T with respect to the Hilbert-Schmidt scalar product. Then, T is contractive between L ( ρ ) and L ( ρ ) , i.e., the following inequality holds for every X ∈ B AB : (16) k T ( X ) k L ( ρ ) ≤ k X k L ( ρ ) . Proof.
Using the property of duality for the ρ -weighted norms of L p -spaces (property 2 ofProposition 3), we can write: k T ( X ) k L ( ρ ) = sup k Y k L ∞ ( ρ ) ≤ tr h T ( X ) ρ / Y ρ / i = sup k Y k ∞ ≤ tr h T ( X ) ρ / Y ρ / i = sup − ≤ Y ≤ tr h T ( X ) ρ / Y ρ / i , where in the first step we have used the fact that, for every ρ ∈ S AB , k·k L ∞ ( ρ ) coincideswith the operator norm.Recalling now that T ∗ is the dual of T with respect to the Hilbert-Schmidt scalarproduct, we have:tr h T ( X ) ρ / Y ρ / i = tr h X T ∗ ( ρ / Y ρ / ) i = tr h X ρ / ρ − / T ∗ ( ρ / Y ρ / ) ρ − / ρ / i . Since we are considering the supremum over the observables verifying − ≤ Y ≤ , ifwe apply to these inequalitites T ∗ ( ρ / · ρ / ), we have − ρ ≤ T ∗ ( ρ / Y ρ / ) ≤ ρ (becauseof the assumption T ∗ ( ρ ) = ρ ).Hence, if we denote Z = ρ − / T ∗ ( ρ / Y ρ / ) ρ − / , it is clear that whenever − ≤ Y ≤ , also − ≤ Z ≤ . Therefore, k T ( X ) k L ( ρ ) = sup − ≤ Y ≤ tr h T ( X ) ρ / Y ρ / i = sup − ≤ Y ≤ tr h X ρ / ρ − / T ∗ ( ρ / Y ρ / ) ρ − / ρ / i ≤ sup − ≤ Z ≤ tr h X ρ / Z ρ / i = k X k L ( ρ ) , where the last equality comes again from the property of duality of weighted L p -norms. (cid:3) In the proof of the previous lemma we have made strong use of the property of dualityof L p ( ρ ). Indeed, considering the L ( ρ )-norm as dual of the operator norm, has beenessential to obtain the desired result. Using similar tools, we can now prove the last stepin the proof of Theorem 1. Step 4.
With the notation of the previous steps, we have (17) k L ( σ AB ) k ∞ ≤ (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B σ AB σ − / A ⊗ σ − / B − AB (cid:13)(cid:13)(cid:13) ∞ . Proof.
The strategy we follow in this proof is the opposite to the one used in the previouslemma, i.e., we study now the L ∞ ( σ A ⊗ σ B )-norm as the dual of the L ( σ A ⊗ σ B )-norm. Since k·k L ∞ ( ρ AB ) coincides with the usual ∞ -norm (operator norm) for every ρ AB ∈ S AB ,we can write k L ( σ AB ) k ∞ = kT σ A ⊗ σ B ( σ AB ) − AB k L ∞ ( σ A ⊗ σ B ) .Using the aforementioned property of duality for the σ A ⊗ σ B -weighted norms of L p -spaces, we have: kT σ A ⊗ σ B ( σ AB ) − AB k L ∞ ( σ A ⊗ σ B ) == sup k O AB k L σA ⊗ σB ) ≤ h O AB , T σ A ⊗ σ B ( σ AB ) − AB i σ A ⊗ σ B = sup k O AB k L σA ⊗ σB ) ≤ tr h ( σ A ⊗ σ B ) / O AB ( σ A ⊗ σ B ) / ( T σ A ⊗ σ B ( σ AB ) − AB ) i = sup k O AB k L σA ⊗ σB ) ≤ tr h σ / A ⊗ σ / B O AB σ / A ⊗ σ / B T σ A ⊗ σ B ( σ AB ) i| {z } R − tr h σ / A ⊗ σ / B O AB σ / A ⊗ σ / B i| {z } S . Let us analyze the terms R and S separately. For R , we have: R = tr h σ / A ⊗ σ / B O AB σ / A ⊗ σ / B T σ A ⊗ σ B ( σ AB ) i = tr (cid:20) ( σ A ⊗ σ B ) / O AB ( σ A ⊗ σ B ) / Z ∞−∞ dt β ( t ) ( σ A ⊗ σ B ) − − it σ AB ( σ A ⊗ σ B ) − it (cid:21) = tr (cid:20) O AB Z ∞−∞ dt β ( t ) ( σ A ⊗ σ B ) − it σ AB ( σ A ⊗ σ B ) it (cid:21) = Z ∞−∞ dt β ( t ) tr h O AB ( σ A ⊗ σ B ) − it σ AB ( σ A ⊗ σ B ) it i = Z ∞−∞ dt β ( t ) tr h ( σ A ⊗ σ B ) it O AB ( σ A ⊗ σ B ) − it σ AB i = tr σ AB Z ∞−∞ dt β ( t ) ( σ A ⊗ σ B ) it O AB ( σ A ⊗ σ B ) − it | {z } e O AB , where in the third and last equality we have used the fact that the integral and the tracecommute, and the fourth equality is due to the cyclicity of the trace. We have also defined: e O AB := Z ∞−∞ dt β ( t ) ( σ A ⊗ σ B ) it O AB ( σ A ⊗ σ B ) − it .If we were able to express S in terms of e O AB , we could simplify the expression thatappears in the supremum above. We can do that in the following way: UPERADDITIVITY OF QUANTUM RELATIVE ENTROPY FOR GENERAL STATES 11 S = tr h σ / A ⊗ σ / B O AB σ / A ⊗ σ / B i = tr (cid:20) σ / A ⊗ σ / B O AB σ / A ⊗ σ / B Z ∞−∞ dt β ( t ) (cid:21) = Z ∞−∞ dt β ( t ) tr h σ / A ⊗ σ / B O AB σ / A ⊗ σ / B i = Z ∞−∞ dt β ( t ) tr h ( σ A ⊗ σ B ) ( σ A ⊗ σ B ) it O AB ( σ A ⊗ σ B ) − it i = tr (cid:20) ( σ A ⊗ σ B ) Z ∞−∞ dt β ( t )( σ A ⊗ σ B ) it O AB ( σ A ⊗ σ B ) − it (cid:21) = tr h ( σ A ⊗ σ B ) e O AB i , where we have used again the properties of cyclicity of the trace and commutativity of theintegral and the trace.Placing now the values for R and S that we have just computed in the supremum ofthe first part of the proof, we have: kT σ A ⊗ σ B ( σ AB ) − AB k L ∞ ( σ A ⊗ σ B ) = sup k O AB k L σA ⊗ σB ) ≤ (cid:16) tr h σ AB e O AB i − tr h σ A ⊗ σ B e O AB i(cid:17) = sup k O AB k L σA ⊗ σB ) ≤ tr h e O AB ( σ AB − σ A ⊗ σ B ) i . This expression looks much simpler than the one we had before. However, we needto prove that (cid:13)(cid:13)(cid:13) e O AB (cid:13)(cid:13)(cid:13) L ( σ A ⊗ σ B ) ≤ e O AB as one of the terms where thesupremum is taken. Indeed, if we consider the map T : A AB → A AB given by O AB Z ∞−∞ dt β ( t )( σ A ⊗ σ B ) it O AB ( σ A ⊗ σ B ) − it ,it is clearly a quantum channel and also verifies T ∗ ( σ A ⊗ σ B ) = σ A ⊗ σ B . Hence, in virtueof Lemma 7, we have (cid:13)(cid:13)(cid:13) e O AB (cid:13)(cid:13)(cid:13) L ( σ A ⊗ σ B ) ≤ k O AB k L ( σ A ⊗ σ B ) ,and, therefore,sup k O AB k L σA ⊗ σB ) ≤ tr h e O AB ( σ AB − σ A ⊗ σ B ) i ≤ sup k Ω AB k L σA ⊗ σB ) ≤ tr[Ω AB ( σ AB − σ A ⊗ σ B )] . In this last supremum over elements of 1-norm, we can undo the previous transfor-mations in order to obtain again an ∞ -norm. First, we need to write the term in thesupremum as a σ A ⊗ σ B -product of two terms:tr[Ω AB ( σ AB − σ A ⊗ σ B )] == tr h ( σ A ⊗ σ B ) / ( σ A ⊗ σ B ) − / σ AB ( σ A ⊗ σ B ) − / ( σ A ⊗ σ B ) / Ω AB i − tr h ( σ A ⊗ σ B ) / Ω AB ( σ A ⊗ σ B ) / i = D Ω AB , ( σ A ⊗ σ B ) − / σ AB ( σ A ⊗ σ B ) − / E σ A ⊗ σ B − h Ω AB , AB i σ A ⊗ σ B = D Ω AB , ( σ A ⊗ σ B ) − / σ AB ( σ A ⊗ σ B ) − / − AB E σ A ⊗ σ B . Finally, using again the property of duality for the norms of L ( σ A ⊗ σ B ) and L ∞ ( σ A ⊗ σ B ), we have:sup k Ω AB k L σA ⊗ σB ) ≤ tr[Ω AB ( σ AB − σ A ⊗ σ B )]= sup k Ω AB k L σA ⊗ σB ) ≤ D Ω AB , ( σ A ⊗ σ B ) − / σ AB ( σ A ⊗ σ B ) − / − AB E σ A ⊗ σ B = (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B σ AB σ − / A ⊗ σ − / B − AB (cid:13)(cid:13)(cid:13) L ∞ ( σ A ⊗ σ B ) = (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B σ AB σ − / A ⊗ σ − / B − AB (cid:13)(cid:13)(cid:13) ∞ , where we have used again the fact that k·k L ∞ ( ρ AB ) coincides with the usual ∞ -norm forevery ρ AB ∈ S AB .In conclusion, kT σ A ⊗ σ B ( σ AB ) − AB k ∞ ≤ (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B σ AB σ − / A ⊗ σ − / B − AB (cid:13)(cid:13)(cid:13) ∞ . (cid:3) By putting together Step 1, Step 2, Step 3 and Step 4, we conclude the proof of Theorem1.
Remark . This result constitutes an extension of the superadditivity property, i.e., theconstant H ( σ AB ) that appears in the statement of the main theorem is 0 when σ AB = σ A ⊗ σ B and is small whenever σ AB ∼ σ A ⊗ σ B . A trivial upper bound can be found withrespect to the trace distance as follows, (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B σ AB σ − / A ⊗ σ − / B − AB (cid:13)(cid:13)(cid:13) ∞ == (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B ( σ AB − σ A ⊗ σ B ) σ − / A ⊗ σ − / B (cid:13)(cid:13)(cid:13) ∞ ≤ (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B ( σ AB − σ A ⊗ σ B ) σ − / A ⊗ σ − / B (cid:13)(cid:13)(cid:13) ≤ (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B (cid:13)(cid:13)(cid:13) ∞ k σ AB − σ A ⊗ σ B k (cid:13)(cid:13)(cid:13) σ − / A ⊗ σ − / B (cid:13)(cid:13)(cid:13) ∞ ≤ σ − k σ AB − σ A ⊗ σ B k . Remark . The term k H ( σ AB ) k ∞ is also closely related to certain forms of decay of cor-relations of states that have already appeared in quantum many body systems, such as LTQO (Local Topological Quantum Order) [16], or the concept of local indistinguishability as a strengthened form of weak clustering in [10].Let us suppose that k H ( σ AB ) k ∞ ≤ λ ( ℓ ) for a certain small scalar λ ( ℓ ) that decayssufficiently fast as a function of the distance ℓ between regions A and B in a many bodysystem, and denote by h f i ϕ the expected value of an observable f ∈ A AB with respect toa state ϕ (usually the ground or thermal state of the system). Then, for every observableof the form O A ⊗ O B ≥
0, if the reduced density matrix on AB of ϕ is σ AB , the previouscondition can be rewritten as (cid:12)(cid:12)(cid:12) h O A O B i ϕ − h O A i ϕ h O B i ϕ (cid:12)(cid:12)(cid:12) ≤ λ h O A i ϕ h O B i ϕ .One can now compare this expression with the definition of decay of correlations (cid:12)(cid:12)(cid:12) h O A O B i ϕ − h O A i ϕ h O B i ϕ (cid:12)(cid:12)(cid:12) ≤ λ ( ℓ ) k O A k ∞ k O B k ∞ ,or LTQO (cid:12)(cid:12)(cid:12) h O A O B i ϕ − h O A i ϕ h O B i ϕ (cid:12)(cid:12)(cid:12) ≤ λ ( ℓ ) h O A i ϕ k O B k ∞ . UPERADDITIVITY OF QUANTUM RELATIVE ENTROPY FOR GENERAL STATES 13 Conclusion
In this work, we have proven an extension of the property of superadditivity of thequantum relative entropy for general states. Our result constitutes an improvement to theusual lower bound for the relative entropy of two bipartite states, given by the property ofmonotonicity, in terms of the relative entropies in the two constituent spaces, whenever thesecond state is near to be a tensor product. Therefore, it might be relevant for situationswhere this property is expected to hold, such as quantum many body systems, in whichit is likely that the Gibbs state satisfies this property in spatially separated systems.In [10], Kastoryano and Brandao proved, for certain Gibbs samplers, the existence of apositive spectral gap for the dissipative dynamics, via a quasi-factorization result of thevariance. This provides a bound for the mixing time of the evolution of the semigroupthat drives the system to thermalization which is polynomial in the system size. We leavefor future work the possibility of using the result of the present paper to obtain a quasi-factorization of the relative entropy in quantum many body systems, which could allow usto prove, under some conditions of decay of correlations on the Gibbs state, the existenceof a positive log-Sobolev constant, obtaining an exponential improvement in the boundfor the mixing time obtained in [10].
Acknowledgment
We are very grateful to D. Sutter and M. Tomamichel, who detected an error in aprevious version of the paper. We also thank M. Junge for fruitful discussions. AC andDPG acknowledge support from MINECO (grant MTM2014-54240-P), from Comunidadde Madrid (grant QUITEMAD+- CM, ref. S2013/ICE-2801), and the European ResearchCouncil (ERC) under the European Union’s Horizon 2020 research and innovation pro-gramme (grant agreement No 648913). AC is partially supported by a La Caixa-SeveroOchoa grant (ICMAT Severo Ochoa project SEV-2011-0087, MINECO). AL acknowledgesfinancial support from the European Research Council (ERC Grant Agreement no 337603),the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN viathe QMATH Centre of Excellence (Grant No. 10059). This work has been partiallysupported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO).
References [1]
R. Bhatia , Matrix Analysis,
Springer Science & Business Media (1997), doi:10.1007/978-1-4612-0653-8.[2]
M. Breitenbecker and H.R. Gruemm , Note on trace inequalities,
Commun. Math. Phys. (1972),276-279, doi:10.1007/BF01645522.[3] F. Cesi , Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs randomfields,
Probab. Theory Related Fields (2001), 569-584, doi:10.1007/PL00008792.[4]
I. Csisz´ar , Information-type measures of difference of probability distributions and indirect observa-tions,
Stud. Sci. Math. Hung. (1967), 299-318.[5] P. Dai Pra, A.M. Paganoni and G. Posta , Entropy inequalities for unbounded spin systems,
Ann.Probab. (2002), 1959-1976, doi:10.1214/aop/1039548378.[6] F. Dupuis and M.M. Wilde , Swiveled R´enyi entropies,
Quantum Inf. Process. (3) (2016), 1309-1345, doi:10.1007/s11128-015-1211-x.[7] R. Gallego, J. Eisert and H. Wilming , Thermodynamic work from operational principles,
NewJ. Phys. (2016), 103017, doi:10.1088/1367-2630/18/10/103017.[8] S. Golden , Lower Bounds for the Helmholtz Function,
Phys. Rev., Series II, (1965), B1127-B1128, doi:10.1103/PhysRev.137.B1127.[9]
F. Hiai and D. Petz , The proper formula for relative entropy and its asymptotics in quantumprobability,
Commun. Math. Phys. (1) (1991), 99-114, doi:10.1007/BF02100287.[10]
M.J. Kastoryano and F.G.S.L. Brand˜ao , Quantum Gibbs Samplers: The Commuting Case,
Commun. Math. Phys. (2016), 915-957, doi:10.1007/s00220-016-2641-8.[11]
H. Kosaki , Application of the complex interpolation method to a von Neumann algebra: non-commutative L p -spaces, J. Funct. Anal. (1984), 29-78, doi:10.1016/0022-1236(84)90025-9. [12] S. Kullback and R.A. Leibler , On information and sufficiency,
Annals of Math. Stat. (1) (1951),79-86, doi:10.1214/aoms/1177729694.[13] E.H. Lieb , Convex trace functions and the Wigner–Yanase–Dyson conjecture,
Adv. Math. (3)(1973), 267-288, doi:10.1016/0001-8708(73)90011-X.[14] F. Martinelli and E. Olivieri , Approach to equilibrium of Glauber dynamics in the one phaseregion II. The general case,
Commun. Math. Phys. (1994), 487-514, doi:10.1007/BF02101930.[15]
K. Matsumoto , Reverse Test and Characterization of Quantum Relative Entropy, preprint (2010),arxiv:1010.1030[16]
S. Michalakis and J. Pytel , Stability of frustration free Hamiltonians,
Commun. Math. Phys. (2) (2013), 277-302, doi:10.1007/s00220-013-1762-6.[17]
M. Ohya and D. Petz , Quantum Entropy and Its Use,
Texts and Monographs in Physics (Springer-Verlag, Berlin) (1993).[18]
M.S. Pinsker , Information and Information Stability of Random Variables and Processes,
HoldenDay (1964).[19]
M.B. Ruskai , Inequalities for traces on Von Neumann algebras,
Commun. Math. Phys. (1972),280-289, doi:10.1007/BF01645523.[20] C.E. Shannon , A mathematical theory of communication,
Bell Syst. Tech. J. (1948), 379-423,623-656, doi:10.1002/j.1538-7305.1948.tb01338.x, 10.1002/j.1538-7305.1948.tb00917.x.[21] D. Sutter, M. Berta and M. Tomamichel , Multivariate Trace Inequalities,
Commun. Math. Phys. (1) (2017), 37-58, doi:10.1007/s00220-016-2778-5.[22]
C.J. Thompson , Inequality with Applications in Statistical Mechanics,
J. Math. Phys. (1965),1812-1813, doi:10.1063/1.1704727.[23] H. Umegaki , Conditional expectation in an operator algebra IV. Entropy and information,
KodaiMath. Sem. Rep. (1962), 59-85, doi:10.2996/kmj/1138844604.[24] A. Wehrl , General properties of entropy,
Rev. Mod. Phys. (2) (1978), 221-260,doi:10.1103/RevModPhys.50.221.[25] M.M. Wilde , Monotonicity of p -norms of multiple operators via unitary swivels, preprint (2016),arxiv:1610.01262.[26] M.M. Wilde , Quantum Information Theory, Second Edition, (2017),
Cambridge University Press ,doi:10.1017/9781316809976.[27]
H. Wilming, R. Gallego and J. Eisert , Axiomatic Characterization of the Quantum RelativeEntropy and Free Energy,
Entropy (6) (2017), 241, doi:10.3390/e19060241.[28] M.M. Wolf
Instituto de Ciencias Matem´aticas (CSIC-UAM-UC3M-UCM), C/ Nicol´as Cabrera13-15, Campus de Cantoblanco, 28049 Madrid, Spain
E-mail address : [email protected] (Lucia) QMATH, Department of Mathematical Sciences, University of Copenhagen, Uni-versitetsparken 5, 2100 Copenhagen, Denmark and NBIA, Niels Bohr Institute, Universityof Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark
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