Superadditivity of quantum relative entropy for general states

SUPERADDITIVITY OF QUANTUM RELATIVE ENTROPY FOR GENERAL STATES

arXiv:1705.03521v1 [quant-ph] 9 May 2017

´ ´ ÍA ANGELA CAPEL, ANGELO LUCIA, AND DAVID PEREZ-GARC

Abstract. The property of superadditivity of the quantum relative entropy states that, in a bipartite system HAB = HA ⊗ HB , for every density operator ρAB one has D(ρAB ||σA ⊗ σB ) ≥ D(ρA ||σA ) + D(ρB ||σB ). In this work, we provide an extension of this inequality for arbitrary density operators σAB .

1. Introduction and notation The quantum relative entropy between two density operators ρ and σ in a finite dimensional Hilbert space, D(ρ||σ), is given by tr[ρ(log ρ − log σ)] if supp(ρ) ⊆ supp(σ) and by +∞ otherwise1. It constitutes a measure of distinguishability between two quantum states and is a fundamental tool in quantum information theory ([14], [24]). The quantum relative entropy is the quantum analogue of the Kullback-Leibler divergence ([9]), 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 distinguishability. The larger the relative entropy of two states is, the more information for discriminating between the hypotheses associated to them can be obtained from an observation. One of the main properties of quantum relative entropy is superadditivity, which states that in a bipartite system HAB = HA ⊗ HB one has: (1)

D(ρAB ||σA ⊗ σB ) ≥ D(ρA ||σA ) + D(ρB ||σB )

for all ρAB , where we use the standard notation ρA = trB [ρAB ] and trB is the partial trace. The main aim of this work is to provide a quantitative extension of (1) for an arbitrary density operator σAB . Note that for all ρAB and σAB , as a consequence of monotonicity of the quantum relative entropy, the following holds: 2D(ρAB ||σAB ) ≥ D(ρA ||σA ) + D(ρB ||σB ).

Therefore we aim to give a constant α ∈ [1, 2] at the LHS of (1) that measures how far σAB is from σA ⊗ σB . Our main result reads: Theorem 1. For any bipartite states ρAB , σAB : where

(1 + 2khk∞ )D(ρAB ||σAB ) ≥ D(ρA ||σA ) + D(ρB ||σB ),

1 −1 −1 , σAB − 1AB , σA ⊗ σB 2 and {·, ·} denotes the anticommutator. Note that h = 0 if σAB = σA ⊗ σB . h=

This result is likely to be relevant for situations where it is natural to assume σAB ∼ σA ⊗ σB . This is the case of (quantum) many body systems where such property is expected to hold for spatially separated regions A, B in the Gibbs state above the critical Date: May 11, 2017. 1It can also be defined in infinite dimensions, as well as generalized von Neumann algebras ([14]). However, in this work, for simplicity we will restrict to finite dimensions. 1

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´ ÍA CAPEL, LUCIA, AND PEREZ-GARC

temperature. Indeed, a classical version of Theorem 1 proven by Cesi [3] and Dai Pra, Paganoni and Posta [5], was the key step to provide the arguably simplest proof of the seminal result of Martinelli and Olivieri [13] connecting the decay of correlations in the Gibbs state of a classical spin model with the mixing time of the associated Glauber dynamics, via a bound on the log-Sobolev constant. 1.1. Notation. We consider a finite dimensional Hilbert space H. We denote the set of bounded linear operators acting on H by B = B(H) (whose elements we denote by lowercase Latin letters: f, g...), and its subset of Hermitian operators by A ⊆ B (whose elements we call observables). The set of positive semidefinite Hermitian operators is denoted 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 ⊗ 1 : B ⊗ Mn → B ⊗ Mn is positive for every n ∈ N, where Mn is the space of complex n × n matrices. We also say that T is trace preserving if tr[T (f )] = tr[f ] for all f ∈ B. Finally, if T verifies all these properties, i.e., is a completely positive and trace preserving map, it is called a quantum channel (for more information on this topic, see [25]). √ We denote by k·k1 the trace norm kf k1 = tr f ∗ f and by k·k∞ the operator norm (kf k∞ = sup{kf (x)kH : kxkH = 1}). In the following section, we will make use of this (Hölder) inequality ([1]): (2)

kf gk1 ≤ kf k1 kgk∞

for every f, g ∈ B.

In most of the paper, we consider a bipartite finite dimensional Hilbert space HAB = HA ⊗ HB . 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 considered act on HAB (resp. HA , HB ). There is a natural inclusion of AA in AAB by identifying AA = AA ⊗ 1B .

1.2. Relative entropy. Let H be a finite dimensional Hilbert space, f, g ∈ A+ , f verifying tr[f ] 6= 0. The quantum relative entropy of f and g is defined by ([22]): 1 (3) D(f ||g) = tr [f (log f − log g)] . tr[f ]

Remark 1. 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: (4)

D(ρ||σ) = tr [ρ(log ρ − log σ)] .

In the following proposition, we collect some well-known properties of the relative entropy, which will be of use in the following section. Proposition 1 (Properties of the relative entropy, [23], [10]). Let HAB be a bipartite finite dimensional Hilbert space, HAB = HA ⊗ HB . Let ρAB , σAB ∈ SAB . The following properties hold: (1) Non-negativity. D(ρAB ||σAB ) ≥ 0 and D(ρAB ||σAB ) = 0 ⇔ ρAB = σAB . (2) Finiteness. D(ρAB ||σAB ) < ∞ if, and only if, supp(ρAB ) ⊆ supp(σAB ), where supp stands for support. (3) Monotonicity. D(ρAB ||σAB ) ≥ D(T (ρAB )||T (σAB )) for every quantum channel T. (4) Factorization. D(ρA ⊗ ρB ||σA ⊗ σB ) = D(ρA ||σA ) + D(ρB ||σB ).

These properties, especially the property of non-negativity, allow to consider the relative entropy as a measure of separation of two states, even though, technically, it is not a distance (with its usual meaning), since it is not symmetric and lacks a triangle inequality.


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Let us prove now the property of superadditivity, whenever σAB = σA ⊗ σB . Proposition 2. Let HAB = HA ⊗ HB and ρAB , σAB ∈ SAB . If σAB = σA ⊗ σB , then D(ρAB ||σAB ) = Iρ (A : B) + D(ρA ||σA ) + D(ρB ||σB ), where Iρ (A : B) = D(ρAB ||ρA ⊗ ρB ) is the mutual information ([18]). As a consequence, D(ρAB ||σA ⊗ σB ) ≥ D(ρA ||σA ) + D(ρB ||σB ). Proof. Since σAB = σA ⊗ σB , we have D(ρAB ||σAB ) = = = =

tr[ρAB (log ρAB − log σAB )] tr[ρAB (log ρAB − log ρA ⊗ ρB + log ρA ⊗ ρB − log σA ⊗ σB )] D(ρAB ||ρA ⊗ ρB ) + D(ρA ⊗ ρB ||σA ⊗ σB ) Iρ (A : B) + D(ρA ||σA ) + D(ρB ||σB ).

Now, since Iρ (A : B) is a relative entropy, it is greater or equal than zero (property 1 of proposition 1), so D(ρAB ||σA ⊗ σB ) ≥ D(ρA ||σA ) + D(ρB ||σB ). We prove now a lemma for observables (non necessarily of trace 1) which yields a relation between the relative entropy of two observables and the relative entropy of some dilations of each of them. In particular, it is a useful tool to express the relative entropy of two observables in terms of the relative entropy of their normalizations (i.e., the quotient of each of them by their trace). Lemma 2. Let H be a finite dimensional Hilbert space and let f, g ∈ A+ such that tr[f ] 6= 0. For all positive real numbers a and b, we have: a (5) D(af ||bg) = D(f ||g) + log . b Proof. 1 D(af ||bg) = (a tr [f (log af − log bg)]) a tr f 1 = (tr[f log a] + tr[f log f ] − tr[f log b] − tr[f log g]) tr f 1 = (tr[f (log f − log g)]) + log a − log b tr f a = D(f ||g) + log , b where, in the first and third equality, we are using the linearity of the trace, and we are denoting log a1 by log a for every a ≥ 0. Since the relative entropy of two density matrices is non-negative (property 1 of proposition 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 ] 6= 0 and tr[g] 6= 0. Then, the following inequality holds: (6)

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]) ≥ 0,


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and we can apply lemma 2: 0 ≤ D(f / tr[f ] || g/ tr[g]) = D(f ||g) + log

tr[g] . tr[f ]

2. Proof of main result We divide the proof of Theorem 1 in three steps. In the first step, we provide a lower bound for the relative entropy of ρAB and σAB in terms of D(ρA ||σA ), D(ρB ||σB ) and an error term, which we will further bound in the following steps. Step 1. For density matrices ρAB , σAB ∈ SAB , it holds that

(7)

D(ρAB ||σAB ) ≥ D(ρA ||σA ) + D(ρB ||σB ) − log tr M,

where M = exp [log σAB − log σA ⊗ σB + log ρA ⊗ ρB ] and equality holds (both sides being equal to zero) if ρAB = σAB . Moreover, if σAB = σA ⊗ σB , then log tr M = 0. Proof. It holds that: D(ρAB ||σAB ) − [D(ρA ||σA ) + D(ρB ||σB )] = = D(ρAB ||σAB ) − D(ρA ⊗ ρB ||σA ⊗ σB )  



   = tr ρAB log ρAB − ( log σAB − log σA ⊗ σB + log ρA ⊗ ρB ) {z } | log M

= D(ρAB ||M ),

where M is defined as in the statement of the step and in the first equality we have used the fourth property of proposition 1. We can now apply corollary 1 to obtain that D(ρAB ||M ) = tr[ρAB (log ρAB − log M )] ≥ − log tr M . It is easy to check, given the definition of M , that M = σAB if ρAB = σAB , so both sides are equal to zero in this case. Also, if σAB = σA ⊗ σB , M is equal to ρA ⊗ ρB . In both cases we have log tr M = 0. Our target now is to bound the error term, log tr M , in terms of the relative entropy of ρAB and σAB times a constant which depends only on A, B and σAB , and represents how far σAB is from being a tensor product. In the second step of the proof, we will bound this term by the trace of the product of a term which contains this ‘distance’ between σAB and σA ⊗ σB and another term which depends on ρAB and not on σAB . However, before that, we need to introduce some concepts and results. First, we recall the Golden-Thompson inequality, proven independently in [6] and [20] (and extended to the infinite dimensional case in [16] and [2]), which says that for Hermitian operators f and g, i i h h (8) tr ef +g ≤ tr ef eg , where we denote by ef the exponential of f , given by ∞ k X f . ef := k! k=0


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The trivial generalization ofthe Golden-Thompson inequality to three operators instead f +g+h f g h of two in the form tr e ≤ tr e e e is false, as Lieb mentioned in [11]. However, in the same paper, he provides a correct generalization of this inequality for three operators. This result has recently been extended in [19]. Theorem 3. Let g a positive operator, and define Z ∞ (9) Tg (f ) = dt (g + t)−1 f (g + t)−1 . 0

Tg is positive-semidefinite if g is. We have that (10)

i h tr[exp(−f + g + h)] ≤ tr eh Tef (eg ) .

This superoperator Tg provides a pseudo-inversion of the operator g with respect to the operator where it is evaluated. In particular, if f and g commute, it is exactly the standard inversion, as we can see in the following corollary. Corollary 2. If f and g commute, then Z ∞ dt (g + t)−2 = f g−1 , Tg (f ) = f 0

and therefore i i h h tr[exp(−f + g + h)] ≤ tr eh e−f eg = tr eh e−f +g . This shows that Lieb’s theorem is really a generalization of Golden-Thompson inequality. We can exploit now some properties of Tg . As a superoperator, itPis self-adjoint and positive definite. Also, in a basis in which g is diagonal, we have g = i gi |iihi| and Tg (|iihj|) =

log(gi ) − log(gj ) |iihj| , gi − gj

Tg (|iihi|) = gi−1 |iihi| .

We can then prove, by checking the formula in the basis in which g is diagonal (as seen in [17]), that Tg (f ) has the following equivalent expression: (11)

Tg (f ) =

Z

∞

dt (Lg + tRg )−1 (f )

0

1 , 1+t

where Lg and Rg are the left and right multiplication by g, respectively: Lg (f ) = gf,

Rg (f ) = f g.

Another tool we need for the proof of our result is the L¨ owner-Heinz theorem ([12] and [7], with a short proof in [8]), which states that the map t 7→ −t−1 is operator monotone and operator concave. Using this result, we can prove the following lemma. Lemma 4. Let ρ and σ be two states. The following inequality holds: 1 Tσ (ρ) ≤ {σ −1 , ρ} , 2 where {f, g} is the anticommutator of f and g, i.e., {f, g} = f g + gf .


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Proof. With the previous notation and making a change of variables s = Tσ (ρ) =

Z

∞

(Lσ + tRσ )−1 (ρ)

=

Z0 ∞

=

Z

1 1

=

Z

t we have 1+t

1 dt 1+t

((1 + t)Lσ + (1 + t)tRσ )−1 (ρ)dt

0

0

0

s 1+ 1−s

s Lσ + 1−s

s 1+ 1−s

Rσ

−1

(ρ)

1 ds (1 − s)2

[(1 − s)Lσ + sRσ ]−1 ds.

Now, in virtue of the L¨ owner-Heinz theorem, −1 [(1 − s)Lσ + sRσ ]−1 ≤ (1 − s)L−1 σ (ρ) + sRσ (ρ).

Therefore, Tσ (ρ) ≤ = =

Z

0

1

−1 (1 − s)L−1 σ (ρ) + sRσ (ρ) ds

1 1 1 −1 Lσ−1 (ρ) + Rσ−1 (ρ) = σ ρ + ρσ −1 2 2 2 1 −1 {σ , ρ} , 2

−1 where we have used in the last line the fact that L−1 σ (ρ) = Lσ−1 (ρ) and Rσ (ρ) = Rσ−1 (ρ).

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(h ρA ⊗ ρB ),

where h=

1 −1 −1 −1 −1 ⊗ σB σAB + σAB σA σA ⊗ σB − 1AB . 2

Proof. We apply Lieb’s theorem to equation 7:   tr M



   = tr explog σAB − log σA ⊗ σB + log ρA ⊗ ρB  {z } | {z } | {z } | g

f

h

≤ tr[σAB TσA ⊗σB (ρA ⊗ ρB )].

Using lemma 4 for TσA ⊗σB (ρA ⊗ ρB ), we obtain: TσA ⊗σB (ρA ⊗ ρB ) ≤

1 −1 −1 , ρA ⊗ ρB , σA ⊗ σB 2

−1 −1 . Since σAB is positive, we have ⊗ σB where we are using the fact that (σA ⊗ σB )−1 = σA

tr M ≤

−1 1 −1 , ρA ⊗ ρB . ⊗ σB tr σAB σA 2


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Therefore, tr M

−1 1 −1 tr σAB σA , ρA ⊗ ρB ⊗ σB 2 1 −1 −1 −1 −1 = ρA ⊗ ρB + tr σAB ρA ⊗ ρB σA ⊗ σB tr σAB σA ⊗ σB 2 = tr[h ρA ⊗ ρB ] + tr[ρA ⊗ ρB ] = tr[h ρA ⊗ ρB ] + 1, ≤

where we are denoting h :=

1 −1 −1 , σAB − 1AB σA ⊗ σB 2

and using the fact that the trace is cyclic and multiplicative with respect to tensor products. Finally, by using the fact log(x) ≤ x − 1, we conclude log tr M ≤ tr M − 1 ≤ tr[h ρA ⊗ ρB ]. In the third and final step of the proof, we need to bound tr[h ρA ⊗ ρB ] in terms of the relative entropy of ρAB and σAB times a constant depending only on h (since h depends on A, B and σAB and represents how entangled σAB is between the regions A and B). The first well-known result we will use in this step is Pinsker’s inequality ([15], [4]). Theorem 5. For ρAB and σAB density matrices, it holds that (13)

kρAB − σAB k21 ≤ 2D(ρAB ||σAB ).

This result will be of use at the end of the proof to finally obtain the relative entropy in the right-hand side of the desired inequality. However, it is important to notice the different scales of the L1 -norm of the difference between ρAB and σAB and the relative entropy of ρAB and σAB in Pinsker’s inequality. Since we are interested in obtaining the relative entropy with exponent one, we will need to increase the degree of the term with the trace we already have and from which we will construct an L1 -norm (since, for the moment, its degree is one). We will see later that the fact that in tr[h ρA ⊗ ρB ] we have ρAB split into two regions, the multiplicativity of the trace with respect to tensor products and the monotonicity of the relative entropy play a decisive role in the proof. Another important fact that we notice in the left-hand side of Pinsker’s inequality is that there is a difference between two states (in fact, the ones appearing in the relative entropy), and, for the moment, we have no difference of this type inside the trace of our expression. To change this fact and obtain something similar to the difference between ρAB and σAB , we need the following lemma, which is an analogous quantum version of a classical result used in [5]. Lemma 6. For every operator OA ∈ BA and OB ∈ BB the following holds: tr[h σA ⊗ OB ] = tr[h OA ⊗ σB ] = 0.

Proof. We only prove tr[h σA ⊗ OB ] = 0,


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since the other equality is completely analogous. 1 −1 −1 tr[h σA ⊗ OB ] = tr σ ⊗ σB , σAB σA ⊗ OB − tr[σA ⊗ OB ] 2 A 1 −1 −1 σAB σA ⊗ OB tr σA ⊗ σB = 2 −1 −1 σA ⊗ OB − tr[σA ] tr[OB ] ⊗ σB + tr σAB σA 1 −1 −1 = ) σAB + tr σAB 1A ⊗ (σB tr 1A ⊗ (OB σB OB ) − tr[OB ] 2 1 −1 −1 = OB − tr[OB ] σB + tr σB σB tr OB σB 2 = tr[OB ] − tr[OB ] = 0, where we have used the fact that operators on disjoint regions commute and some properties of the trace previously mentioned. We are also employing the fact that, for every fA ∈ BA and gAB ∈ BAB , the following holds: tr[fA ⊗ 1B gAB ] = tr[fA gA ]. We are now ready to prove the third and final step in the proof of theorem 1. Step 3. With the notation of theorem 1, (14)

tr[h ρA ⊗ ρB ] ≤ 2khk∞ D(ρAB ||σAB ).

Proof. In virtue of the previous lemma, it is clear that tr[h ρA ⊗ ρB ] = tr[h (ρA − σA ) ⊗ (ρB − σB )], since ρA and ρB have support on A and B, respectively. Now, we use the multiplicativity with respect to tensor products of the trace norm and a Hölder’s inequality between the trace norm and the operator norm. Thus, tr[h (ρA − σA ) ⊗ (ρB − σB )] ≤ khk∞ k(ρA − σA ) ⊗ (ρB − σB )k1 = khk∞ kρA − σA k1 kρB − σB k1 . Finally, Pinsker’s inequality (theorem 5) implies that p p kρA − σA k1 ≤ 2D(ρA ||σA ), kρB − σB k1 ≤ 2D(ρB ||σB ). Therefore,

p kρA − σA k1 kρB − σB k1 ≤ 2 D(ρA ||σA )D(ρB ||σB ) ≤ 2D(ρAB ||σAB ),

where in the last inequality we have used monotonicity of the relative entropy with respect to the partial trace (proposition 1). By putting together step 1, step 2 and step 3, we conclude the proof of theorem 1. Acknowledgement AC and DPG acknowledge support from MINECO (grant MTM2014-54240-P), from Comunidad de Madrid (grant QUITEMAD+- CM, ref. S2013/ICE-2801), and the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 648913). AC is partially supported by a La Caixa-Severo Ochoa grant (ICMAT Severo Ochoa project SEV-2011-0087, MINECO). AL


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acknowledges financial support from the European Research Council (ERC Grant Agreement no 337603), the Danish Council for Independent Research (Sapere Aude) and VILLUM FONDEN via the QMATH Centre of Excellence (Grant No. 10059). This work has been partially supported by ICMAT Severo Ochoa project SEV-2015-0554 (MINECO). References [1] R. Bhatia, Matrix Analysis, Springer Science & Business Media 169 (1997), doi:10.1007/978-1-46120653-8 [2] M. Breitenbecker and H.R. Gruemm, Note on trace inequalities, Commun. Math. Phys. 26 (1972), 276-279, doi:10.1007/BF01645522 [3] F. Cesi, Quasi-factorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields, Probab. Theory Related Fields 120 (2001), 569-584, doi:10.1007/PL00008792 ´ r, Information-type measures of difference of probability distributions and indirect observa[4] I. Csisza tions, Stud. Sci. Math. Hung. 2 (1967), 299-318 [5] P. Dai Pra, A.M. Paganoni and G. Posta, Entropy inequalities for unbounded spin systems, Ann. Probab. 30 (2002), 1959-1976, doi:10.1214/aop/1039548378 [6] S. Golden, Lower Bounds for the Helmholtz Function, Phys. Rev., Series II, 137 (1965), B1127B1128, doi:10.1103/PhysRev.137.B1127 [7] E. Heinz, Beitr¨ age zur St¨ orungstheorie der Spektralzerlegung, Math. Ann. 123 (1951), 415-438, doi:10.1007/BF02054965 [8] T. Kato, Notes on some inequalities for linear operators, Math. Ann. 125 (1952), 208-212, doi:10.1007/BF01343117 [9] S. Kullback and R.A. Leibler, On information and sufficiency, Annals of Math. Stat. 22(1) (1951), 79-86, doi:10.1214/aoms/1177729694 [10] E.H. Lieb and M.B. Ruskai, Proof of the strong subadditivity of quantum mechanical entropy, J. Math. Phys. 14 (1973), 1938-1941, doi:10.1063/1.1666274 [11] E.H. Lieb, Convex trace functions and the Wigner–Yanase–Dyson conjecture, Adv. Math. 11(3) (1973), 267-288, doi:10.1016/0001-8708(73)90011-X ¨ ¨ wner, Uber [12] K. Lo monotone Matrixfunktionen, Math. Z. 38 (1934), 177-216, doi:10.1007/BF01170633 [13] F. Martinelli and E. Olivieri, Approach to equilibrium of Glauber dynamics in the one phase region II. The general case, Commun. Math. Phys. 161 (1994), 487-514, doi:10.1007/BF02101930 [14] M. Ohya and D. Petz, Quantum Entropy and Its Use, Texts and Monographs in Physics (SpringerVerlag, Berlin) (1993) [15] M.S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden Day (1964) [16] M.B. Ruskai, Inequalities for traces on Von Neumann algebras, Commun. Math. Phys. 26 (1972), 280-289, doi:10.1007/BF01645523 [17] M.B. Ruskai, Lieb’s simple proof of concavity of (A, B) 7→ TrAp K † B 1−p K and remarks on related inequalities, Int. J. Quantum Inform. 03 (2005), 579-590, doi:10.1142/S0219749905001109 [18] C.E. Shannon, A mathematical theory of communication, Bell Syst. Tech. J. 27 (1948), 379-423, 623-656, [19] D. Sutter, M. Berta and M. Tomamichel, Multivariate Trace Inequalities, Commun. Math. Phys. (2016), doi:10.1007/s00220-016-2778-5 [20] C.J. Thompson, Inequality with Applications in Statistical Mechanics, J. Math. Phys. 6 (1965), 1812-1813, doi:10.1063/1.1704727 [21] A. Uhlmann, Relative entropy and the Wigner-Yanase-Dyson-Lieb concavity in an interpolation theory, Commun. Math. Phys. 54(1) (1977), 21-32 [22] H. Umegaki, Conditional expectation in an operator algebra IV. Entropy and information, Kodai Math. Sem. Rep. 14 (1962), 59-85 [23] A. Wehrl, General properties of entropy, Rev. Mod. Phys. 50(2) (1978), 221-260, doi:10.1103/RevModPhys.50.221 [24] M.M. Wilde, Quantum Information Theory, Second Edition, (2017), Cambridge University Press, doi:10.1017/9781316809976 [25] M.M. Wolf, Quantum Channels and Operations. Guided tour, (2012), https://wwwm5.ma.tum.de/foswiki/pub/M5/Allgemeines/MichaelWolf/QChannelLecture.pdf ´ ticas (CSIC-UAM-UC3M-UCM), C/ Nicola ´ s Cabrera (Capel) Instituto de Ciencias Matema 13-15, Campus de Cantoblanco, 28049 Madrid, Spain E-mail address: [email protected]

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(Lucia) QMATH, Department of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, 2100 Copenhagen, Denmark and NBIA, Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark E-mail address: [email protected] ´ lisis Matema ´ tico, Universidad Complutense de Madrid, (Pérez-Garc´ıa) Departamento de Ana ´ ticas (CSIC-UAM-UC3M-UCM), C/ 28040 Madrid, Spain and Instituto de Ciencias Matema ´ s Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain Nicola E-mail address: [email protected]