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Convexity of quasi-entropy type functions: Lieb’s and Ando’s convexity theorems revisited

Fumio Hiai, Dėnes Petz

Published Jan 1, 2013• 8 citations
OpenAlex

Abstract

Given a positive function f on (0, ∞) and a non-zero real parameter θ, we consider a function I θ f (A, B, X) = Tr X ∗ (f(LAR −1 B)RB) −θ (X) in three matrices A, B> 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity of Iθ f (A, B, X). Similar discussions are given for quasi-entropies and quantum skew informations.

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Given a positive function f on (0, ∞) and a non-zero real parameter θ, we consider a function I θ f (A, B, X) = Tr X ∗ (f(LAR −1 B)RB) −θ (X) in three matrices A, B> 0 and X. This generalizes the notion of monotone metrics on positive definite matrices, and in the literature θ = ±1 has been typical. We investigate how operator monotony of f is sufficient and/or necessary for joint convexity/concavity of Iθ f (A, B, X). Similar discussions are given for quasi-entropies and quantum skew informations.
We revisit and improve joint concavity/convexity and monotonicity properties of quasi-entropies due to Petz in a new fashion. Then we characterize equality cases in the monotonicity inequalities (the data-processing inequalities) of quasi-entropies in several ways as follows: Let Φ:B(H)→B(K) be a trace-preserving map such that Φ* is a Schwarz map. When f is an operator monotone or operator convex function on [0, ∞), we present several equivalent conditions for the equality SfK(Φ(ρ)‖Φ(σ))=SfΦ*(K)(ρ‖σ) to hold for given positive operators ρ, σ on H and K∈B(K). The conditions include equality cases in the monotonicity versions of Lieb’s concavity and Ando’s convexity theorems. Specializing the map Φ we have equivalent conditions for equality cases in Lieb’s concavity and Ando’s convexity. Similar equality conditions are discussed also for monotone metrics and χ2-divergences. We further consider some types of linear preserver problems for those quantum information quantities.
Carlen-Frank-Lieb conjecture and monotonicity of α−z\alpha-zα−z R\'enyi relative entropy
In this paper we study the joint convexity/concavity of the trace functions Ψp,q,s(A,B)=Tr(Bq2K∗ApKBq2)s,  p,q,s∈R, \Psi_{p,q,s}(A,B)=\text{Tr}(B^{\frac{q}{2}}K^*A^{p}KB^{\frac{q}{2}})^s,~~p,q,s\in \mathbb{R}, Ψp,q,s​(A,B)=Tr(B2q​K∗ApKB2q​)s,  p,q,s∈R, where AAA and BBB are positive definite matrices and KKK is any fixed invertible matrix. We will give full range of (p,q,s)∈R3(p,q,s)\in\mathbb{R}^3(p,q,s)∈R3 for Ψp,q,s\Psi_{p,q,s}Ψp,q,s​ to be jointly convex/concave for all KKK. As a consequence, we confirm a conjecture of Carlen, Frank and Lieb. In particular, we confirm a weaker conjecture of Audenaert and Datta and obtain the full range of (α,z)(\alpha,z)(α,z) for α\alphaα-zzz Renyi relative entropies to be monotone under completely positive trace preserving maps. We also give simpler proofs of many known results, including the concavity of Ψp,0,1/p\Psi_{p,0,1/p}Ψp,0,1/p​ for 0<p<10<p<10<p<1 which was first proved by Epstein using complex analysis. The key is to reduce the problem to the joint convexity/concavity of the trace functions Ψp,1−p,1(A,B)=TrK∗ApKB1−p,  −1≤p≤1, \Psi_{p,1-p,1}(A,B)=\text{Tr} K^*A^{p}KB^{1-p},~~-1\le p\le 1, Ψp,1−p,1​(A,B)=TrK∗ApKB1−p,  −1≤p≤1, using a variational method.
We consider both known and not previously studied trace functions with applications in quantum physics. By using perspectives we obtain convexity statements for different notions of residual entropy, including the entropy gain of a quantum channel as studied by Holevo and others. We give new and simplified proofs of the Carlen-Lieb theorems concerning concavity or convexity of certain trace functions by making use of the theory of operator monotone functions. We then apply these methods in a study of new types of trace functions. Keywords: Trace function, convexity, entropy gain, residual entropy, operator monotone function.
We consider a generalization of relative entropy derived from the Wigner–Yanase–Dyson entropy and give a simple, self-contained proof that it is convex. Moreover, special cases yield the joint convexity of relative entropy, and for Tr K* ApK B1-pLieb's joint concavity in (A, B) for 0 < p < 1 and Ando's joint convexity for 1 < p ≤ 2. This approach allows us to obtain conditions for equality in these cases, as well as conditions for equality in a number of inequalities which follow from them. These include the monotonicity under partial traces, and some Minkowski type matrix inequalities proved by Carlen and Lieb for [Formula: see text]. In all cases, the equality conditions are independent of p; for extensions to three spaces they are identical to the conditions for equality in the strong subadditivity of relative entropy.
Just as the classical f-divergences parameterized on convex functions generalize the classical relative entropy, the quantum quasi-entropies generalize the quantum relative entropy parameterized on operator convex functions. Quantum quasi-entropies satisfy a version of the monotonicity property and are jointly convex in their arguments. We provide the equality conditions for these inequalities for a family of operator convex functions that includes the von Neumann entropies as a special case. Quantum quasi-entropies are defined with an arbitrarily chosen matrix and since the inequalities are true for any choice of such matrices, we show that these inequalities can be interpreted as operator inequalities.
Many trace inequalities can be expressed either as concavity/convexity theorems or as monotonicity theorems. A classic example is the joint convexity of the quantum relative entropy which is equivalent to the Data Processing Inequality. The latter says that quantum operations can never increase the relative entropy. The monotonicity versions often have many advantages, and often have direct physical application, as in the example just mentioned. Moreover, the monotonicity results are often valid for a larger class of maps than, say, quantum operations (which are completely positive). In this paper we prove several new monotonicity results, the first of which is a monotonicity theorem that has as a simple corollary a celebrated concavity theorem of Epstein. Our starting points are the monotonicity versions of the Lieb Concavity and the Lieb Convexity Theorems. We also give two new proofs of these in their general forms using interpolation. We then prove our new monotonicity theorems by several duality arguments.