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Linear and Multilinear Algebra

On the matrix Heron means and Rényi divergences

Trung Hoa Dinh, ‎Raluca Dumitru, Jose A‎. ‎Franco

Published May 13, 2020• 6 citations
Publisher

Abstract

Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean A♯1−αB=A1/2(A−1/2BA−1/2)1−αA1/2, 0≤α≤1 and its ‘naïve’ extension given by Qα,z(A,B)=(B1−α2zAαzB1−α2z)z, which is a matrix function in the definition of the quantum α-z-Rényi divergence. In connection to these results, for positive semidefinite matrices, we show that the inequality ∥A+B+2rA♯1−αB∥p≤∥A+B+2rQα,z(A,B)∥p, holds for p = 1, 2, α=1,z, and r≥0, among other related inequalities.

DOI
10.1080/03081087.2020.1763239

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Bhatia, Lim, and Yamazaki studied the norm minimality of several Kubo-Ando means of positive semidefinite matrices. Recently, Hiai proved a norm minimality result involving the the weighted geometric mean A♯1−αB=A1/2(A−1/2BA−1/2)1−αA1/2, 0≤α≤1 and its ‘naïve’ extension given by Qα,z(A,B)=(B1−α2zAαzB1−α2z)z, which is a matrix function in the definition of the quantum α-z-Rényi divergence. In connection to these results, for positive semidefinite matrices, we show that the inequality ∥A+B+2rA♯1−αB∥p≤∥A+B+2rQα,z(A,B)∥p, holds for p = 1, 2, α=1,z, and r≥0, among other related inequalities.
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Quantum Algorithms for Computing Maximal Quantum fff-divergence and Kubo-Ando means
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Barycentric Rényi divergences were introduced in [Mosonyi, Bunth, Vrana, Linear Algebra and its Applications, 2024] as an alternative to standard Kubo-Ando constructions to define multivariate quantum Rényi divergences. They are defined via a variational expression and depend on a finite collection of quantum relative entropies DqxD^{q_x}Dqx​. When all the relative entropies are monotone under CPTP maps then so are the corresponding barycentric Rényi divergences, and when all the relative entropies are additive then the corresponding barycentric Rényi divergences are subadditive under tensor product. Additivity has only been established before for the case where all DqxD^{q_x}Dqx​ are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) was known to admit an explicit expression. Here we settle the problem of additivity by showing that for any choice of monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on invertible inputs. This in turn implies that if at least two of the DqxD^{q_x}Dqx​ are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.
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We use nearly parallel pure states to characterize positive linear functionals φφφ on Mn\mathbb{M}_nMn​ as positive multiples of the trace if and only if φ(A♮B)≤φ(A)φ(B)φ(A \natural B) \leq \sqrt{φ(A) φ(B)}φ(A♮B)≤φ(A)φ(B)​ for all positive definite matrices AAA and BBB. Here A♮B=(A−1#B)1/2A(A−1#B)1/2A \natural B = (A^{-1} \# B)^{1/2} A (A^{-1} \# B)^{1/2}A♮B=(A−1#B)1/2A(A−1#B)1/2 represents the spectral geometric mean. For further clarification, we establish novel characterizations through the inequality φ(A♮B)≤φ((A+B)/2)φ(A \natural B) \leq φ((A+B)/2)φ(A♮B)≤φ((A+B)/2) for all positive definite matrices AAA and BBB. We also present a trace inequality related to quantum fidelity that applies to all positive definite matrices, and demonstrate that it does not characterize the trace.