We present several theorems on strict and strong convexity, and higher order differential formulae for sandwiched quasi-relative entropy (a parametrized version of the classical fidelity). These are crucial for establishing global linear convergence of the gradient projection algorithm for optimization problems for these functions. The case of the classical fidelity is of special interest for the multimarginal optimal transport problem (the n-coupling problem) for Gaussian measures. © 2018 World Scientific Publishing Company.

Rajendra Bhatia

Department of Mathematics

T. Jain

Y. Lim

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Reviews in Mathematical Physics

On the space of positive definite matrices, we consider distance functions of the form d(A, B) = [tr A(A, B) - tr G(A, B)] 1 / 2, where A(A, B) is the arithmetic mean and G(A, B) is one of the different versions of the geometric mean. When G(A, B) = A1 / 2B1 / 2 this distance is ‖ A1 / 2- B1 / 2‖ 2, and when G(A,B)=(A1/2BA1/2)1/2 it is the Bures–Wasserstein metric. We study two other cases: G(A,B)=A1/2(A-1/2BA-1/2)1/2A1/2, the Pusz–Woronowicz geometric mean, and G(A,B)=exp(logA+logB2), the log Euclidean mean. With these choices, d(A, B) is no longer a metric, but it turns out that d2(A, B) is a divergence. We establish some (strict) convexity properties of these divergences. We obtain characterisations of barycentres of m positive definite matrices with respect to these distance measures. One of these leads to a new interpretation of a power mean introduced by Lim and Palfia, as a barycentre. The other uncovers interesting relations between the log Euclidean mean and relative entropy. © 2019, Springer Nature B.V.

Letters in Mathematical Physics

Matrix versions of the Hellinger distance

Linear Algebra and Its Applications

Norm inequalities related to the matrix geometric mean

On variational expressions for quantum relative entropies

Foundations and Trends¬Æ in Machine Learning

Convex Optimization: Algorithms and Complexity

Communications in Mathematical Physics

Strong Converse for the Classical Capacity of Entanglement-Breaking and Hadamard Channels via a Sandwiched Rényi Relative Entropy

Journal of Mathematical Physics

Strong convexity of sandwiched entropies and related optimization problems

Journal	Reviews in Mathematical Physics
Publisher	World Scientific Publishing Co. Pte Ltd
ISSN	0129055X