Characterisations of Matrix and Operator-Valued $\Phi$-Entropies, and Operator Efron-Stein Inequalities

Abstract

We derive new characterisations of the matrix $\mathrm{\Phi }$-entropy functionals introduced in [Electron.~J.~Probab., 19(20): 1--30, 2014]. Notably, all known equivalent characterisations of the classical $\Phi$-entropies have their matrix correspondences. Next, we propose an operator-valued generalisation of the matrix $\Phi$-entropy functionals, and prove their subadditivity under Löwner partial ordering. Our results demonstrate that the subadditivity of operator-valued $\Phi$-entropies is equivalent to the convexity of various related functions. This result can be used to demonstrate an interesting result in quantum information theory: the matrix $\Phi$-entropy of a quantum ensemble is monotone under unital quantum channels. Finally, we derive the operator Efron-Stein inequality to bound the operator-valued variance of a random matrix.

21 pages. Text partially overlaps with arXiv:1506.06801. Accepted in Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences

References in corpus (2)

• Moment inequalities for functions of independent random variables
• On the joint convexity of the Bregman divergence of matrices

Cited by in corpus (3)

• Exponential Decay of Matrix $\Phi$-Entropies on Markov Semigroups with Applications to Dynamical Evolutions of Quantum Ensembles
• Matrix Poincaré, Φ-Sobolev inequalities, and quantum ensembles
• Nonlinear Matrix Concentration via Semigroup Methods