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

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