### Characterisations of Matrix and Operator-Valued $Φ$-Entropies, and Operator Efron-Stein Inequalities

**2016-01-31**

1602.00233 | math-ph

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 L\"owner 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.

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