Modulus of convexity for operator convex functions

Abstract

Given an operator convex function $f(x)$, we obtain an operator-valued lower bound for $cf(x)+(1-c)f(y)-f(cx+(1-c)y)$, $c\in [0,1]$. The lower bound is expressed in terms of the matrix Bregman divergence. A similar inequality is shown to be false for functions that are convex but not operator convex.

5 pages, change of title. The new version shows that the main result of the original paper cannot be extended to convex functions that are not operator convex!

References in corpus (4)

• Structure of states which satisfy strong subadditivity of quantum entropy with equality
• Faithful Squashed Entanglement
• Quantum f-divergences and error correction
• A Matrix Convexity Approach to Some Celebrated Quantum Inequalities

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• Reconstructing quantum states from local data
• Continuity of quantum entropic quantities via almost convexity
• Integral formula for quantum relative entropy implies data processing inequality
• Bounds on the Concavity of Quantum Entropy
• Continuity bounds for quantum entropies arising from a fundamental entropic inequality