Defining quantum divergences via convex optimization

Hamza Fawzi1 and Omar Fawzi2

1DAMTP, University of Cambridge, United Kingdom
2Univ Lyon, ENS Lyon, UCBL, CNRS, Inria, LIP, F-69342, Lyon Cedex 07, France

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Abstract

We introduce a new quantum Rényi divergence $D^{\#}_{\alpha}$ for $\alpha \in (1,\infty)$ defined in terms of a convex optimization program. This divergence has several desirable computational and operational properties such as an efficient semidefinite programming representation for states and channels, and a chain rule property. An important property of this new divergence is that its regularization is equal to the sandwiched (also known as the minimal) quantum Rényi divergence. This allows us to prove several results. First, we use it to get a converging hierarchy of upper bounds on the regularized sandwiched $\alpha$-Rényi divergence between quantum channels for $\alpha > 1$. Second it allows us to prove a chain rule property for the sandwiched $\alpha$-Rényi divergence for $\alpha > 1$ which we use to characterize the strong converse exponent for channel discrimination. Finally it allows us to get improved bounds on quantum channel capacities.

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Cited by

[1] Kun Fang and Zi-Wen Liu, "No-go theorems for quantum resource purification: new approach and channel theory", arXiv:2010.11822.

[2] Dawei Ding, Sumeet Khatri, Yihui Quek, Peter W. Shor, Xin Wang, and Mark M. Wilde, "Bounding the forward classical capacity of bipartite quantum channels", arXiv:2010.01058.

[3] Bjarne Bergh, Robert Salzmann, and Nilanjana Datta, "The $\alpha \to 1$ limit of the Sharp Quantum Rényi Divergence", arXiv:2102.06576.

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