Defining quantum divergences via convex optimization

Hamza Fawzi; Omar Fawzi

doi:10.22331/q-2021-01-26-387

Defining quantum divergences via convex optimization

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

Published:	2021-01-26, volume 5, page 387
Eprint:	arXiv:2007.12576v2
Doi:	https://doi.org/10.22331/q-2021-01-26-387
Citation:	Quantum 5, 387 (2021).

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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.

► BibTeX data

@article{Fawzi2021definingquantum,
  doi = {10.22331/q-2021-01-26-387},
  url = {https://doi.org/10.22331/q-2021-01-26-387},
  title = {Defining quantum divergences via convex optimization},
  author = {Fawzi, Hamza and Fawzi, Omar},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {5},
  pages = {387},
  month = jan,
  year = {2021}
}

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