Optimal quantum networks and one-shot entropies

TL;DR

This paper introduces a semidefinite programming approach for optimizing quantum networks, linking performance to max relative entropy, and extends the concept of conditional min-entropy to quantum causal networks, with applications in quantum dynamics and non-causal games.

Contribution

It develops a general optimization framework for quantum networks, including causal and non-causal structures, and extends quantum entropy concepts to these networks, enabling new applications.

Findings

Optimal performance equals max relative entropy.

Extended conditional min-entropy to quantum causal networks.

Applied to quantum dynamics and non-causal quantum games.

Abstract

We develop a semidefinite programming method for the optimization of quantum networks, including both causal networks and networks with indefinite causal structure. Our method applies to a broad class of performance measures, defined operationally in terms of interactive tests set up by a verifier. We show that the optimal performance is equal to a max relative entropy, which quantifies the informativeness of the test. Building on this result, we extend the notion of conditional min-entropy from quantum states to quantum causal networks. The optimization method is illustrated in a number of applications, including the inversion, charge conjugation, and controlization of an unknown unitary dynamics. In the non-causal setting, we show a proof-of-principle application to the maximization of the winning probability in a non-causal quantum game.