Time-reversal of rank-one quantum strategy functions

TL;DR

This paper proves a time-reversal property for a class of linear functions on quantum strategies, showing that maximum values are preserved under time reversal, with implications for quantum entropy relationships.

Contribution

It introduces a novel time-reversal property for rank-one quantum strategy functions, providing new insights into quantum information processing and entropy relations.

Findings

Maximum values of certain quantum strategy functions are invariant under time reversal.

Provides an alternative proof for the relationship between quantum min- and max-entropy.

Generalizes known entropy relationships using the time-reversal property.

Abstract

The quantum strategy (or quantum combs) framework is a useful tool for reasoning about interactions among entities that process and exchange quantum information over the course of multiple turns. We prove a time-reversal property for a class of linear functions, defined on quantum strategy representations within this framework, that corresponds to the set of rank-one positive semidefinite operators on a certain space. This time-reversal property states that the maximum value obtained by such a function over all valid quantum strategies is also obtained when the direction of time for the function is reversed, despite the fact that the strategies themselves are generally not time reversible. An application of this fact is an alternative proof of a known relationship between the conditional min- and max-entropy of bipartite quantum states, along with generalizations of this relationship.