Operational applications of the diamond norm and related measures in quantifying the non-physicality of quantum maps

TL;DR

This paper introduces operational measures based on the diamond norm to quantify the non-physicality of quantum maps, with applications in quantum dynamics, error mitigation, and quantum discrimination tasks.

Contribution

It presents new operational measures for non-physical quantum maps, linking them to the diamond norm and providing practical tools for quantum information processing.

Findings

The measures quantify the simulation cost of unphysical maps.

They reveal advantages of non-CP maps in quantum discrimination.

The measures reduce to the diamond norm for trace-preserving maps.

Abstract

Although quantum channels underlie the dynamics of quantum states, maps which are not physical channels -- that is, not completely positive -- can often be encountered in settings such as entanglement detection, non-Markovian quantum dynamics, or error mitigation. We introduce an operational approach to the quantitative study of the non-physicality of linear maps based on different ways to approximate a given linear map with quantum channels. Our first measure directly quantifies the cost of simulating a given map using physically implementable quantum channels, shifting the difficulty in simulating unphysical dynamics onto the task of simulating linear combinations of quantum states. Our second measure benchmarks the quantitative advantages that a non-completely-positive map can provide in discrimination-based quantum games. Notably, we show that for any trace-preserving map, the quantities both reduce to a fundamental distance measure: the diamond norm, thus endowing this norm with new operational meanings in the characterisation of linear maps. We discuss applications of our results to structural physical approximations of positive maps, quantification of non-Markovianity, and bounding the cost of error mitigation.