Ancilla dimension in quantum channel discrimination

TL;DR

This paper investigates the limitations of using ancillas with output dimensions for optimal quantum channel discrimination, providing counterexamples and characterizing entanglement measures related to channel reversibility.

Contribution

It demonstrates that ancillas with output dimension equal to the input are not always sufficient for optimal discrimination, and characterizes entanglement measures linked to channel reversibility.

Findings

Counterexamples show output-dimension ancillas are insufficient for some channels.

Characterization of operators maximizing entanglement negativity.

Reversibility of channels linked to preservation of weak entanglement measures.

Abstract

Single-shot quantum channel discrimination is a fundamental task in quantum information theory. It is well known that entanglement with an ancillary system can help in this task, and furthermore that an ancilla with the same dimension as the input of the channels is always sufficient for optimal discrimination of two channels. A natural question to ask is whether the same holds true for the output dimension. That is, in cases when the output dimension of the channels is (possibly much) smaller than the input dimension, is an ancilla with dimension equal to the output dimension always sufficient for optimal discrimination? We show that the answer to this question is "no" by construction of a family of counterexamples. This family contains instances with arbitrary finite gap between the input and output dimensions, and still has the property that in every case, for optimal discrimination, it is necessary to use an ancilla with dimension equal to that of the input. The proof relies on a characterization of all operators on the trace norm unit sphere that maximize entanglement negativity. In the case of density operators we generalize this characterization to a broad class of entanglement measures, which we call weak entanglement measures. This characterization allows us to conclude that a quantum channel is reversible if and only if it preserves entanglement as measured by any weak entanglement measure, with the structure of maximally entangled states being equivalent to the structure of reversible maps via the Choi isomorphism. We also include alternate proofs of other known characterizations of channel reversibility.