Entanglement cost of bipartite quantum channel discrimination under positive partial transpose operations

TL;DR

This paper develops a resource-theoretic framework for bipartite quantum channel discrimination, introducing entanglement cost measures, SDP-based success probabilities, and symmetry reductions, with applications to various quantum channels.

Contribution

It introduces the concept of entanglement cost for bipartite channel discrimination and develops SDP methods and symmetry principles for its computation.

Findings

SDP formulations for success probabilities in bipartite channel discrimination

Efficient computation of one-shot PPT entanglement cost

Application of the framework to depolarizing, SWAP, and Werner--Holevo channels

Abstract

Quantum channel discrimination is a fundamental task in quantum information processing. In the one-shot regime, discrimination between two candidate channels is characterized by the diamond norm. Beyond this basic setting, however, many scenarios in distributed quantum information processing remain unresolved, motivating notions of distinguishability that capture the power of the available resources. In this work, we formulate a theory of testers for bipartite channel discrimination, leading to the concept of the entanglement cost of bipartite channel discrimination: the minimum Schmidt rank $k$ of a shared maximally entangled state required for local protocols to achieve the globally optimal success probability. We introduce $k$-injectable testers as a tester-based description of entanglement-assisted local discrimination and, in particular, study the class of $k$-injectable positive-partial-transpose (PPT) testers, which constitutes a numerically tractable relaxation of the practically relevant class of LOCC testers. For every $k$, we derive a semidefinite program (SDP) for the optimal success probability, which in turn yields an efficiently computable one-shot PPT entanglement cost. To render these optimization problems numerically feasible, we prove a symmetry-reduction principle for covariant channel pairs, thereby reducing the effective dimension of the associated SDPs. Finally, by dualizing the SDP, we derive bounds on the composite channel-discrimination problem and illustrate our framework with proof-of-principle examples based on the depolarizing channel, the depolarized SWAP channel, and the Werner--Holevo channels.