Computing entanglement costs of non-local operations on the basis of algebraic geometry

TL;DR

This paper introduces an algebraic geometry framework to optimize entanglement costs in distributed quantum operations, providing new analytical tools and resolving open problems in the field.

Contribution

It develops a systematic algebraic geometry approach to simplify entanglement cost optimization and generalizes previous results for non-local quantum operations.

Findings

Unified generalization of entanglement cost results

Resolved open problem on local state discrimination

Strengthened hierarchy for entanglement trade-offs

Abstract

In the study of distributed quantum information processing, it is crucial to minimize the entanglement consumption by optimizing local operations. We develop a framework based on algebraic geometry to systematically simplify the optimization over separable (SEP) channels, which serve as widely used models for local operations. We apply this framework to computing one-shot entanglement cost for implementing non-local operations under SEP channels, in both probabilistic and zero-error settings. First, we present a unified generalization of previous analytical results on the entanglement cost. Via the generalization, we resolve an open problem posed by Yu et al. regarding the entanglement cost of local state discrimination. Second, we strengthen the Doherty--Parrilo--Spedalieri hierarchy and determine the trade-off between the entanglement cost and the success probability of implementing various operations -- such as entanglement distillation, non-local unitary channels, measurements, state verification, and multipartite entanglement distribution.