Localization of joint quantum measurements on $\mathbb{C}^d \otimes \mathbb{C}^d$ by entangled resources with Schmidt number at most $d$

TL;DR

This paper characterizes when joint quantum measurements on bipartite systems can be implemented with limited entanglement, revealing fundamental constraints and providing a complete classification for certain cases.

Contribution

It offers a protocol-independent algebraic characterization of localizable PVMs and resolves a conjecture on two-qubit measurement localization.

Findings

Rank-1 PVMs with a maximally entangled element can be localized with Schmidt number at most d if they form a maximally entangled basis.

Two-qubit rank-1 PVMs can be localized with two-qubit entanglement, resolving a conjecture of Gisin and Del Santo.

The characterization extends to ideal two-qudit measurements, strengthening previous results.

Abstract

Localizable measurements are joint quantum measurements that can be implemented using only non-adaptive local operations and shared entanglement. We provide a protocol-independent characterization of localizable projection-valued measures (PVMs) by exploiting algebraic structures that any such measurement must satisfy. We first show that a rank-1 PVM on $\mathbb{C}^d\otimes\mathbb{C}^d$ containing an element with the maximal Schmidt rank can be localized using entanglement of a Schmidt number at most $d$ if and only if it forms a maximally entangled basis corresponding to a nice unitary error basis. This reveals strong limitations imposed by non-adaptive local operations, in contrast to the adaptive setting where any joint measurement is implementable. We then completely characterize two-qubit rank-1 PVMs that can be localized with two-qubit entanglement, resolving a conjecture of Gisin and Del Santo, and finally extend our characterization to ideal two-qudit measurements, strengthening earlier results.