Classification of joint quantum measurements based on entanglement cost of localization

TL;DR

The paper develops a finite-resource framework to localize joint quantum measurements using entanglement, building finite versions of teleportation-based schemes and linking localization cost to measurement complexity via the Clifford hierarchy. It analytically classifies all two-qubit measurements localizable with up to three ebits, revealing rich symmetry-connected families such as Bell-state-related bases, the elegant joint measurement, and twisted variants, and proving connections to established structures. It further sketches systematic generalizations to higher levels, higher dimensions, and multipartite settings, supported by numerical explorations that uncover extensive families of localizable measurements. The work provides a principled, representation-theoretic lens on nonlocal measurements and establishes a framework for exploring network nonlocality and resource costs in quantum information protocols.

Abstract

Despite their importance in quantum theory, joint quantum measurements remain poorly understood. An intriguing conceptual and practical question is whether joint quantum measurements on separated systems can be performed without bringing them together. Remarkably, by using shared entanglement, this can be achieved perfectly when disregarding the post-measurement state. However, existing localization protocols typically require unbounded entanglement. In this work, we address the fundamental question: "Which joint measurements can be localized with a finite amount of entanglement?" We develop finite-resource versions of teleportation-based schemes and analytically classify all two-qubit measurements that can be localized in the first steps of these hierarchies. These include several measurements with exceptional properties and symmetries, such as the Bell state measurement and the elegant joint measurement. This leads us to propose a systematic classification of joint measurements based on entanglement cost, which we argue directly connects to the complexity of implementing those measurements. We illustrate how to numerically explore higher levels and construct generalizations to higher dimensions and multipartite settings.

Figures (2)

• Figure 1: A joint measurement $M_{AB} = \{M_c\}_c$ is $n$-ebit localizable if and only if the Born statistics $p(c|M_c,\psi)$ for any state $\psi$ can be reconstructed from the outcomes $a$ and $b$ of Alice and Bob's local operations ${\mathsf{A}}_a$ and ${\mathsf{B}}_b$ on their respective subsystems $A$ and $B$, with the assistance of $n$ copies of the maximally entangled state $\phi_+$ (see Definition \ref{['def:localizable']}).
• Figure 2: The first (\ref{['fig:1ebit']}) and second (\ref{['fig:3ebits']}) level of the finite-consumption version of the Vaidman localization protocol. The classical labels $a_0$, $a_1$ and $b_0$ denote the outcomes of Bell state measurements performed in the local lab of respectively Alice and Bob. $\ketbra{i}$ denotes a measurement in the computational basis. If the measurement $M$ is localizable, there exists a deterministic decoding $D$ which identifies all eigenstates of the measurement $\psi_c$ from the local data $i,a_i,b$ of Alice and Bob.

Theorems & Definitions (9)

• Definition 1: $n$-ebit localizable
• Definition 2: Equivalent measurements
• Theorem 1: Localizable measurements with one ebit
• Theorem 2: Localizable measurements with three ebits
• Conjecture 1: Localisable measurements with nine ebits
• Lemma 1
• proof
• Lemma 2: Uniqueness of solutions to Eq. \ref{['eq:app:1ebit']}
• proof