Entanglement Localization and Optimal Measurement

TL;DR

This paper introduces a generic optimality condition for entanglement localization in many-body systems, proposing a canonical measurement approach that simplifies calculation and aids in defining entanglement length.

Contribution

It presents a canonical measurement method for entanglement localization that is operationally meaningful, easy to compute, and optimal in key cases, advancing understanding of entanglement in many-body systems.

Findings

The canonical measurement approach (LECM) is optimal in important spin-1/2 system cases.

LECM provides a practical way to calculate localized entanglement without complex optimization.

Numerical results on the $j_1-j_2$ spin model illustrate the behavior of LECM.

Abstract

The entanglement can be localized between two noncomplementary parts of a many-body system by performing measurements on the rest of the system. This localized entanglement (LE) depends on the chosen basis set of measurement (BSM). We derive here a generic optimality condition for the LE, which, besides being helpful in studying tripartite systems in pure states, can also be of use in studying mixed states of general bipartite systems. We further discuss a canonical way of localizing entanglement, where the BSM is not chosen arbitrarily, but is fully determined by the properties of the system. The LE obtained in this way, we call the localized entanglement by canonical measurement (LECM), is not only operationally meaningful and easy to calculate in practice (without needing any demanding optimization procedure), it provides a nice way to define the entanglement length in many-body systems. For spin-1/2 systems, the LECM is shown to be optimal in some important cases. At the end, some numerical results are presented for $j_1-j_2$ spin model to demonstrate how the LECM behaves.