Controlling gain with loss: Bounds on localizable entanglement in multi-qubit systems

TL;DR

This paper explores bounds on localizable entanglement in multi-qubit systems, analyzing various states and models, and examines how entanglement can be concentrated through measurements despite noise and disorder.

Contribution

It provides analytical bounds for specific states, numerical analysis for general states, and investigates the effects of noise and disorder on localizable entanglement in quantum spin models.

Findings

Bounds on localizable entanglement for GHZ and W states derived analytically.

Localizable entanglement tends to equal bipartite entanglement in large Dicke states.

Localized entanglement exhibits a cubic dependence on lost entanglement in spin models.

Abstract

We investigate the relation between the amount of entanglement localized on a chosen subsystem of a multi-qubit system via local measurements on the rest of the system, and the bipartite entanglement that is lost during this measurement process. We study a number of paradigmatic pure states, including the generalized GHZ, the generalized W, Dicke, and the generalized Dicke states. For the generalized GHZ and W states, we analytically derive bounds on localizable entanglement in terms of the entanglement present in the system prior to the measurement. Also, for the Dicke and the generalized Dicke states, we demonstrate that with increasing system size, localizable entanglement tends to be equal to the bipartite entanglement present in the system over a specific partition before measurement. We extend the investigation numerically in the case of arbitrary multi-qubit pure states. We also analytically determine the modification of these results, including the proposed bounds, in situations where these pure states are subjected to single-qubit phase-flip noise on all qubits. Additionally, we study one-dimensional paradigmatic quantum spin models, namely the transverse-field XY model and the XXZ model in an external field, and numerically demonstrate a cubic dependence of the localized entanglement on the lost entanglement. We show that this relation is robust even in the presence of disorder in the strength of the external field.