Estimating localizable entanglement from witnesses

TL;DR

This paper develops practical lower bounds for localizable entanglement in noisy quantum states using witness-based and measurement-based approaches, enabling estimation with limited measurements and analyzing their robustness against noise.

Contribution

It introduces two novel lower bounds for localizable entanglement that require minimal measurement and no full state tomography, enhancing experimental feasibility.

Findings

Witness-based bound depends only on expectation value of an entanglement witness.

Measurement-based bound is analytically derived for graph states under Pauli noise.

Both bounds show robustness against local physical noise.

Abstract

Computing localizable entanglement for noisy many-particle quantum states is difficult due to the optimization over all possible sets of local projection measurements. Therefore, it is crucial to develop lower bounds, which can provide useful information about the behaviour of localizable entanglement, and which can be determined by measuring a limited number of operators, or by performing least number of measurements on the state, preferably without performing a full state tomography. In this paper, we adopt two different yet related approaches to obtain a witness-based, and a measurement-based lower bounds for localizable entanglement. The former is determined by the minimal amount of entanglement that can be present in a subsystem of the multipartite quantum state, which is consistent with the expectation value of an entanglement witness. Determining this bound does not require any information about the state beyond the expectation value of the witness operator, which renders this approach highly practical in experiments. The latter bound of localizable entanglement is computed by restricting the local projection measurements over the qubits outside the subsystem of interest to a suitably chosen basis. We discuss the behaviour of both lower bounds against local physical noise on the qubits, and discuss their dependence on noise strength and system size. We also analytically determine the measurement-based lower bound in the case of graph states under local uncorrelated Pauli noise.