Entanglement measures for two-particle quantum histories

TL;DR

This paper introduces a method to quantify entanglement in bipartite quantum histories using Feynman's sum-over-histories formalism, linking entanglement to the Schmidt rank of the propagator matrix, and demonstrates its non-classical features with Hardy’s interferometers.

Contribution

It proposes a novel entanglement measure for bipartite quantum histories based on the Schmidt decomposition of the Feynman propagator matrix, extending the concept of entanglement across time.

Findings

Entangled histories occur when the Schmidt rank exceeds 1.

The approach allows quantification of entanglement from weak measurements.

Entangled histories challenge local hidden variable theories.

Abstract

Quantum entanglement is a key resource, which grants quantum systems the ability to accomplish tasks that are classically impossible. Here, we apply Feynman's sum-over-histories formalism to interacting bipartite quantum systems and introduce entanglement measures for bipartite quantum histories. Based on the Schmidt decomposition of the matrix comprised of the Feynman propagator complex coefficients, we prove that bipartite quantum histories are entangled if and only if the Schmidt rank of this matrix is larger than 1. The proposed approach highlights the utility of using a separable basis for constructing the bipartite quantum histories and allows for quantification of their entanglement from the complete set of experimentally measured sequential weak values. We then illustrate the non-classical nature of entangled histories with the use of Hardy's overlapping interferometers and explain why local hidden variable theories are unable to correctly reproduce all observable quantum outcomes. Our theoretical results elucidate how the composite tensor product structure of multipartite quantum systems is naturally extended across time and clarify the difference between quantum histories viewed as projection operators in the history Hilbert space or viewed as chain operators and propagators in the standard Hilbert space.