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Photonics for Quantum (PfQ)

Characterizing entanglement from correlations: the multi-partite case

James Schneeloch

Published Aug 27, 2021• 0 citations
Publisher

Abstract

In this talk we will discuss how entanglement is defined with respect to separability, and how the strategies for quantifying two-party entanglement can be extended to multiple parties. Though residual entanglement (i.e., what is left when two-party entanglement is subtracted away) is a popular concept for measuring three-party entanglement, we criticize this and expand on a new resource-based measure known as the tri-partite entanglement of formation. Though there remain multiple fundamental questions to be answered on measures of multi-partite entanglement, we show how the tri-partite entanglement of formation can be bounded from below using experimental correlations in much the same way as was done for two parties. Moreover, we give examples showing how much tripartite entanglement may be present in entangled photon triplets generated in nonlinear optics, and provide a general witness for N-partite entanglement. In our development of the N-partite witness, we also discovered something interesting about the nature of quantum correlations. Though there is no quantum-imposed limit to the correlations two parties can share (e.g., in both position and in momentum), there is such a limit for three or more parties.

DOI
10.1117/12.2611224

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In this talk we will discuss how entanglement is defined with respect to separability, and how the strategies for quantifying two-party entanglement can be extended to multiple parties. Though residual entanglement (i.e., what is left when two-party entanglement is subtracted away) is a popular concept for measuring three-party entanglement, we criticize this and expand on a new resource-based measure known as the tri-partite entanglement of formation. Though there remain multiple fundamental questions to be answered on measures of multi-partite entanglement, we show how the tri-partite entanglement of formation can be bounded from below using experimental correlations in much the same way as was done for two parties. Moreover, we give examples showing how much tripartite entanglement may be present in entangled photon triplets generated in nonlinear optics, and provide a general witness for N-partite entanglement. In our development of the N-partite witness, we also discovered something interesting about the nature of quantum correlations. Though there is no quantum-imposed limit to the correlations two parties can share (e.g., in both position and in momentum), there is such a limit for three or more parties.
Abstract This chapter takes a closer look at entanglement in multipartite systems. An examination of pure tripartite systems serves a starting point from which we discover the equivalence classes of Greenberger-Horne-Zeilinger (GHZ) states and W states. We then continue with the mixed-state case and discuss the notions of full separability, partition-separability and biseparability versus genuine tripartite entanglement, before stating the GHZ theorem in the formulation following Mermin. For systems of three or more parties we then formulate the definitions for k-separability and genuine multipartite entanglement (GME), as well as for k-producibility and entanglement depth, followed by prominent examples for GME states such as the generalized GHZ states and Dicke states. We give an overview of the problem of detecting GME using various linear and non-linear GME witnesses, lifted witnesses, PPT mixers, before discussing challenges for the characterization and quantification of multipartite entanglement. We close by analysing the phenomenon of entangled entanglement.
Quantifying entanglement is vital to understand entanglement as a resource in quantum information processing, and many entanglement measures have been suggested for this purpose. When mathematically defining an entanglement measure, we should consider the distinguishability between entangled and separable states, the invariance under local transformation, the monotonicity under local operations and classical communication, and the convexity. These are reasonable requirements but may be insufficient, in particular when taking into account the usefulness of quantum states in multi-party quantum information processing. Therefore, if we want to investigate multipartite entanglement as a resource, then it can be necessary to consider the usefulness of quantum states in multi-party quantum information processing when we define a multipartite entanglement measure. In this paper, we define new multipartite entanglement measures for three-qubit systems based on the three-party teleportation capability, and show that these entanglement measures satisfy the requirements for being genuine multipartite entanglement measures. We also generalize our entanglement measures for N-qubit systems, where [Formula: see text], and discuss that these quantities may be good candidates to measure genuine multipartite entanglement.
The degree to which a pure quantum state is entangled can be characterized by the distance or angle to the nearest unentangled state. This geometric measure of entanglement is explored for bi-partite and multi-partite pure and mixed states. It is determined analytically for arbitrary two-qubit mixed states, generalized Werner, and isotropic states, and is also applied to certain multi-partite mixed states, including two distinct multi-partite bound entangled states. Moreover, the ground-state entanglement of the XY model in a transverse field is calculated and shown to exhibit singular behavior near the quantum critical line. Along the way, connections are pointed out between the geometric measure of entanglement, the Hartree approximation, entanglement witnesses, correlation functions, and the relative entropy of entanglement.
We study a strong form of genuine multipartite nonlocality (GMNL), where an nnn-partite scenario admits a quantum strategy generating outcomes belonging to a restricted set with certainty, whereas every strategy built from nonclassical resources shared among at most k<nk<nk<n parties fails this criterion with positive probability. This generalizes what is referred to as strong contextuality/nonlocality, or pseudotelepathy, beyond ruling out local models. For the simplest five-party Bell scenario of two binary-outcome measurement settings per party, we use the inflation technique to demonstrate that a game with a perfect quantum strategy witnesses a robust separation from models built from bipartite resources under local operations and shared randomness (LOSR), establishing the behavior is LOSR-GMNL. We provide broader context for this result by investigating its simulability with resources shared among three or four parties under different resource models and alternative definitions of GMNL. A central open question is whether nnn-party quantum resources can achieve such a separation from models built from (n−1)(n-1)(n−1)-party resources, for n≥3n\ge 3n≥3. We make progress here by constructing a family of robustly LOSR-GMNL games with εεε-optimal quantum strategies experimentally indistinguishable in the limit from true probability-one-winning strategies for any desired level of precision.
Quantum-entangled measurements are known to enable multi-party behaviors that are impossible with unentangled measurements on nonlocal resources, even those that are super-quantum and bound only by the no-signaling principle. This advantage can be witnessed by the entanglement swapping protocol, along with corresponding impossibility results for "nonlocality swapping". However, the advantage assumes the absence of pre-existing nonlocal resources shared by the swapped-to parties; it no longer holds if all pairs of parties are allowed to share bipartite nonlocal resources. Here, we consider a resource-theoretic perspective in which bipartite nonclassical resources are free resources that can be shared by any pair of parties in a multipartite network, and ask whether quantum entangled measurements can still provide an advantage over certain basic measurements, known as wirings, of nonsignaling nonlocal resources. We resolve this question in the affirmative by demonstrating an explicit four-party behavior that can be achieved with bipartite quantum resources subject to entangled measurements, and cannot be achieved if the bipartite resources are allowed to be more general nonsignaling nonlocal "boxes" so long as the measurements are restricted to local wirings, even also allowing for globally shared classical randomness. Furthermore, the argument generalizes: the same separation can be witnessed for K+2 parties with access to K-partite nonlocal resources for any K > 2. We also examine these results in different contexts, such as the star network configurations and scenarios not admitting globally shared classical randomness, further enhancing understanding of the capabilities of entangled measurements in multi-party configurations.