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New Tests of Multipartite Entanglement in Bell-Type Experiments

Koji Nagata, Tadao Nakamura

Published Nov 1, 2015• 0 citations
OpenAlex

Abstract

In trial, we especially consider inequalities for confirming multipartite entanglement from experimental data obtained in Bell-type experiments. We present new entanglement witness inequalities. Some physical situation is that we measure $\sigma_x$, $\sigma_y$, and $\sigma_z$ per side. Our analysis discovers a new multipartite entangled state and it is experimentally feasible. If the reduction factor $V$ of the interferometric contrast observed in a $N$-particle correlation experiment is $V>0.4$, then a measured state is full $N$-partite entanglement in a significant specific case. It is not revealed by previous Bell-type experimentally feasible methods presented in \cite{NKI}, which states if $V>0.5$ then the significant specific type state is full $N$-partite entanglement.

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In trial, we especially consider inequalities for confirming multipartite entanglement from experimental data obtained in Bell-type experiments. We present new entanglement witness inequalities. Some physical situation is that we measure σx\sigma_xσx​, σy\sigma_yσy​, and σz\sigma_zσz​ per side. Our analysis discovers a new multipartite entangled state and it is experimentally feasible. If the reduction factor VVV of the interferometric contrast observed in a NNN-particle correlation experiment is V>0.4V>0.4V>0.4, then a measured state is full NNN-partite entanglement in a significant specific case. It is not revealed by previous Bell-type experimentally feasible methods presented in \cite{NKI}, which states if V>0.5V>0.5V>0.5 then the significant specific type state is full NNN-partite entanglement.
We show that the rich structure of multipartite entanglement can be tested following a device-independent approach. Specifically we present Bell inequalities for distinguishing between different types of multipartite entanglement, without placing any assumptions on the measurement devices used in the protocol, in contrast with usual entanglement witnesses. We first address the case of three qubits and present Bell inequalities that can be violated by W states but not by Greenberger-Horne-Zeilinger states, and vice versa. Next, we devise 'subcorrelation Bell inequalities' for any number of parties, which can provably not be violated by a broad class of multipartite entangled states (generalizations of Greenberger-Horne-Zeilinger states), but for which violations can be obtained for W states. Our results give insight into the nonlocality of W states. The simplicity and robustness of our tests make them appealing for experiments.
We derive tight quadratic inequalities for all kinds of hybrid separable-inseparable n-particle density operators on an arbitrary dimensional space. This methodology enables us to derive a tight quadratic inequality as tests for full n-partite entanglement in various Bell-type correlation experiments on the systems that may not be identified as a collection of qubits, e.g., those involving photons measured by incomplete detectors. It is also proved that when the two measured observables are assumed to precisely anticommute, a stronger quadratic inequality can be used as a witness of full n-partite entanglement.
Entanglement witnesses provide tools to detect entanglement in experimental situations without the need of having full tomographic knowledge about the state. If one estimates in an experiment an expectation value smaller than zero, one can directly infer that the state has been entangled, or specifically multi-partite entangled, in the first place. In this paper, we emphasize that all these tests—based on the very same data—give rise to quantitative estimates in terms of entanglement measures: 'If a test is strongly violated, one can also infer that the state was quantitatively very much entangled'. We consider various measures of entanglement, including the negativity, the entanglement of formation, and the robustness of entanglement, in the bipartite and multipartite setting. As examples, we discuss several experiments in the context of quantum state preparation that have recently been performed.
In a recent paper, Bancal et al. [Phys. Rev. Lett. 106, 250404 (2011)] put forward the concept of device-independent witnesses of genuine multipartite entanglement. These witnesses are capable of verifying genuine multipartite entanglement produced in a laboratory without resorting to any knowledge of the dimension of the state space or of the specific form of the measurement operators. As a by-product they found a multiparty three-setting Bell inequality which makes it possible to detect genuine nnn-partite entanglement in a noisy nnn-qubit Greenberger-Horne-Zeilinger (GHZ) state for visibilities as low as 2/32/32/3 in a device-independent way. In this paper, we generalize this inequality to an arbitrary number of settings, demonstrating a threshold visibility of $2/\ensuremath{\pi}~0.6366$ for number of settings going to infinity. We also present a pseudotelepathy Bell inequality achieving the same threshold value. We argue that our device-independent witnesses are optimal in the sense that for nnn odd the above value cannot be beaten with nnn-party-correlation Bell inequalities.
Genuine multipartite entanglement represents the strongest type of entanglement, which is an essential resource for quantum information processing. Standard methods to detect genuine multipartite entanglement, e.g., entanglement witnesses, state tomography, or quantum state verification, require full knowledge of the Hilbert space dimension and precise calibration of measurement devices, which are usually difficult to acquire in an experiment. The most radical way to overcome these problems is to detect entanglement solely based on the Bell-like correlations of measurement outcomes collected in the experiment, namely, device independently. However, it is difficult to certify genuine entanglement of practical multipartite states in this way, and even more difficult to quantify it, due to the difficulty in identifying optimal multipartite Bell inequalities and protocols tolerant to state impurity. In this Letter, we explore a general and robust device-independent method that can be applied to various realistic multipartite quantum states in arbitrary finite dimension, while merely relying on bipartite Bell inequalities. Our method allows us both to certify the presence of genuine multipartite entanglement and to quantify it. Several important classes of entangled states are tested with this method, leading to the detection of genuinely entangled states. We also certify genuine multipartite entanglement in weakly entangled Greenberger-Horne-Zeilinger states, showing that the method applies equally well to less standard states.
We show how to estimate a broad class of multipartite entanglement measures from Bell-basis measurement data. In addition to lowering the experimental requirements relative to previously known methods of estimating these measures, our proposed scheme also enables a simpler analysis of the number of measurement repetitions required to achieve an $\ensuremath{\epsilon}$-close approximation of the measures, which we provide for each. We focus our analysis on the recently introduced concentratable entanglements [Beckey et al., Phys. Rev. Lett. 127, 140501 (2021)] because many other well-known multipartite entanglement measures are recovered as special cases of this family of measures. We extend the definition of the concentratable entanglements to mixed states and show how to construct lower bounds on the mixed state concentratable entanglements that can also be estimated using only Bell-basis measurement data. Finally, we demonstrate the feasibility of our methods by classically simulating their implementation on a noisy Rydberg atom quantum computer.