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arXiv:1701.08253 · Physical Review A

Improvement in device-independent witnessing of genuine tripartite entanglement by local marginals

Some Sankar Bhattacharya, Biswajit Paul, Arup Roy, Amit Mukherjee, Chellasamy Jebarathinam, Manik Banik

Published Apr 24, 2017• 7 citations
arXivHTMLPDF

Abstract

We consider the problem of determining the presence of genuine tripartite entanglement through the violation of Mermin inequality. Although the violation of Mermin inequality cannot certify the presence of genuine nonlocality, it can certify the presence of genuine tripartite entanglement whenever the violation is strictly greater than $2\sqrt{2}$. Here we show that Mermin inequality suffices as a genuine entanglement witness even when its value is $2\sqrt{2}$ if at least two of the local marginal distributions are not completely random, provided the local Hilbert space dimension of at least one of the subsystems is 2. Thus local marginals suffice as a semi-device-independent genuine entanglement witness. This is intriguing in a sense, as the local properties of a composite system can help to identify its global property. Furthermore, analyzing another quantity constructed from Mermin polynomials, we show that the genuine entanglement certification task for the correlations with Mermin inequality violation equal to $2\sqrt{2}$ can actually be made fully device independent.

DOI
10.1103/physreva.95.042130

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We consider the problem of determining the presence of genuine tripartite entanglement through the violation of Mermin inequality. Although the violation of Mermin inequality cannot certify the presence of genuine nonlocality, it can certify the presence of genuine tripartite entanglement whenever the violation is strictly greater than 222\sqrt{2}22​. Here we show that Mermin inequality suffices as a genuine entanglement witness even when its value is 222\sqrt{2}22​ if at least two of the local marginal distributions are not completely random, provided the local Hilbert space dimension of at least one of the subsystems is 2. Thus local marginals suffice as a semi-device-independent genuine entanglement witness. This is intriguing in a sense, as the local properties of a composite system can help to identify its global property. Furthermore, analyzing another quantity constructed from Mermin polynomials, we show that the genuine entanglement certification task for the correlations with Mermin inequality violation equal to 222\sqrt{2}22​ can actually be made fully device independent.
We consider the problem of determining whether genuine multipartite entanglement was produced in an experiment, without relying on a characterization of the systems observed or of the measurements performed. We present an n-partite inequality that is satisfied by all correlations produced by measurements on biseparable quantum states, but which can be violated by n-partite entangled states, such as Greenberger-Horne-Zeilinger states. In contrast to traditional entanglement witnesses, the violation of this inequality implies that the state is not biseparable independently of the Hilbert space dimension and of the measured operators. Violation of this inequality does not imply, however, genuine multipartite nonlocality. We show more generically how the problem of identifying genuine tripartite entanglement in a device-independent way can be addressed through semidefinite programming.
We demonstrate that, in the tripartite scenario with all parties' local events being space-like separated, Hardy-type nonlocality reveals a stronger nonlocal correlation than those captured by Mermin-type inequalities, an important distinction previously unrecognized. To substantiate this assertion, we develop a general framework for tripartite correlations by extending the notion of Settings Independence and Outcome Independence beyond their bipartite formulation. This framework highlights the pivotal role of Hardy-type reasoning in detecting genuine multipartite nonlocality. Furthermore, we show that the tripartite Hardy-nonlocality enables self-testing of a broad class of pure nonmaximally genuine entangled tripartite states, offering a key advantage over Bell inequality-based methods by allowing certification even under nonmaximal violations. Unlike Bell functionals, which typically self-test only a single extremal point, the Hardy relation self-tests a set of extremal correlations for any nonzero Hardy probability. This, in turn, facilitates the device-independent certification of randomness from Hardy-type correlations. We find that the maximum certifiable randomness is ${log}_{2}7\ensuremath{\approx}2.8073$-bits, highlighting both the practical and foundational significance of Hardy-based techniques for quantum randomness generation.
Experimental demonstration of entanglement needs to have a precise control of experimentalist over the system on which the measurements are performed as prescribed by an appropriate entanglement witness. To avoid such trust problem, recently device-independent entanglement witnesses (\emph{DIEW}s) for genuine tripartite entanglement have been proposed where witnesses are capable of testing genuine entanglement without precise description of Hilbert space dimension and measured operators i.e apparatus are treated as black boxes. Here we design a protocol for enhancing the possibility of identifying genuine tripartite entanglement in a device independent manner. We consider three mixed tripartite quantum states none of whose genuine entanglement can be detected by applying standard \emph{DIEW}s, but their genuine tripartite entanglement can be detected by applying the same when distributed in some suitable entanglement swapping network.
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.
It has recently been shown that by broadcasting the subsystems of a bipartite quantum state, one can activate Bell nonlocality and significantly improve noise tolerance bounds for device-independent entanglement certification. In this work we strengthen these results and explore new aspects of this phenomenon. First, we prove new results related to the activation of Bell nonlocality. We construct Bell inequalities tailored to the broadcast scenario, and show how broadcasting can lead to even stronger notions of Bell nonlocality activation. In particular, we exploit these ideas to show that bipartite states admitting a local hidden-variable model for general measurements can lead to genuine tripartite nonlocal correlations. We then study device-independent entanglement certification in the broadcast scenario, and show through semidefinite programming techniques that device-independent entanglement certification is possible for the two-qubit Werner state in essentially the entire range of entanglement. Finally, we extend the concept of EPR steering to the broadcast scenario, and present novel examples of activation of the two-qubit isotropic state. Our results pave the way for broadcast-based device-independent and semi-device-independent protocols.
While quantum networks have been extensively studied as a natural extension of the standard Bell scenario with a richer correlation structure, general constructions of nonlinear Bell inequalities with self-testing properties are still largely lacking. In this work, we present a general method for constructing such inequalities in the simplest network scenario, in which two independent sources distribute bipartite quantum states to three spatially separated observers. These inequalities allow for arbitrary numbers of binary measurements and are maximally violated by maximally entangled states of the corresponding local dimensions together with sets of pairwise anticommuting Clifford observables. Importantly, their maximal quantum values can be determined analytically, which makes them particularly promising for device-independent applications. In particular, we prove that these Bell inequalities can be used to device-independently certify the underlying quantum network, including both the quantum states produced by the sources and the observables measured by all parties. To the best of our knowledge, this is the first self-testing result for quantum networks that relies solely on the maximal violation of a nonlocality witness.