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arXiv:1704.08162 · Physical Review A

Tripartite-entanglement detection through tripartite quantum steering in one-sided and two-sided device-independent scenarios

Chellasamy Jebarathinam, Debarshi Das, Arup Roy, Amit Mukherjee, Some Sankar Bhattacharya, Bihalan Bhattacharya, Alberto Riccardi, Debasis Sarkar

Published Aug 1, 2018• 22 citations
arXivHTMLPDF

Abstract

In the present work we study tripartite quantum steering of quantum correlations arising from two local dichotomic measurements on each side in the two types of partially device-independent scenarios: the one-sided device-independent scenario where one of the parties performs untrusted measurements while the other two parties perform trusted measurements and the two-sided device-independent scenario where one of the parties performs trusted measurements while the other two parties perform untrusted measurements. We demonstrate that tripartite steering in the two-sided device-independent scenario is weaker than tripartite steering in the one-sided device-independent scenario by using two families of quantum correlations. That is, these two families of quantum correlations in the two-sided device-independent framework detect tripartite entanglement through tripartite steering for a larger region than that in the one-sided device-independent framework. It is shown that tripartite steering in the two-sided device-independent scenario implies the presence of genuine tripartite entanglement of $2\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}2$ quantum system, even if the correlation does not exhibit genuine nonlocality or genuine steering.

DOI
10.1103/physreva.98.022101

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In the present work we study tripartite quantum steering of quantum correlations arising from two local dichotomic measurements on each side in the two types of partially device-independent scenarios: the one-sided device-independent scenario where one of the parties performs untrusted measurements while the other two parties perform trusted measurements and the two-sided device-independent scenario where one of the parties performs trusted measurements while the other two parties perform untrusted measurements. We demonstrate that tripartite steering in the two-sided device-independent scenario is weaker than tripartite steering in the one-sided device-independent scenario by using two families of quantum correlations. That is, these two families of quantum correlations in the two-sided device-independent framework detect tripartite entanglement through tripartite steering for a larger region than that in the one-sided device-independent framework. It is shown that tripartite steering in the two-sided device-independent scenario implies the presence of genuine tripartite entanglement of 2\ifmmode×\else\texttimes\fi2\ifmmode×\else\texttimes\fi22\ifmmode\times\else\texttimes\fi{}2\ifmmode\times\else\texttimes\fi{}22\ifmmode×\else\texttimes\fi2\ifmmode×\else\texttimes\fi2 quantum system, even if the correlation does not exhibit genuine nonlocality or genuine steering.
Einstein-Podolsky-Rosen (EPR) steering is a form of quantum nonlocality which is intermediate between entanglement and Bell nonlocality. EPR steering is a resource for quantum key distribution that is device independent on only one side in that it certifies bipartite entanglement when one party's device is not characterized while the other party's device is fully characterized. In this work, we introduce two types of genuine tripartite EPR steering, and derive two steering inequalities to detect them. In a semi-device-independent scenario where only the dimensions of two parties are assumed, the correlations which violate one of these inequalities also certify genuine tripartite entanglement. It is known that Alice can demonstrate bipartite EPR steering to Bob if and only if her measurement settings are incompatible. We demonstrate that quantum correlations can also detect tripartite EPR steering from Alice to Bob and Charlie, even if Charlie's measurement settings are compatible.
Quantum steering captures the ability of one party, through local measurements on a shared entangled state, to affect the conditional states held by distant parties in a way that admits no local explanation. Its device-independent (DI) quantification -- requiring no characterization of any state or measurement -- has been developed in the bipartite setting, notably through the framework of assemblage moment matrices (AMMs) [Phys. Rev. Lett. 116, 240401 (2016)], but has remained largely unexplored beyond it. Here, we extend the AMM framework to tripartite steering, covering both the scenario in which two parties jointly steer a third one (2-steer-1) and that in which a single party steers the remaining two (1-steer-2). For each scheme, we construct a semidefinite program that, from the observed correlations -- or merely from the violation of a Bell inequality -- yields lower bounds on the steering robustness of the underlying assemblage. As a by-product, our bounds also certify, in a DI manner, the incompatibility of the measurements performed by the steering parties.
Quantum steering is a kind of quantum feature between quantum entanglement and Bell non-locality. Due to its special asymmetry, quantum steering plays an important role in many aspects of quantum information processing and thus has been widely concerned. There are many contributions on quantum steering in bipartite systems up to now, while tripartite systems have more complex structures than bipartite systems and then have more diversified steering scenarios. In this paper, two types of quantum steering scenarios are introduced for a tripartite quantum system, named “one-sided device-independent steering" and “two-sided device-independent steering". Each of them contains two levels according to the separability of the local hidden states and the independence of the local hidden variables, respectively. Based on giving the mathematical definitions of these steering scenarios, some necessary and sufficient conditions for a state to be unsteerable are obtained and sufficient conditions for a state to be steerable are also established.
We consider the task of certification of genuine entanglement of tripartite states. We first present an all-versus-nothing proof of tripartite Einstein-Podolsky-Rosen (EPR) steering by demonstrating the non-existence of a local hidden state (LHS) model in the tripartite network as a motivation to our main result. A full logical contradiction of the predictions of the LHS model with quantum mechanical outcome statistics for any three-qubit generalized Greenberger-Horne-Zeilinger (GGHZ) states and pure W-class states is shown, using which, one can distinguish between the GGHZ and W-class states in the two-sided device-independent (2SDI) steering scenario. We next formulate a 2SDI fine-grained steering inequality for the tripartite scenario. We show that the maximum quantum violation of this FGI can be used to certify genuine entanglement of three-qubit pure states.
Quantum steering, as a manifestation of nonlocal quantum correlations, plays a crucial role in enabling various quantum information processing tasks. However, practical implementations are often hindered by significant challenges arising from imperfect or untrusted measurement devices. This study investigates the impact of measurement inaccuracies on quantum steering, with a particular focus on errors in the untrusted party's measurement devices. We first analyze how such errors affect the evaluation of steering inequalities, and then derive bipartite steering inequalities based on correlation matrices under imperfect measurements. Our findings show that even small measurement errors can significantly compromise the certification of quantum steerability, an effect that becomes particularly pronounced as the system dimension increases. Furthermore, by extending the proposed steering inequality to a modified tripartite scenario via correlation matrices, we demonstrate that the influence of measurement imperfections is far more severe in multipartite quantum steering than in the bipartite case. Our results underscore the critical need to account for measurement imperfections in experimental quantum steering and provide a theoretical framework for characterizing and mitigating these effects in high-dimensional quantum systems.
Loophole-free quantum nonlocality often demands experiments with high complexity (defined by all parties' settings and outcomes) and multiple efficient detectors. Here, we identify the fundamental efficiency and complexity thresholds for quantum steering using two-qubit entangled states. Remarkably, it requires only one photon detector on the untrusted side, with efficiency ε>1/Xε> 1/Xε>1/X, where X≥2X \geq 2X≥2 is the number of settings on that side. This threshold applies to all pure entangled states, in contrast to analogous Bell-nonlocality tests, which require almost unentangled states to be loss-tolerant. We confirm these predictions in a minimal-complexity (X=2X = 2X=2 for the untrusted party and a single three-outcome measurement for the trusted party), detection-loophole-free photonic experiment with $ε= (51.6 \pm 0.4)\% $.