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Isolating genuine nonclassicality in tripartite quantum correlations

Chellasamy Jebarathinam

Published Jul 21, 2014• 0 citations
OpenAlex

Abstract

tions under the constraints of the HLHV model [3] and fully LHV model [15]. Analogous to genuine quantum discord [13], these measures detect the presence of genuine nonclassicality in Svetlichny-local correlations as well. We obtain a 3-decomposition that any correlation in the Svetlichny-box polytope which is a subpolytope of full NS polytope can be written as a convex combination of a Svetlichny-box, a maximally three-way contextual box, and a box which does not have Svetlichny and Mermin discord. Svetlichny and Mermin discord quantify the components of Svetlichny-box and three-way contextual box respectively in the 3-decomposition. Thus, Svetlichny and Mermin discord quantify genuine nonclassicality of Svetlichny-local quantum correlations originating from Svetlichny nonlocality and three-way contextuality respectively. We identify the set of genuinely nonclassical biseparable and separable three-qubit states using Svetlichny and Mermin discord.

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Quantum correlations forms a subset of the set of nonsignaling boxes. This allows us to characterize quantum correlations as a convex combination of the extremal boxes of the nonsignaling polytope which are Popescu-Rohrlich boxes (maximally nonlocal boxes) and local deterministic boxes. There exists multiple decomposition of quantum correlations in the context of the nonsignaling polytope. I find that the existence of Popescu-Rohrlich box decomposition for local boxes associates two notions of discord which capture nonclassicality of quantum correlations originating from Bell nonlocality and EPR-steering. I introduce, Bell and Mermin discord, and demonstrate that any bipartite nonsignaling box admits a three-way decomposition. This decomposition allows us to isolate the origin of nonclassicality into three disjoint sources: a Popescu-Rohrlich box, a maximally local box that detects EPR-steering, and a classical correlation. Interestingly, I show that all non-null quantum discord states which are neither classical-quantum states nor quantum-classical states can give rise to nonclassical correlations which have non-null Bell and/or Mermin discord for suitable noncommuting measurements. I introduce two notions of genuine discord, which are the generalizations of Bell and Mermin discord to the multipartite boxes, to characterize the presence of genuine nonclassicality in multipartite quantum correlations.

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Does Svetlichny-box polytope fully characterize quantum correlations?

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In the nonsignaling framework, nonclassicality in correlation arising from two spatially separated input-output devices gets manifested, solely, through its \emph{nonlocal} behavior. Study of correlations based on this said feature is commonly known as local-nonlocal paradigm. While in two-party scenario correlations can be of only two types either local or nonlocal, the situation gets more involved for multi-party scenario, \emph{e.g.,} for tripartite scenario, correlations can be of three types: fully local, two-way local, and genuinely nonlocal. Nonsignaling correlations having quantum realization are termed physical. Fully local and certain quantum realizable two-way local tripartite correlations always have a quantum realization with tripartite biseparable states if there is no restriction on the local Hilbert-space dimensions. In this work, we study the quantum simulation of fully local and two-way local tripartite correlations with restricted local Hilbert-space dimensions, in particular we consider $\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2$ quantum systems. Interestingly, in this restricted simulation scenario we find that simulation of certain fully local and two-way local correlations necessarily requires \emph{genuine quantumness} in the three qubit states. This, going beyond the standard nonlocality paradigm, captures a new notion of genuine nonclassicality even in the fully local and two-way local correlations. To explore this newly introduced notion of genuine nonclassicality, we propose two quantities of interest, called \emph{Svetlichny strength} and \emph{Mermin strength}, and extensively study their properties.

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We introduce the measures, Bell discord (BD) and Mermin discord (MD), to characterize bipartite quantum correlations in the context of nonsignaling (NS) polytopes. These measures divide the full NS polytope into four regions depending on whether BD and/or MD is zero. This division of the NS polytope allows us to obtain a 3-decomposition that any bipartite box with two binary inputs and two binary outputs can be decomposed into Popescu-Rohrlich (PR) box, a maximally local box, and a local box with BD and MD equal to zero. BD and MD quantify two types of nonclassicality of correlations arising from all quantum correlated states which are neither classical-quantum states nor quantum-classical states. BD and MD serve us the semi-device-independent witnesses of nonclassicality of local boxes in that nonzero value of these measures imply incompatible measurements and nonzero quantum discord only when the dimension of the measured states is fixed. The 3-decomposition serves us to isolate the origin of the two types of nonclassicality into a PR-box and a maximally local box which is related to EPR-steering, respectively. We consider a quantum polytope that has an overlap with all the four regions of the full NS polytope to figure out the constraints of quantum correlations.

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According to the studies of genuine tripartite nonlocality in discrete variable quantum systems conducted so far, Svetlichny inequality is considered as the best Bell-type inequality to detect genuine (three way) nonlocality of pure tripartite genuine entangled states. In the present work, we have considered another Bell-type inequality (which has been reported as the 99th facet of NS 2 local polytope in Bancal et al (2013 Phys. Rev. A 88 014102 ), to reveal genuine tripartite nonlocality of generalized GHZ (Greenberger–Horne–Zeilinger) class and a subclass of extended GHZ class states Acín et al (2000 Phys. Rev. Lett. 85 1560 ) thereby proving the conjecture given by Bancal et al (2013 Phys. Rev. A 88 014102 ) for the GGHZ class and the subclass of extended GHZ states. We compare the violation of this inequality with Svetlichny inequality which reveals the efficiency of the former inequality over the latter to demonstrate genuine nonlocality using the above classes of quantum states. Even in some cases discord monogamy score can be used as a better measure of quantum correlation over Svetlichny inequality for those classes of pure states. Besides, the 99th facet inequality is found efficient not only for revealing genuine nonlocal behavior of correlations emerging in systems using pure entangled states but also in some cases of mixed entangled states over Svetlichny inequality and some well known measures of entanglement.

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Genuine multpartite quantum nonlocality can be quantified by the communication cost needed to reproduce the nonlocal correlation by classical communication models. This prompts the question as to how one may provide such an operational characterization for the nonclassicality of local multipartite correlations arising from genuinely quantum states possessing quantum discord. To answer this question, we consider a classical simulation protocol where one of the parties pre-share a random variable with the other parties who may also share arbitrary randomness. We demonstrate that genuine nonclassicality of local multipartite correlations is necessary for implying genuine superlocality which is the requirement for a larger dimension of the random variable in the above classical simulation protocol than that of quantum state that can generate the nonclassical correlations.

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Definitions of multipartite nonlocality

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In a multipartite setting, it is possible to distinguish quantum states that are genuinely $n$-way entangled from those that are separable with respect to some bipartition. Similarly, the nonlocal correlations that can arise from measurements on entangled states can be classified into those that are genuinely $n$-way nonlocal, and those that are local with respect to some bipartition. Svetlichny introduced an inequality intended as a test for genuine tripartite nonlocality. This work introduces two alternative definitions of $n$-way nonlocality, which we argue are better motivated both from the point of view of the study of nature, and from the point of view of quantum information theory. We show that these definitions are strictly weaker than Svetlichny's, and introduce a series of suitable Bell-type inequalities for the detection of three-way nonlocality. Numerical evidence suggests that all three-way entangled pure quantum states can produce three-way nonlocal correlations.

arXiv:1112.2626•Physical Review A•Jul 22, 2013•218 citations
tions under the constraints of the HLHV model [3] and fully LHV model [15]. Analogous to genuine quantum discord [13], these measures detect the presence of genuine nonclassicality in Svetlichny-local correlations as well. We obtain a 3-decomposition that any correlation in the Svetlichny-box polytope which is a subpolytope of full NS polytope can be written as a convex combination of a Svetlichny-box, a maximally three-way contextual box, and a box which does not have Svetlichny and Mermin discord. Svetlichny and Mermin discord quantify the components of Svetlichny-box and three-way contextual box respectively in the 3-decomposition. Thus, Svetlichny and Mermin discord quantify genuine nonclassicality of Svetlichny-local quantum correlations originating from Svetlichny nonlocality and three-way contextuality respectively. We identify the set of genuinely nonclassical biseparable and separable three-qubit states using Svetlichny and Mermin discord.
Quantum correlations forms a subset of the set of nonsignaling boxes. This allows us to characterize quantum correlations as a convex combination of the extremal boxes of the nonsignaling polytope which are Popescu-Rohrlich boxes (maximally nonlocal boxes) and local deterministic boxes. There exists multiple decomposition of quantum correlations in the context of the nonsignaling polytope. I find that the existence of Popescu-Rohrlich box decomposition for local boxes associates two notions of discord which capture nonclassicality of quantum correlations originating from Bell nonlocality and EPR-steering. I introduce, Bell and Mermin discord, and demonstrate that any bipartite nonsignaling box admits a three-way decomposition. This decomposition allows us to isolate the origin of nonclassicality into three disjoint sources: a Popescu-Rohrlich box, a maximally local box that detects EPR-steering, and a classical correlation. Interestingly, I show that all non-null quantum discord states which are neither classical-quantum states nor quantum-classical states can give rise to nonclassical correlations which have non-null Bell and/or Mermin discord for suitable noncommuting measurements. I introduce two notions of genuine discord, which are the generalizations of Bell and Mermin discord to the multipartite boxes, to characterize the presence of genuine nonclassicality in multipartite quantum correlations.
In the nonsignaling framework, nonclassicality in correlation arising from two spatially separated input-output devices gets manifested, solely, through its \emph{nonlocal} behavior. Study of correlations based on this said feature is commonly known as local-nonlocal paradigm. While in two-party scenario correlations can be of only two types either local or nonlocal, the situation gets more involved for multi-party scenario, \emph{e.g.,} for tripartite scenario, correlations can be of three types: fully local, two-way local, and genuinely nonlocal. Nonsignaling correlations having quantum realization are termed physical. Fully local and certain quantum realizable two-way local tripartite correlations always have a quantum realization with tripartite biseparable states if there is no restriction on the local Hilbert-space dimensions. In this work, we study the quantum simulation of fully local and two-way local tripartite correlations with restricted local Hilbert-space dimensions, in particular we consider C2⊗C2⊗C2\mathbb{C}^2\otimes\mathbb{C}^2\otimes\mathbb{C}^2C2⊗C2⊗C2 quantum systems. Interestingly, in this restricted simulation scenario we find that simulation of certain fully local and two-way local correlations necessarily requires \emph{genuine quantumness} in the three qubit states. This, going beyond the standard nonlocality paradigm, captures a new notion of genuine nonclassicality even in the fully local and two-way local correlations. To explore this newly introduced notion of genuine nonclassicality, we propose two quantities of interest, called \emph{Svetlichny strength} and \emph{Mermin strength}, and extensively study their properties.
We introduce the measures, Bell discord (BD) and Mermin discord (MD), to characterize bipartite quantum correlations in the context of nonsignaling (NS) polytopes. These measures divide the full NS polytope into four regions depending on whether BD and/or MD is zero. This division of the NS polytope allows us to obtain a 3-decomposition that any bipartite box with two binary inputs and two binary outputs can be decomposed into Popescu-Rohrlich (PR) box, a maximally local box, and a local box with BD and MD equal to zero. BD and MD quantify two types of nonclassicality of correlations arising from all quantum correlated states which are neither classical-quantum states nor quantum-classical states. BD and MD serve us the semi-device-independent witnesses of nonclassicality of local boxes in that nonzero value of these measures imply incompatible measurements and nonzero quantum discord only when the dimension of the measured states is fixed. The 3-decomposition serves us to isolate the origin of the two types of nonclassicality into a PR-box and a maximally local box which is related to EPR-steering, respectively. We consider a quantum polytope that has an overlap with all the four regions of the full NS polytope to figure out the constraints of quantum correlations.
According to the studies of genuine tripartite nonlocality in discrete variable quantum systems conducted so far, Svetlichny inequality is considered as the best Bell-type inequality to detect genuine (three way) nonlocality of pure tripartite genuine entangled states. In the present work, we have considered another Bell-type inequality (which has been reported as the 99th facet of NS 2 local polytope in Bancal et al (2013 Phys. Rev. A 88 014102 ), to reveal genuine tripartite nonlocality of generalized GHZ (Greenberger–Horne–Zeilinger) class and a subclass of extended GHZ class states Acín et al (2000 Phys. Rev. Lett. 85 1560 ) thereby proving the conjecture given by Bancal et al (2013 Phys. Rev. A 88 014102 ) for the GGHZ class and the subclass of extended GHZ states. We compare the violation of this inequality with Svetlichny inequality which reveals the efficiency of the former inequality over the latter to demonstrate genuine nonlocality using the above classes of quantum states. Even in some cases discord monogamy score can be used as a better measure of quantum correlation over Svetlichny inequality for those classes of pure states. Besides, the 99th facet inequality is found efficient not only for revealing genuine nonlocal behavior of correlations emerging in systems using pure entangled states but also in some cases of mixed entangled states over Svetlichny inequality and some well known measures of entanglement.
Genuine multpartite quantum nonlocality can be quantified by the communication cost needed to reproduce the nonlocal correlation by classical communication models. This prompts the question as to how one may provide such an operational characterization for the nonclassicality of local multipartite correlations arising from genuinely quantum states possessing quantum discord. To answer this question, we consider a classical simulation protocol where one of the parties pre-share a random variable with the other parties who may also share arbitrary randomness. We demonstrate that genuine nonclassicality of local multipartite correlations is necessary for implying genuine superlocality which is the requirement for a larger dimension of the random variable in the above classical simulation protocol than that of quantum state that can generate the nonclassical correlations.
In a multipartite setting, it is possible to distinguish quantum states that are genuinely nnn-way entangled from those that are separable with respect to some bipartition. Similarly, the nonlocal correlations that can arise from measurements on entangled states can be classified into those that are genuinely nnn-way nonlocal, and those that are local with respect to some bipartition. Svetlichny introduced an inequality intended as a test for genuine tripartite nonlocality. This work introduces two alternative definitions of nnn-way nonlocality, which we argue are better motivated both from the point of view of the study of nature, and from the point of view of quantum information theory. We show that these definitions are strictly weaker than Svetlichny's, and introduce a series of suitable Bell-type inequalities for the detection of three-way nonlocality. Numerical evidence suggests that all three-way entangled pure quantum states can produce three-way nonlocal correlations.