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arXiv:1912.10976 · Physical Review A

Sharing nonlocality and nontrivial preparation contextuality using the same family of Bell expressions

Asmita Kumari, A. K. Pan

Published Dec 24, 2019• 54 citations
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Abstract

In the work by Silva et al. [Phys. Rev. Lett. 114, 250401 (2015)], the sharing of nonlocality by multiple observers was demonstrated through the quantum violation of Clauser-Horne-Shimony-Halt inequality. We provide a scheme for the sharing of nonlocality and nontrivial preparation contextuality sequentially through the quantum violation of a family of Bell's inequalities, where Alice and Bob perform ${2}^{n\ensuremath{-}1}$ and $n$ numbers of measurements of dichotomic observables, respectively. For this, we consider that Alice always performs projective measurement and multiple Bobs sequentially perform unsharp measurement. We show that when Bob's choices of measurement settings are unbiased, a maximum of two Bobs can sequentially share the nonlocality through the violation of our inequalities. Further, we show that the local bound of the aforementioned family of inequalities gets reduced if nontrivial preparation noncontextuality assumptions are further imposed. Then there is a chance to share the nontrivial preparation contextuality for a greater number of Bobs than that of nonlocality. We demonstrate that the nontrivial preparation contextuality can be sequentially shared by arbitrary numbers of Bob for unbiased choices of his measurement settings.

DOI
10.1103/physreva.100.062130

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In the work by Silva et al. [Phys. Rev. Lett. 114, 250401 (2015)], the sharing of nonlocality by multiple observers was demonstrated through the quantum violation of Clauser-Horne-Shimony-Halt inequality. We provide a scheme for the sharing of nonlocality and nontrivial preparation contextuality sequentially through the quantum violation of a family of Bell's inequalities, where Alice and Bob perform 2n\ensuremath−1{2}^{n\ensuremath{-}1}2n\ensuremath−1 and nnn numbers of measurements of dichotomic observables, respectively. For this, we consider that Alice always performs projective measurement and multiple Bobs sequentially perform unsharp measurement. We show that when Bob's choices of measurement settings are unbiased, a maximum of two Bobs can sequentially share the nonlocality through the violation of our inequalities. Further, we show that the local bound of the aforementioned family of inequalities gets reduced if nontrivial preparation noncontextuality assumptions are further imposed. Then there is a chance to share the nontrivial preparation contextuality for a greater number of Bobs than that of nonlocality. We demonstrate that the nontrivial preparation contextuality can be sequentially shared by arbitrary numbers of Bob for unbiased choices of his measurement settings.
This paper illustrates a direct connection between quantum steering and non-trivial preparation contextuality. In two party-two measurement per party-two outcomes per measurement (2−2−2)(2-2-2)(2−2−2) Bell scenario, any argument of Bell nonlocality is a proof of trivial preparation contextuality; however, the converse may not hold. If one of the parties (say, Alice) performs the measurements of more than two dichotomic observables, then it is possible to find a set of non-trivial functional relations between Alice's observables. We argue that the existence of a suitable set of such non-trivial relations between Alice's observables may warrant the unsteerability of quantum states at the end of another spatially separated party (say, Bob). Interestingly, such constraints can be read as non-trivial preparation non-contextuality assumptions in an ontological model. We further demonstrate two types of Bell inequalities that can be converted into linear steering inequalities using the aforementioned non-trivial conditions on Alice's observables. Such steering inequalities can also be considered as non-trivial preparation noncontextual inequalities. Since the local bound of the family of Bell expression gets reduced under the additional non-trivial conditions, it provides a test of quantum steering and nonlocality from the same family of Bell expressions depending upon its violation of the non-trivial preparation non-contextual or the local bound, thereby establishing a direct connection between quantum steering and non-trivial preparation contextuality.
Based on the quantum violation of the bipartite Bell inequality, it has been established that the sharing of nonlocality can be demonstrated for at most two sequential observers at one end and at most one pair of observers at both ends. In this work we study the sharing of nonlocality and preparation contextuality based on a bipartite Bell inequality, involving arbitrary nnn measurements by one party and ${2}^{n\ensuremath{-}1}$ measurements by another party. Such a Bell inequality has two bounds, the local bound and the preparation noncontextual bound, which is smaller than the local bound. We show that while nonlocality can be shared only by the first pair of the sequential observers, the preparation contextuality can be shared by an arbitrary pair of independent sequential observers at both ends.
Bipartite entangled states whose violations of the Clauser-Horne-Shimony-Holt Bell inequality can be observed by a single Alice and arbitrarily many sequential Bobs exist [Brown and Colbeck, Phys. Rev. Lett. 125, 090401 (2020)]. Here we consider their analogs for tripartite systems: a tripartite entangled state is shared among Alice, Bob, and multiple Charlies. The first Charlie measures his qubit and then passes his qubit to the next Charlie, who measures again with other measurements, and so on. The goal is to maximize the number of Charlies that can observe some kind of nonlocality with the single Alice and Bob. It has been shown that at most two Charlies can share genuine nonlocality of the Greenberger-Horne-Zeilinger state via the violation of the Svetlichny inequality with Alice and Bob [S. Saha et al., Quantum Inf. Process. 18, 42 (2019); Zhang and Fei, Phys. Rev. A 103, 032216 (2021)]. In this work, we show that arbitrarily many Charlies can have standard nonlocality (via violations of the Mermin inequality) and some other kind of genuine nonlocality (which is known as genuinely nonsignal nonlocality) with a single Alice and single Bob.
There exist bipartite entangled states whose violations of Clauser-Horne-Shimony-Holt (CHSH) Bell inequality can be observed by a single Alice and arbitrarily many sequential Bobs [Phys. Rev. Lett. 125, 090401 (2020)]. Here we consider its analogues for tripartite systems: a tripartite entangled state is shared among Alice, Bob and multiple Charlies. The first Charlie measures his qubit and then passes his qubit to the next Charlie who measures again with other measurements and so on. The goal is to maximize the number of Charlies that can observe some kind of nonlocality with the single Alice and Bob. It has been shown that at most two Charlies could share genuine nonlocality of the Greenberger-Horne-Zeilinger (GHZ) state via the violation of Svetlichny inequality with Alice and Bob [Quantum Inf. Process. 18, 42 (2019) and Phys. Rev. A 103, 032216 (2021)]. In this work, we show that arbitrarily many Charlies can have standard nonlocality (via violations of Mermin inequality) and some other kind of genuine nonlocality (which is known as genuinely nonsignal nonlocality) with the single Alice and single Bob.
Contextuality and measurement incompatibility are two fundamental aspects of nonclassicality, and their manifestations in observed quantum correlations are often deeply interconnected. Recently, measurement incompatibility has been studied in connection with nonlocality, particularly in terms of their robustness under various quantum channels. This line of investigation helps establish a connection between the channels that break nonlocality and those that break incompatibility. In this study, we focus on an asymmetric bipartite Bell scenario involving three and four inputs on Alice and Bob sides, respectively, with each of these inputs having dichotomous outcomes. Under the assumption of locality, the observed statistics in this asymmetric scenario obeys the Elegant Bell inequality (EBI). Here, we use a different version of the EBI that relies on the assumption of the preparation noncontextuality. By taking the violation of this noncontextual version of EBI as a witness of preparation contextuality we establish a connection between the channels that break contextuality and the channels that break triple-wise measurement incompatibility. Our results suggest that any channel which breaks EBI contextuality will also break Clauser-Horne-Shimony-Holt (CHSH) nonlocality; however, the reverse does not hold. We also show that a depolarising channel that breaks N-wise incompatibility can also break a certain form of contextuality, witnessed by a generalised inequality involving N measurements on one wing of a bipartite Bell scenario.
The prevailing consensus is that the sequential sharing of nonlocality in a Bell experiment requires generalized unsharp measurements, given that a sharp measurement inevitably destroys the entanglement of the shared state. In contrast, a recent work [A. Steffinlongo and A. Tavakoli, Projective measurements are sufficient for recycling nonlocality, Phys. Rev. Lett. 129, 230402 (2022)PRLTAO0031-900710.1103/PhysRevLett.129.230402] demonstrated the sharing of nonlocality up to two sequential observers by employing projective measurement aided with local randomness. Here, we introduce a form of local randomness-assisted qubit projective measurement protocol that enables the sharing of nonlocality by an arbitrary number of sequential observers (Bobs) with a single spatially separated observer (Alice). We achieve this by inspecting the diversity involved in implementing generalized measurements that harness nonlocality by preserving a sufficient amount of entanglement of the shared two-qubit entangled state. Furthermore, we highlight the crucial interplay between the degrees of measurement incompatibility of Alice and Bob in demonstrating the unbounded sharing of nonlocality.