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arXiv:1609.05555 · Physical Review A

Conditional steering under the von Neumann scenario

Kaushiki Mukherjee, Biswajit Paul, Sumana Karmakar, Debasis Sarkar, Amit Mukherjee, Some Sankar Bhattacharya, Arup Roy

Published Aug 16, 2017• 0 citations
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Abstract

In Phys. Lett. A 166, 293 (1992), Popescu and Rohrlich characterized nonlocality of pure $n$-partite entangled systems by studying bipartite violation of local realism when $n\ensuremath{-}2$ number of parties perform projective measurements on their particles. A pertinent question in this scenario is whether similar characterization is possible for $n$-partite mixed entangled states also. In the present work we have followed an analogous approach so as to explore whether given a tripartite mixed entangled state the conditional bipartite states obtained by performing projective measurement on the third party demonstrate a weaker form of nonlocality, quantum steering. We also compare this phenomenon of conditional steering with existing notions of tripartite correlations.

DOI
10.1103/physreva.96.022122

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In Phys. Lett. A 166, 293 (1992), Popescu and Rohrlich characterized nonlocality of pure nnn-partite entangled systems by studying bipartite violation of local realism when n\ensuremath−2n\ensuremath{-}2n\ensuremath−2 number of parties perform projective measurements on their particles. A pertinent question in this scenario is whether similar characterization is possible for nnn-partite mixed entangled states also. In the present work we have followed an analogous approach so as to explore whether given a tripartite mixed entangled state the conditional bipartite states obtained by performing projective measurement on the third party demonstrate a weaker form of nonlocality, quantum steering. We also compare this phenomenon of conditional steering with existing notions of tripartite correlations.
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.
The phenomenon of quantum steering in bipartite quantum systems can be reduced to the question of whether or not the first party can perform measurements such that the conditional states on the second party can be explained by a local hidden-state model. Clearly, the answer to this depends on the measurements which the first party is able to perform. We introduce a local hidden-state model explaining the conditional states for all generalized measurements on Bell-diagonal states of two qubits. More precisely, it is known for the restricted case of projective measurements and Bell-diagonal states characterized by the correlation matrix TTT that a local hidden-state model exists if and only if ${R}_{T}=2\ensuremath{\pi}{N}_{T}\left|det(T)\right|\ensuremath{\ge}1$, where NT{N}_{T}NT​ is defined by an integral over the Bloch sphere. For generalized measurements described by positive operator-valued measures we construct a model if ${R}_{T}\ensuremath{\ge}6/5$. Our work paves the way for a systematic study of steerability of quantum states with generalized measurements beyond the highly symmetric Werner states.
Phenomena that slip beyond the grasp of our classical intuition reveal uniquely quantum effects that deepen our understanding of the physical world and enable advances in information processing, particularly in quantum communication and computation. One such phenomenon is quantum steering, whereby measurements performed by one party influence the conditional states of another when the two share an entangled quantum system. If the observed correlations cannot be explained by a local hidden state model, the state is said to be steerable. In this work, we investigate the typicality of this behavior: given a generic two-qubit state and mmm Haar-random projective measurements, what is the probability of observing steering? We derive analytical expressions for the steering probability PS\mathcal{P}_SPS​ of Werner states in two- and three-setting scenarios, the latter restricted to coplanar projective measurements on the Bloch sphere. For larger numbers of settings and various random states ensembles, we perform numerical analyses showing that PS\mathcal{P}_SPS​ increases systematically with the number of measurements and substantially exceeds the corresponding probabilities associated with Bell nonlocality. Our results demonstrate that random states with minimal environmental coupling exhibit a high probability of steering for finite mmm and approach genuine typicality, PS=100%\mathcal{P}_S=100\%PS​=100%, as the number of settings increases. We provide a detailed characterization of PS\mathcal{P}_SPS​ across different state ensembles and specific families, including Bell-diagonal and Werner states, identifying those with the greatest non-classical potential and highlighting their relevance for protocols in which steering serves as a key resource.
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.
Steerability is a characteristic of quantum correlations lying between entanglement and Bell nonlocality. Understanding how these steering correlations can be shared between different parties has profound applications in ensuring the security of quantum communication protocols. Here we show that at most two bipartite reduced states of a three-qubit state can violate the three-setting CJWR linear steering inequality, contrary to the two-setting linear steering inequality. This result explains that quantum steering correlations have limited shareability properties apart from the conventional 'nonshareable' monogamy constraint. In contrast to the two-measurement-setting scenario, the three-setting scenario turns out to be more useful for developing a deeper understanding of the shareability of tripartite steering correlations. Apart from the distribution of steering correlations, several relations between reduced bipartite steering, different measures of bipartite entanglement of reduced states, and genuine tripartite entanglement are presented here. The results enable the detection of different kinds of tripartite entanglement.