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arXiv:1404.7050 · Physical Review A

Fine-grained Einstein-Podolsky-Rosen–steering inequalities

Tanumoy Pramanik, Marc Kaplan, A. S. Majumdar

Published Nov 24, 2014• 98 citations
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Abstract

We derive a steering inequality based on a fine-grained uncertainty relation to capture Einstein-Podolsky-Rosen steering for bipartite systems. Our steering inequality improves over previous ones since it can experimentally detect all steerable two-qubit Werner state with only two measurement settings on each side. According to our inequality, pure entangled states are maximally steerable. Moreover, by slightly changing the setting, we can express the amount of violation of our inequality as a function of their violation of the Clauser-Horne-Shimony-Holt inequality. Finally, after deriving a monogamy relation we prove that the amount of violation of our steering inequality is, up to a constant factor, a lower bound on the key rate of a one-sided device-independent quantum key distribution protocol secure against individual attacks.

DOI
10.1103/physreva.90.050305

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We derive a steering inequality based on a fine-grained uncertainty relation to capture Einstein-Podolsky-Rosen steering for bipartite systems. Our steering inequality improves over previous ones since it can experimentally detect all steerable two-qubit Werner state with only two measurement settings on each side. According to our inequality, pure entangled states are maximally steerable. Moreover, by slightly changing the setting, we can express the amount of violation of our inequality as a function of their violation of the Clauser-Horne-Shimony-Holt inequality. Finally, after deriving a monogamy relation we prove that the amount of violation of our steering inequality is, up to a constant factor, a lower bound on the key rate of a one-sided device-independent quantum key distribution protocol secure against individual attacks.
Monogamy inequalities for the way bipartite Einstein-Podolsky-Rosen (EPR) steering can be distributed among NNN systems are derived. One set of inequalities is based on witnesses with two measurement settings, and may be used to demonstrate correlation of outcomes between two parties, that cannot be shared with more parties. It is shown that the monogamy for steering is directional. Two parties cannot independently demonstrate steering of a third system, using the same two-setting steering witness, but it is possible for one party to steer two independent systems. This result explains the monogamy of two-setting Bell inequality violations and the sensitivity of the continuous variable (CV) EPR criterion to losses on the steering party. We generalize to mmm settings. A second type of monogamy relation gives the quantitative amount of sharing possible, when the number of parties is less than or equal to mmm, and takes a form similar to the Coffman-Kundu-Wootters relation for entanglement. The results enable characterization of the tripartite steering for CV Gaussian systems and qubit Greenberger-Horne-Zeilinger and WWW states.
We derive a fine-grained uncertainty relation for the measurement of two incompatible observables on a single quantum system of continuous variables, and show that continuous-variable systems are more uncertain than discrete-variable systems. Using the derived fine-grained uncertainty relation, we formulate a stronger steering criterion that is able to reveal the steerability of NOON states that has hitherto not been possible using other criteria. We further obtain a monogamy relation for our steering inequality which leads to an, in principle, improved lower bound on the secret key rate of a one-sided device independent quantum key distribution protocol for continuous variables.
Abstract Steering, a quantum property stronger than entanglement but weaker than non-locality in the quantum correlation hierarchy, is a key resource for one-sided device-independent quantum key distribution applications, in which only one of the communicating parties is trusted. A fine-grained steering inequality was introduced in (2014 Phys. Rev. A 90 050305), enabling for the first time the detection of steering in all steerable two-qubit Werner states using only two measurement settings. Here, we numerically and experimentally investigate this inequality for generalized Werner states and successfully detect steerability in a wide range of two-photon polarization-entangled Bell local states generated by a parametric down-conversion source.
Quantum steering, as a cornerstone of quantum information, is usually used to witness the quantum correlation of bipartite and multi-partite states. Here, we experimentally demonstrate the quantum steering inequality of two-qubit mixed states based on the fine-grained uncertainty relation. Our experimental results show that the steering inequality has potent sensitivity to Werner states and Bell diagonal states. The steering strategy exhibits a strong ability to identify that Werner states are steerable when the decoherence coefficient a>12. Compared to the steering inequality obtained by another stratagem, the steering witness criteria of mixed states based on the fine-grained uncertainty relation demonstrated in our experiment has better precision and accuracy. Moreover, the detection efficiency in our measurement setup is only required to be 50% to close the detection loophole, which means our approach needs less detector efficiency to certificate the steerability of mixed states.
The two-qubit maximally steerable mixed state (MSMS), defined as one that violates to the most degree a steering inequality for any fixed linear entropy, is derived mathematically here based on the two-setting (three-setting) linear steering inequality and the two-setting Clauser-Horne-Shimony-Holt--like steering inequality. Interestingly, the form of the MSMS based on such different steering inequalities obtained here is identical to the two-qubit maximally nonlocal mixed state (MNMS). It is clearly shown that finding any two-qubit state, of which the state mixedness exceeds 2/32/32/3, thus violating the three-setting steering inequalities, is impossible. The violation of the inequalities with the Werner state and with the maximally entangled mixed state, respectively, as well as the relations between their optimal violation and the linear entropy, is also discussed comprehensively. In particular, within the range $\ensuremath{\varepsilon}(\ensuremath{\rho})\ensuremath{\in}[0,2/3]$ of the fixed linear entropy, the Werner state reaches the same violation as the MSMS does of the three-setting linear steering inequality, but not of the two-setting steering inequalities (for the latter inequalities the violation with the Werner state is generally less than that with the MSMS). For the MEMS, the optimal violation is always lower than that of the MSMS for any fixed linear entropy.
We consider the problem of one-sided device-independent self-testing of any pure entangled two-qubit state based on steering inequalities which certify the presence of quantum steering. In particular, we note that in the 2-2-2 steering scenario (involving two parties, two measurement settings per party, and two outcomes per measurement setting), the maximal violation of a fine-grained steering inequality can be used to witness certain extremal steerable correlations, which certify all pure two-qubit entangled states. We demonstrate that the violation of the analogous Clauser-Horne-Shimony-Holt inequality of steering or the nonvanishing value of a quantity constructed using a correlation function called mutual predictability, together with the maximal violation of the fine-grained steering inequality, can be used to self-test any pure entangled two-qubit state in a one-sided device-independent way.