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arXiv:0806.0096 · Physical Review A

More efficient Bell inequalities for Werner states

Tamás Vértesi

Published Sep 17, 2008• 141 citations
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Abstract

In this paper we study the nonlocal properties of two-qubit Werner states parametrized by the visibility parameter $0\ensuremath{\leqslant}p\ensuremath{\leqslant}1$. A family of Bell inequalities is constructed that proves the two-qubit Werner states to be nonlocal for the parameter range $0.7056<p\ensuremath{\leqslant}1$. This is slightly wider than the range $0.7071<p\ensuremath{\leqslant}1$, corresponding to the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. This answers a question posed by Gisin in the positive, i.e., there exist Bell inequalities which are more efficient than the CHSH inequality in the sense that they are violated by a wider range of two-qubit Werner states.

DOI
10.1103/physreva.78.032112

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In this paper we study the nonlocal properties of two-qubit Werner states parametrized by the visibility parameter 0\ensuremath⩽p\ensuremath⩽10\ensuremath{\leqslant}p\ensuremath{\leqslant}10\ensuremath⩽p\ensuremath⩽1. A family of Bell inequalities is constructed that proves the two-qubit Werner states to be nonlocal for the parameter range 0.7056<p\ensuremath⩽10.7056<p\ensuremath{\leqslant}10.7056<p\ensuremath⩽1. This is slightly wider than the range 0.7071<p\ensuremath⩽10.7071<p\ensuremath{\leqslant}10.7071<p\ensuremath⩽1, corresponding to the violation of the Clauser-Horne-Shimony-Holt (CHSH) inequality. This answers a question posed by Gisin in the positive, i.e., there exist Bell inequalities which are more efficient than the CHSH inequality in the sense that they are violated by a wider range of two-qubit Werner states.
We find two two-qubit bipartite states ${\ensuremath{\rho}}_{1},{\ensuremath{\rho}}_{2}$ such that arbitrarily many copies of one or the other cannot exhibit nonlocal correlations in a two-setting--two-outcome Bell scenario. However, the bipartite state ${\ensuremath{\rho}}_{1}\ensuremath{\bigotimes}{\ensuremath{\rho}}_{2}$ violates the Clauser-Horne-Shimony-Holt (CHSH) Bell inequality [J. F. Clauser, M. A. Horne, A. Shimony, and R. A. Holt, Phys. Rev. Lett. 23, 880 (1969).] by an amount of 2.023. We also identify a CHSH-local state $\ensuremath{\rho}$ such that ${\ensuremath{\rho}}^{\ensuremath{\bigotimes}2}$ is CHSH inequality-violating. The tools employed can be easily adapted to find instances of nonlocality activation in arbitrary Bell scenarios.
We construct a Bell inequality from the Clauser-Horne-Shimony-Holt inequality for two qubits that provides a stronger bound on the correlations of entangled states than allowed by the CHSH inequality. The argument involved here can be generalized to nnn qubits. The inequalities obtained are violated by all the generalized Greenberger-Horne-Zeilinger states of multiqubits.
We investigate the violation of Bell-type inequalities for two-qubit Werner-like states parametrized by the positive parameter $0\ensuremath{\leqslant}p\ensuremath{\leqslant}1$. We use an unbalanced homodyne detection scheme to obtain the quantum mechanical probabilities. A violation of the Bell-Wigner and Janssens inequalities is obtained for a large range of the parameter ppp. The range given by these inequalities is greater than the one given by the Clauser-Horne inequality. The range in which a violation is attained actually coincides with the range where the Werner-like states are known to be nonseparable, i.e., for p>1∕3p>1∕3p>1∕3. However, the improvement over the Clauser-Horne inequality is achieved at the price of restricting the class of possible local hidden variable theories.
A correlation inequality is derived from local realism and a supplementary assumption. Unlike Clauser-Horne (CH) inequality [or Clauser-Horne-Shimony-Holt (CHSH) inequality] which is violated by quantum mechanics by a factor of 2\sqrt 22​, this inequality is violated by a factor of 1.5. Thus the magnitude of violation of this inequality is approximately 20.7% larger than the magnitude of violation of previous inequalities. Moreover, unlike CH (or CHSH) inequality which requires the measurement of five detection probabilities, the present inequality requires the measurement of only two detection probabilities. This inequality can therefore be used to test locality more simply than CH or CHSH inequality.
We study a recently proposed Einstein–Podolsky–Rosen steering inequality (Cavalcanti et al 2015 J. Opt. Soc. Am. B 32 A74–A81 ). Analogous to Clauser–Horne–Shimony–Holt (CHSH) inequality for Bell nonlocality, in the simplest scenario, i.e., two parties, two measurements per party and two outcomes per measurement, this newly proposed inequality has been proved to be necessary and sufficient for steering. In this article we find the optimal violation amount of this inequality in quantum theory. Interestingly, the optimal violation amount matches with optimal quantum violation of CHSH inequality, i.e., Cirel'son quantity. We further study the optimal violation of this inequality for different classes of 2-qubit quantum states.
It is well-known that in certain scenarios weakly entangled states can generate stronger nonlocal effects than their maximally entangled counterparts. In this paper, we consider violations of the Clauser-Horne-Shimony-Holt (CHSH) inequality when one party has inefficient detectors, a scenario known as an asymmetric Bell experiment. For any fixed detection efficiency, we derive a simple upper bound on the entanglement needed to violate the inequality by more than some specified amount $\ensuremath{\kappa}\ensuremath{\ge}0$. When $\ensuremath{\kappa}=0$, the amount of entanglement in all states violating the inequality goes to zero as the detection efficiency approaches 5050%50 from above. We finally consider the scenario in which detection inefficiency arises for only one choice of local measurement. In this case, it is shown that the CHSH inequality can always be violated for any nonzero detection efficiency and any choice of noncommuting measurements.