Optimal quantum violation of Clauser–Horne–Shimony–Holt like steering 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.

[1]  Rupert Ursin,et al.  Loophole-free Einstein–Podolsky–Rosen experiment via quantum steering , 2011, 1111.0760.

[2]  David Jennings,et al.  Quantum steering ellipsoids. , 2013, Physical review letters.

[3]  S. Walborn,et al.  Revealing hidden Einstein-Podolsky-Rosen nonlocality. , 2011, Physical review letters.

[4]  B. S. Cirel'son.  Quantum generalizations of Bell's inequality , 1980 .

[5]  Sibasish Ghosh,et al.  Degree of Complementarity Determines the Nonlocality in Quantum Mechanics , 2012, 1206.6054.

[6]  Manik Banik.  Measurement incompatibility and Schrödinger-Einstein-Podolsky-Rosen steering in a class of probabilistic theories , 2015, 1502.05779.

[7]  E. Schrödinger.  Probability relations between separated systems , 1936, Mathematical Proceedings of the Cambridge Philosophical Society.

[8]  A. Shimony,et al.  Proposed Experiment to Test Local Hidden Variable Theories. , 1969 .

[9]  Miguel Navascués,et al.  Quantifying Einstein-Podolsky-Rosen steering. , 2013, Physical review letters.

[10]  Michael J. W. Hall,et al.  Einstein–Podolsky–Rosen steering and the steering ellipsoid , 2014, 1411.1517.

[11]  Eric G. Cavalcanti,et al.  Analog of the Clauser-Horne-Shimony-Holt inequality for steering , 2015 .

[12]  E. Schrödinger.  Discussion of Probability Relations between Separated Systems , 1935, Mathematical Proceedings of the Cambridge Philosophical Society.

[13]  David Jennings,et al.  Quantum steering ellipsoids, extremal physical states and monogamy , 2014, 1403.0418.

[14]  Ou,et al.  Realization of the Einstein-Podolsky-Rosen paradox for continuous variables. , 1992, Physical review letters.

[15]  Eric G. Cavalcanti,et al.  Einstein-Podolsky-Rosen steering inequalities from entropic uncertainty relations , 2013, 1303.7432.

[16]  A C Doherty,et al.  Steering, entanglement, nonlocality, and the Einstein-Podolsky-Rosen paradox. , 2007, Physical review letters.

[17]  Paul Busch,et al.  Indeterminacy relations and simultaneous measurements in quantum theory , 1985 .

[18]  Albert Einstein,et al.  Can Quantum-Mechanical Description of Physical Reality Be Considered Complete? , 1935 .

[19]  H. M. Wiseman,et al.  Experimental criteria for steering and the Einstein-Podolsky-Rosen paradox , 2009, 0907.1109.

[20]  M. Wolf,et al.  Measurements incompatible in quantum theory cannot be measured jointly in any other no-signaling theory. , 2009, Physical review letters.

[21]  Reid,et al.  Demonstration of the Einstein-Podolsky-Rosen paradox using nondegenerate parametric amplification. , 1989, Physical review. A, General physics.

[22]  P. Busch,et al.  The quantum theory of measurement , 1991 .

[23]  Eric G. Cavalcanti,et al.  Entanglement verification and steering when Alice and Bob cannot be trusted , 2012, 1210.6051.

[24]  N. Brunner,et al.  One-way Einstein-Podolsky-Rosen Steering , 2014, 1402.3607.

[25]  Tamás Vértesi,et al.  Joint measurability, Einstein-Podolsky-Rosen steering, and Bell nonlocality. , 2014, Physical review letters.

[26]  Marc Kaplan,et al.  Fine-grained Einstein-Podolsky-Rosen-steering inequalities , 2014 .

[27]  V. Scarani,et al.  One-sided device-independent quantum key distribution: Security, feasibility, and the connection with steering , 2011, 1109.1435.

[28]  M D Reid,et al.  Genuine multipartite Einstein-Podolsky-Rosen steering. , 2012, Physical review letters.

[29]  A. C. Doherty,et al.  Entanglement, einstein-podolsky-rosen correlations, bell nonlocality, and steering , 2007, 0709.0390.

[30]  P. Busch,et al.  Unsharp reality and joint measurements for spin observables. , 1986, Physical review. D, Particles and fields.

[31]  H. M. Wiseman,et al.  Optimal measurements for tests of Einstein-Podolsky-Rosen steering with no detection loophole using two-qubit Werner states , 2014 .

[32]  Charles H. Bennett,et al.  Quantum cryptography without Bell's theorem. , 1992, Physical review letters.

[33]  Otfried Gühne,et al.  Joint measurability of generalized measurements implies classicality. , 2014, Physical review letters.

[34]  J. Bell.  On the Einstein-Podolsky-Rosen paradox , 1964 .