Bell inequalities for three systems and arbitrarily many measurement outcomes

We present a family of Bell inequalities for three parties and arbitrarily many outcomes, which can be seen as a natural generalization of the Mermin-Bell inequality. For a small number of outcomes, we verify that our inequalities define facets of the polytope of local correlations. We investigate the quantum violations of these inequalities, in particular with respect to the Hilbert space dimension. We provide strong evidence that the maximal quantum violation can be reached only using systems with local Hilbert space dimension exceeding the number of measurement outcomes. This suggests that our inequalities can be used as multipartite dimension witnesses.

[1]  N. Gisin,et al.  A framework for the study of symmetric full-correlation Bell-like inequalities , 2012, 1201.2055.

[2]  Tamás Vértesi,et al.  Testing the structure of multipartite entanglement with Bell inequalities. , 2011, Physical review letters.

[3]  Travis Norsen,et al.  Bell's theorem , 2011, Scholarpedia.

[4]  H. Buhrman,et al.  A Generalized Grothendieck Inequality and Nonlocal Correlations that Require High Entanglement , 2011 .

[5]  Melvyn Ho,et al.  Device-independent certification of entangled measurements. , 2011, Physical review letters.

[6]  T. Vértesi,et al.  Multisetting Bell-type inequalities for detecting genuine multipartite entanglement , 2011, 1102.4320.

[7]  Stefano Pironio,et al.  Device-independent witnesses of genuine multipartite entanglement. , 2011, Physical review letters.

[8]  Miguel Navascues,et al.  Certifying entangled measurements in known Hilbert spaces , 2011, 1101.5361.

[9]  Nicolas Brunner,et al.  Semi-device-independent bounds on entanglement , 2010, 1012.1513.

[10]  Adrian Kent,et al.  Private randomness expansion with untrusted devices , 2010, 1011.4474.

[11]  Nicolas Gisin,et al.  Detecting genuine multipartite quantum nonlocality: a simple approach and generalization to arbitrary dimensions. , 2010, Physical review letters.

[12]  Detecting fullN-particle entanglement in arbitrarily-high-dimensional systems with Bell-type inequalities , 2010, 1010.3762.

[13]  Rodrigo Gallego,et al.  Device-independent tests of classical and quantum dimensions. , 2010, Physical review letters.

[14]  Tamás Vértesi,et al.  Maximal violation of a bipartite three-setting, two-outcome Bell inequality using infinite-dimensional quantum systems , 2010 .

[15]  N. Gisin,et al.  Looking for symmetric Bell inequalities , 2010, 1004.4146.

[16]  Stefano Pironio,et al.  Random numbers certified by Bell’s theorem , 2009, Nature.

[17]  Stefano Pironio,et al.  Convergent Relaxations of Polynomial Optimization Problems with Noncommuting Variables , 2009, SIAM J. Optim..

[18]  S. Massar,et al.  Device-independent state estimation based on Bell’s inequalities , 2009, 0907.2170.

[19]  D. Deng,et al.  Reexamination of a multisetting Bell inequality for qudits , 2009, 0903.4964.

[20]  T. V'ertesi,et al.  Quantum bounds on Bell inequalities , 2008, 0810.1615.

[21]  V. Scarani,et al.  Testing the dimension of Hilbert spaces. , 2008, Physical review letters.

[22]  A. Acín,et al.  A convergent hierarchy of semidefinite programs characterizing the set of quantum correlations , 2008, 0803.4290.

[23]  T. V'ertesi,et al.  Efficiency of higher-dimensional Hilbert spaces for the violation of Bell inequalities , 2007, 0712.4320.

[24]  T. V'ertesi,et al.  Generalized Clauser-Horne-Shimony-Holt inequalities maximally violated by higher-dimensional systems , 2007, 0712.4225.

[25]  C. J. Foster,et al.  Bell inequalities for continuous-variable correlations. , 2007, Physical review letters.

[26]  V. Scarani,et al.  Device-independent security of quantum cryptography against collective attacks. , 2007, Physical review letters.

[27]  Yeong-Cherng Liang,et al.  Bounds on quantum correlations in Bell-inequality experiments , 2006, quant-ph/0608128.

[28]  A. Acín,et al.  Bounding the set of quantum correlations. , 2006, Physical review letters.

[29]  Stefano Pironio,et al.  Maximally Non-Local and Monogamous Quantum Correlations , 2006, Physical review letters.

[30]  Nicolas Gisin,et al.  Optimal bell tests do not require maximally entangled states. , 2005, Physical review letters.

[31]  Stefano Pironio.  Lifting Bell inequalities , 2005 .

[32]  Adrian Kent,et al.  No signaling and quantum key distribution. , 2004, Physical review letters.

[33]  J. Bell,et al.  Speakable and Unspeakable in Quantum Mechanics: Preface to the first edition , 2004 .

[34]  J. Lofberg,et al.  YALMIP : a toolbox for modeling and optimization in MATLAB , 2004, 2004 IEEE International Conference on Robotics and Automation (IEEE Cat. No.04CH37508).

[35]  C. Śliwa.  Symmetries of the Bell correlation inequalities , 2003, quant-ph/0305190.

[36]  Ll Masanes.  Tight Bell inequality for d-outcome measurements correlations , 2003, Quantum Inf. Comput..

[37]  F. Verstraete,et al.  Entanglement versus bell violations and their behavior under local filtering operations. , 2001, Physical review letters.

[38]  J. Latorre,et al.  Quantum nonlocality in two three-level systems , 2001, quant-ph/0111143.

[39]  S. Massar,et al.  Bell inequalities for arbitrarily high-dimensional systems. , 2001, Physical review letters.

[40]  M. Żukowski,et al.  Bell's theorem for general N-qubit states. , 2001, Physical review letters.

[41]  M. Wolf,et al.  All-multipartite Bell-correlation inequalities for two dichotomic observables per site , 2001, quant-ph/0102024.

[42]  D. Kaszlikowski,et al.  Violations of local realism by two entangled N-dimensional systems are stronger than for two qubits , 2000, Physical review letters.

[43]  Mann,et al.  Maximal violation of Bell inequalities for mixed states. , 1992, Physical review letters.

[44]  Ekert,et al.  Quantum cryptography based on Bell's theorem. , 1991, Physical review letters.

[45]  Mermin Nd.  Simple unified form for the major no-hidden-variables theorems. , 1990 .

[46]  Kiel T. Williams,et al.  Extreme quantum entanglement in a superposition of macroscopically distinct states. , 1990, Physical review letters.

[47]  A. Zeilinger,et al.  Speakable and Unspeakable in Quantum Mechanics , 1989 .

[48]  M. Kafatos.  Bell's theorem, quantum theory and conceptions of the universe , 1989 .

[49]  J. Bell,et al.  Speakable and Unspeakable in Quatum Mechanics , 1988 .

[50]  R. Stephenson.  A and V , 1962, The British journal of ophthalmology.