Bounds on quantum correlations in Bell-inequality experiments

Bell-inequality violation is one of the most widely known manifestations of entanglement in quantum mechanics; indicating that experiments on physically separated quantum mechanical systems cannot be given a local realistic description. However, despite the importance of Bell inequalities, it is not known in general how to determine whether a given entangled state will violate a Bell inequality. This is because one can choose to make many different measurements on a quantum system to test any given Bell inequality and the optimization over measurements is a high-dimensional variational problem. In order to better understand this problem we present algorithms that provide, for a given quantum state and Bell inequality, both a lower bound and an upper bound on the maximal violation of the inequality. In many cases these bounds determine measurements that would demonstrate violation of the Bell inequality or provide a bound that rules out the possibility of a violation. Both bounds apply techniques from convex optimization and the methodology for creating upper bounds allows them to be systematically improved. Examples are given to illustrate how these algorithms can be used to conclude definitively if some quantum states violate a given Bell inequality.

[1]  A. Cabello.  Proposed experiment to test the bounds of quantum correlations. , 2003, Physical review letters.

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

[3]  Experimental evidence for bounds on quantum correlations. , 2003, Physical review letters.

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

[5]  N. Gisin.  Bell's inequality holds for all non-product states , 1991 .

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

[7]  J. Eisert,et al.  Complete hierarchies of efficient approximations to problems in entanglement theory , 2004 .

[8]  D. Avis,et al.  On the relationship between convex bodies related to correlation experiments with dichotomic observables , 2006, quant-ph/0605148.

[9]  N. Gisin,et al.  Grothendieck's constant and local models for noisy entangled quantum states , 2006, quant-ph/0606138.

[10]  Tobias J. Hagge,et al.  Physics , 1929, Nature.

[11]  Werner,et al.  Quantum states with Einstein-Podolsky-Rosen correlations admitting a hidden-variable model. , 1989, Physical review. A, General physics.

[12]  Pablo A. Parrilo,et al.  Semidefinite programming relaxations for semialgebraic problems , 2003, Math. Program..

[13]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[14]  E. Schrödinger.  Die gegenwärtige Situation in der Quantenmechanik , 1935, Naturwissenschaften.

[15]  Audra E. Kosh,et al.  Linear Algebra and its Applications , 1992 .

[16]  Harry Buhrman,et al.  Quantum Entanglement and Communication Complexity , 2000, SIAM J. Comput..

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

[18]  Nicolas Gisin,et al.  BELL'S INEQUALITIES DETECT EFFICIENT ENTANGLEMENT , 2003 .

[19]  S. Massar,et al.  Causality and Tsirelson's bounds , 2004, quant-ph/0409066.

[20]  Jean B. Lasserre,et al.  Global Optimization with Polynomials and the Problem of Moments , 2000, SIAM J. Optim..

[21]  Stephen P. Boyd,et al.  Semidefinite Programming , 1996, SIAM Rev..

[22]  M. Wolf,et al.  Bell’s inequalities for states with positive partial transpose , 1999, quant-ph/9910063.

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

[24]  N. Gisin,et al.  From Bell's theorem to secure quantum key distribution. , 2005, Physical review letters.

[25]  N Gisin,et al.  Quantum communication between N partners and Bell's inequalities. , 2001, Physical review letters.

[26]  Tsuyoshi Ito,et al.  Bell inequalities stronger than the Clauser-Horne-Shimony-Holt inequality for three-level isotropic states , 2006 .

[27]  A. Fine.  Hidden Variables, Joint Probability, and the Bell Inequalities , 1982 .

[28]  S. Popescu,et al.  Generic quantum nonlocality , 1992 .

[29]  S. Popescu,et al.  Which states violate Bell's inequality maximally? , 1992 .

[30]  P. Parrilo,et al.  Complete family of separability criteria , 2003, quant-ph/0308032.

[31]  James Demmel,et al.  Minimizing Polynomials via Sum of Squares over the Gradient Ideal , 2004, Math. Program..

[32]  N. Gisin,et al.  A relevant two qubit Bell inequality inequivalent to the CHSH inequality , 2003, quant-ph/0306129.

[33]  M. Horodecki,et al.  Violating Bell inequality by mixed spin- {1}/{2} states: necessary and sufficient condition , 1995 .

[34]  Karl Svozil,et al.  Generalizing Tsirelson's bound on Bell inequalities using a min-max principle. , 2004, Physical review letters.

[35]  J. Barrett.  Nonsequential positive-operator-valued measurements on entangled mixed states do not always violate a Bell inequality , 2001, quant-ph/0107045.

[36]  Michel Deza,et al.  Geometry of cuts and metrics , 2009, Algorithms and combinatorics.

[37]  Sophie Laplante,et al.  Simulating quantum correlations as a distributed sampling problem (9 pages) , 2005, quant-ph/0507120.

[38]  M. Żukowski,et al.  Bell's inequalities and quantum communication complexity. , 2004, Physical review letters.

[39]  B. M. Fulk.  MATH , 1992 .

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

[41]  G. Kimura.  The Bloch Vector for N-Level Systems , 2003, quant-ph/0301152.

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

[43]  K. Svozil,et al.  Testing the bounds on quantum probabilities , 2003, quant-ph/0306092.

[44]  Marek Zukowski,et al.  Quantum communication complexity protocol with two entangled qutrits. , 2002, Physical review letters.

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

[46]  M. Horodecki,et al.  Teleportation, Bell's inequalities and inseparability , 1996, quant-ph/9606027.

[47]  N. Gisin,et al.  Maximal violation of Bell's inequality for arbitrarily large spin , 1992 .

[48]  M. Horne,et al.  Experimental Consequences of Objective Local Theories , 1974 .

[49]  M. Horodecki,et al.  Mixed-State Entanglement and Distillation: Is there a “Bound” Entanglement in Nature? , 1998, quant-ph/9801069.

[50]  N. Khaneja,et al.  Characterization of the Positivity of the Density Matrix in Terms of the Coherence Vector Representation , 2003, quant-ph/0302024.

[51]  Peter J Seiler,et al.  SOSTOOLS: Sum of squares optimization toolbox for MATLAB , 2002 .

[52]  Popescu,et al.  Bell's inequalities versus teleportation: What is nonlocality? , 1994, Physical review letters.

[53]  Jan-Ake Larsson.  Modeling the singlet state with local variables , 1999 .

[54]  Barbara M Terhal,et al.  Symmetric extensions of quantum States and local hidden variable theories. , 2003, Physical review letters.

[55]  Michael M. Wolf,et al.  Bell inequalities and entanglement , 2001, Quantum Inf. Comput..

[56]  A. Doherty,et al.  Better Bell-inequality violation by collective measurements , 2006, quant-ph/0604045.

[57]  Pérès.  Separability Criterion for Density Matrices. , 1996, Physical review letters.