Graph-theoretical Bounds on the Entangled Value of Non-local Games

We introduce a novel technique to give bounds to the entangled value of non-local games. The technique is based on a class of graphs used by Cabello, Severini and Winter in 2010. The upper bound uses the famous Lovasz theta number and is efficiently computable; the lower one is based on the quantum independence number, which is a quantity used in the study of entanglement-assisted channel capacities and graph homomorphism games.

[1]  T. Fritz,et al.  A Combinatorial Approach to Nonlocality and Contextuality , 2012, Communications in Mathematical Physics.

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

[3]  Subhash Khot,et al.  On the power of unique 2-prover 1-round games , 2002, Proceedings 17th IEEE Annual Conference on Computational Complexity.

[4]  David E. Roberson,et al.  Quantum homomorphisms , 2016, J. Comb. Theory, Ser. B.

[5]  David E. Roberson,et al.  Graph Homomorphisms for Quantum Players , 2014, TQC.

[6]  Simone Severini,et al.  New Separations in Zero-Error Channel Capacity Through Projective Kochen–Specker Sets and Quantum Coloring , 2013, IEEE Transactions on Information Theory.

[7]  Simone Severini,et al.  On the Quantum Chromatic Number of a Graph , 2007, Electron. J. Comb..

[8]  Asher Peres,et al.  Two simple proofs of the Kochen-Specker theorem , 1991 .

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

[10]  Debbie W. Leung,et al.  Improving zero-error classical communication with entanglement , 2009, Physical review letters.

[11]  László Lovász,et al.  On the Shannon capacity of a graph , 1979, IEEE Trans. Inf. Theory.

[12]  A. Winter,et al.  (Non-)Contextuality of Physical Theories as an Axiom , 2010, 1010.2163.

[13]  Claude E. Shannon,et al.  The zero error capacity of a noisy channel , 1956, IRE Trans. Inf. Theory.

[14]  Julia Kempe,et al.  Unique Games with Entangled Provers are Easy , 2008, 2008 49th Annual IEEE Symposium on Foundations of Computer Science.

[15]  A. Winter,et al.  Graph-theoretic approach to quantum correlations. , 2014, Physical review letters.