On the power of non-local boxes

A non-local box is a virtual device that has the following property: given that Alice inputs a bit at her end of the device and that Bob does likewise, it produces two bits, one at Alice's end and one at Bob's end, such that the XOR of the outputs is equal to the AND of the inputs. This box, inspired from the CHSH inequality, was first proposed by Popescu and Rohrlich to examine the question: given that a maximally entangled pair of qubits is non-local, why is it not maximally non-local? We believe that understanding the power of this box will yield insight into the non-locality of quantum mechanics. It was shown recently by Cerf, Gisin, Massar and Popescu, that this imaginary device is able to simulate correlations from any measurement on a singlet state. Here, we show that the non-local box can in fact do much more: through the simulation of the magic square pseudo-telepathy game and the Mermin-GHZ pseudo-telepathy game, we show that the non-local box can simulate quantum correlations that no entangled pair of qubits can, in a bipartite scenario and even in a multi-party scenario. Finally we show that a single non-local box cannot simulate all quantum correlations and propose a generalization for a multi-party non-local box. In particular, we show quantum correlations whose simulation requires an exponential amount of non-local boxes, in the number of maximally entangled qubit pairs.

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

[3]  Gilles Brassard,et al.  Minimum entangled state dimension required for pseudo-telepathy , 2004, Quantum Inf. Comput..

[4]  M. Newman.  Independent Sets and Eigenspaces , 2004 .

[5]  Nicolas Gisin,et al.  Entanglement and non-locality are different resources , 2004, quant-ph/0412109.

[6]  Gilles Brassard,et al.  Recasting mermin's multi-player game into the framework of pseudo-telepathy , 2005, Quantum Inf. Comput..

[7]  P. K. Aravind,et al.  Bell’s Theorem Without Inequalities and Only Two Distant Observers , 2001, OFC 2001.

[8]  N J Cerf,et al.  Simulating maximal quantum entanglement without communication. , 2005, Physical review letters.

[9]  N. S. Barnett,et al.  Private communication , 1969 .

[10]  Dave Bacon,et al.  Classical simulation of quantum entanglement without local hidden variables , 2001 .

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

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

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

[14]  Stefan Wolf,et al.  Oblivious transfer and quantum non-locality , 2005, Proceedings. International Symposium on Information Theory, 2005. ISIT 2005..

[15]  S. Massar,et al.  Quantum entanglement can be simulated without communication , 2004, quant-ph/0410027.

[16]  Gilles Brassard,et al.  Cost of Exactly Simulating Quantum Entanglement with Classical Communication , 1999 .

[17]  H. Buhrman,et al.  Limit on nonlocality in any world in which communication complexity is not trivial. , 2005, Physical review letters.

[18]  N. Mermin.  Quantum mysteries revisited , 1990 .

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

[20]  G. Brassard,et al.  Quantum Pseudo-Telepathy , 2004, quant-ph/0407221.

[21]  S. Popescu,et al.  Quantum nonlocality as an axiom , 1994 .

[22]  Cerf,et al.  Classical teleportation of a quantum Bit , 2000, Physical review letters.

[23]  D. Bacon,et al.  Communication cost of simulating Bell correlations. , 2003, Physical review letters.

[24]  Peter Høyer,et al.  Consequences and limits of nonlocal strategies , 2004, Proceedings. 19th IEEE Annual Conference on Computational Complexity, 2004..