Multidimensional Bell Inequalities and Quantum Cryptography

ARTUR K. EKERT

Stephen Mark Barnett, Simon J. D. Phoenix

We introduce a suitably constructed Bell inequality as a test of the integrity of a single-particle quantum cryptography scheme.

François Arnault

We give a multidimensional generalization of the complete set of Bell-correlation inequalities given by Werner and Wolf (2001 Phys. Rev. A 64 032112) and by Zˆukowski and Brukner (2002 Phys. Rev. Lett. 88 210401), for the two-dimensional case. Our construction applies to the n -party, two-observable case, where each observable is d -valued. The inequalities obtained involve homogeneous polynomials. They define the facets of a polytope in a complex vector space of dimension d n . We detail the inequalities obtained in the case d = 3 and, from them, we recover known inequalities. We finally explain how the violations of our inequalities by quantum mechanics can be computed and could be observed, when using unitary observables.

Ingemar Bengtsson

Zhengjun Cao, Lihua Liu

Quantum entanglement is of great importance to quantum cryptography and computation. So far, all experimental demonstrations of entanglement are designed to check Bell's inequality which is based on Bell's formulation for EPR paradox. In this note, we specify the assumptions needed in Bell's mathematical argument. We then show the contradictions among these assumptions. As a result, it becomes very easy to see that Bell's inequality is trivial.