Security in Quantum Cryptography vs. Nonlocal Hidden Variables

Abstract

Abstract. In order to prove equivalence of quantum mechanics with nonlocal hidden-variable theories of a Bohm type one assumes that all the possible measurements belong to a restricted class: (a) we measure only positions of particles and (b) have no access to exact values of initial conditions for Bohm's trajectories. However, in any computer simulation based on Bohm's equations one relaxes the assumption (b) and yet obtains agreement with quantum predictions concerning the results of positional measurements. Therefore a theory where (b) is relaxed, although in principle allowing for measurements of a more general type, cannot be experimentally falsified within the current experimental paradigm. Such generalized measurements have not been invented, or have been invented but the information is qualified, but we cannot exclude their possibility on the basis of known experimental data. Since the measurements would simultaneously allow for eavesdropping in standard quantum cryptosystems, the arguments for security of quantum cryptography become logically circular: Bohm-type theories do not allow for eavesdropping because they are fully equivalent to quantum mechanics, but the equivalence follows from the assumption that we cannot measure hidden variables, which would be equivalent to the possibility of eavesdropping... Here we break the vicious circle by a simple modification of entangled-state protocols that makes them secure even if our enemies have more imagination and know how to measure hidden-variable initial conditions with arbitrary precision. Keywords THE FUNDAMENTAL PROBLEM OF QUANTUM CRYPTOGRAPHY?There is a general agreement that quantum mechanics can be replaced by a nonlocal hidden-variable theory [1]. Such a theory involves additional degrees of freedom that we assume to be beyond our reach. However, statistics of the standard measurements would not be changed even if we could access the hidden variables. The proof is trivial: In any computer simulation based on Bohm trajectories we know the initial conditions in question, and yet obtain the correct quantum probabilities [2] satisfying no-eavesdropping criteria. Paradoxically, the criteria are fulfilled even though the eavesdropping is possible [3][4][5][6][7]. There is no contradiction with Bell's theorem because Bohm's hidden variables are nonlocal. This nonlocality has to be taken into account in the computer simulation