M ay 2 00 6 Information-theoretic temporal Bell inequality and quantum computation

Separation between classical and quantum phenomena plays a crucial role in understanding the weirdness of quantum physics and has recently been extensively investigated, especially in the context of quantum information science. One way of drawing a rigid distinction between them is to show the violation of the Bell inequality [1]. In this paper, we develop a Bell-type inequality that distinguishes between classical and quantum computations. Here, in order to allow arguments based on conditional entropy between measurement outcomes of two different observables, we employ the information-theoretic Bell inequalities formulated by Braunstein and Caves [2, 3]. Furthermore, instead of discussing correlations between observables of distantly located physical systems, we focus on one and the same physical system and analyze correlations between measurement outcomes at different times as in the argument on the temporal Bell inequality initiated by Leggett and Garg [4]. The information-theoretic temporal Bell inequality is formulated for classical algorithms. It is satisfied by any classical algorithm but can be violated by quantum ones. As an example to show the violation of the inequality, we discuss the so-called “database” search problem, which is formally stated as follows: Given a black box (oracle) that calculates the function F of x ∈ {0, . . . , 2n−1} such that, for unknown s,