@MISC{Fivel94theprime, author = {Daniel I. Fivel}, title = {The Prime Factorization Property of Entangled Quantum States}, year = {1994} }

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Abstract

Completely entangled quantum states are shown to factorize into tensor products of entangled states whose dimensions are powers of prime numbers. The entangled states of each prime-power dimension transform among themselves under a finite Heisenberg group. We consider processes in which factors are exchanged between entangled states and study canonical ensembles in which these processes occur. It is shown that the Riemann zeta function is the appropriate partition function and that the Riemann hypothesis makes a prediction about the high temperature contribution of modes of large dimension. Completely entangled quantum states are of considerable interest not only in the foundations of quantum mechanics[1, 2], but also because of the possibility of their use in secure communication schemes[3, 4]. It has been shown[5] that any completely entangled two-particle state is expressible in the following form: Let |n, ν〉, n = 1, 2, · · ·, N be any basis of a Hilbert space of dimension N. Here ν = 1, 2 is usually taken to be a particle label, but more generally it may label any pair of Hilbert spaces of the same dimension. Let U indicate an arbitrary anti-unitary transformation on the N-dimensional Hilbert space and define |n U, ν 〉 = U|n, ν〉. (1) Then the state