There is entanglement in the primes

Abstract

Large series of prime numbers can be superposed on a single quantum register and then analyzed in full parallelism. The construction of this Prime state is efficient, as it hinges on the use of a quantum version of any efficient primality test. We show that the Prime state turns out to be very entangled as shown by the scaling properties of purity, Renyi entropy and von Neumann entropy. An analytical approximation to these measures of entanglement can be obtained from the detailed analysis of the entanglement spectrum of the Prime state, which in turn produces new insights in the Hardy-Littlewood conjecture for the pairwise distribution of primes. The extension of these ideas to a Twin Prime state shows that this new state is even more entangled than the Prime state, obeying majorization relations. We further discuss the construction of quantum states that encompass relevant series of numbers and opens the possibility of applying quantum computation to Arithmetics in novel ways.Comment: 30 pages, 11 Figs. Addition of two references and correction of typo