Entanglement and analytical continuation: an intimate relation told by the Riemann zeta function

Abstract

We propose measurements on a quantum system to realize the Riemann zeta function ζ. A single system, that is classical interference, suffices to create the Dirichlet representation of ζ. In contrast, we need measurements performed on two entangled quantum systems to extend ζ into the critical strip of complex space where the non-trivial zeros of ζ are located. As a consequence, we can view these zeros as a result of a Schrödinger cat which is by its very construction similar to, but in its details very different from, the superposition formed by two coherent states of identical amplitudes but opposite phases. This interpretation suggests that entanglement in quantum mechanics is the analogue of analytic continuation of complex analysis.