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.

Dirichlet series as interfering probability amplitudes for quantum measurements

“…We recall [17] that a similar combination of phase factors appears in the time evolution created by the interaction Hamiltonian…”

“…Since in [17] we have proposed a method how to realize such an interaction we now assume that Ĥ I can indeed be implemented. Hence, we find for the initial state…”

“…However, we emphasize that our interpretation in terms of interfering probability amplitudes of joint measurements works for both formulae. The results for the Riemann-Siegel representation are given in [17].…”

“…as the probability amplitudes of the state Ψ | 〉as suggested in [17]. This choice provides three significant advantages: (i) Since na nb Ψ Ψ = , we start from a product state which gets entangled by the time-evolution.…”

“…Unfortunately, in this case we cannot employ the two techniques discussed in the previous sections, because they are limited to an individual system. However, in [17] we have already outlined an approach how to implement (11) is convergent only if 1 σ < . Hence, the line of absolute convergence of (B.3) and therefore of a ζ is at¯1 σ = which is to the right of the line of convergence 0 0 σ = .…”