Dirichlet series as interfering probability amplitudes for quantum measurements

Abstract

We show that all Dirichlet series, linear combinations of them and their analytical continuations represent probability amplitudes for measurements on time-dependent quantum systems. In particular, we connect an arbitrary Dirichlet series to the time evolution of an appropriately prepared quantum state in a non-linear oscillator with logarithmic energy spectrum. However, the realization of a superposition of two Dirichlet sums and its analytical continuation requires two quantum systems which are entangled, and a joint measurement. We illustrate our approach of implementing arbitrary Dirichlet series in quantum systems using the example of the Riemann zeta function and relate its non-trivial zeros to the interference of two quantum states reminiscent of a Schrödinger cat.

Keywords

• Dirichlet series
• non-linear oscillator
• logarithmic energy spectrum
• Riemann zeta function
• analytic continuation
• Berry–Keating representation