The hyperbolic, the arithmetic and the quantum phase

TL;DR

This paper introduces a novel finite-dimensional quantum phase framework linking phase locking with cyclotomy and Ramanujan sums, revealing classical-like phase variability and phase entanglement in specific quantum states.

Contribution

It presents a new approach to quantum phase based on mathematical concepts, highlighting phase locking, squeezing, and entanglement phenomena in finite-dimensional Hilbert spaces.

Findings

Phase variability peaks at prime power dimensions.

Phase squeezing is possible for certain quantum states.

Phase entanglement is introduced for Kloosterman pairs.

Abstract

We develop a new approach of the quantum phase in an Hilbert space of finite dimension which is based on the relation between the physical concept of phase locking and mathematical concepts such as cyclotomy and the Ramanujan sums. As a result, phase variability looks quite similar to its classical counterpart, having peaks at dimensions equal to a power of a prime number. Squeezing of the phase noise is allowed for specific quantum states. The concept of phase entanglement for Kloosterman pairs of phase-locked states is introduced.