Replay to "Comment on 'Quantum phase for an arbitrary system with finite-dimensional Hilbert space'

TL;DR

This paper clarifies the distinction between relative and absolute phase observables, demonstrating that the quantum expectation of relative phase is highly discontinuous and depends on the number-theoretic properties of frequencies.

Contribution

It highlights the crucial difference between relative and absolute phase observables and reveals the discontinuous nature of the quantum expectation of the relative phase.

Findings

Quantum expectation of relative phase is highly discontinuous.

Phase expectation depends on the number-theoretic nature of frequencies.

Clarifies the distinction between relative and absolute phase observables.

Abstract

We point out the crucial difference between the relative and absolute phase observables treated in our contribution \cite{1} and in the Comment by Hall and Pegg \cite{HP} respectively. The main contribution of our work is to show that the quantum expectation of the relative phase is highly discontinuous function of the frequency and to point out interesting dependence of the phase on the number-theoretic nature of the frequencies.