An interpretation and understanding of complex modular values

TL;DR

This paper explores the interpretation of complex modular values in quantum mechanics, emphasizing the importance of modulus and argument separation, and relating these to phase factors and pointer probabilities.

Contribution

It introduces a novel interpretation of modular values using dynamic phase factors and relates their modulus and argument to measurable quantum phases and probabilities.

Findings

Modular values are expressed by average dynamic phase factors with complex probabilities.

The modulus of modular values relates to changes in qubit pointer probabilities.

The argument of modular values combines geometric and intrinsic phases.

Abstract

In contrast to that a weak value of an observable is usually divided into real and imaginary parts, here we show that separation into modulus and argument is important for modular values. We first show that modular values are expressed by the average of dynamic phase factors with complex conditional probabilities. We then relate, using the polar decomposition, the modulus of the modular value to the relative change in the qubit pointer post-selection probabilities, and relate the argument of the modular value to the summation of a geometric phase and an intrinsic phase.