Complementarity and uncertainty in a two-way interferometer

TL;DR

This paper introduces a new operator-based framework to analyze the complementarity and uncertainty in a two-way interferometer, linking these fundamental quantum concepts through variance relations.

Contribution

It presents a novel operator approach for predictability and visibility, connecting complementarity with uncertainty in a two-way interferometer.

Findings

Predictability and visibility are represented by Hermitian and non-Hermitian operators.

The total variance of predictability and visibility relates to their complementarity.

A new variance-based uncertainty relation links complementarity and uncertainty.

Abstract

In terms of operator, the two complementary quantities, the predictability and visibility, are reinvestigated in a two-way interferometer. One Hermitian operator and one non-Hermitian operator (composed of two Hermitian operators) are introduced for the predictability and visibility, respectively. The predictability and visibility can not be measured exactly simultaneously, due to the non-commutation between the two operators. The sum of the variances of the predictability and visibility (the total variance), is used to measure the uncertainty, which is linked to the complementarity relation through the equation, $(\delta_P)^2+(\delta_Vf)^2+P^2+V^2=2$ . This new description for the predictability and visibility connects the complementarity and the uncertainty relations, although neither of them can be derived directly from the other.