Operational path-phase complementarity in single-photon interferometry

TL;DR

This paper explores how path-phase complementarity manifests in single-photon interferometry setups, revealing duality relations between which-way and which-phase information in predictive and retrodictive measurement protocols.

Contribution

It introduces two experimental protocols demonstrating operational implications of path-phase complementarity, including duality relations in predictive and retrodictive contexts.

Findings

Duality relation between WW and WP probabilities in predictive setup

Simultaneous guessing of initial states in retrodictive protocol

Operational meaning of complementarity in single-photon interferometry

Abstract

We examine two setups that reveal different operational implications of path-phase complementarity for single photons in a Mach-Zehnder interferometer (MZI). In both setups, the which-way (WW) information is recorded in the polarization state of the photon serving as a "flying which-way detector". In the "predictive" variant, using a {\em fixed} initial state, one obtains duality relation between the probability to correctly predict the outcome of either a which-way (WW) or which-phase (WP) measurement (equivalent to the conventional path-distinguishibility-visibility). In this setup, only one or the other (WW or WP) prediction has operational meaning in a single experiment. In the second, "retrodictive" protocol, the initial state is secretly selected {\em for each photon} by one party, Alice, among a set of initial states which may differ in the amplitudes and phases of the photon in each arm of the MZI. The goal of the other party, Bob, is to retrodict the initial state by measurements on the photon. Here, a similar duality relation between WP and WW probabilities, governs their {\em simultaneous} guesses in {\em each experimental run}.