On the connection between Complementarity and Uncertainty Principles in the Mach-Zehnder interferometric setting

TL;DR

This paper explores the fundamental connection between complementarity and uncertainty principles in Mach-Zehnder interferometry, demonstrating their equivalence and analyzing entropic uncertainty relations for different regimes.

Contribution

It establishes the equivalence of trade-off relations in interferometry with various uncertainty principles and investigates entropic measures to identify states of minimum uncertainty.

Findings

Trade-off relation is equivalent to Schrödinger-Robertson and Landau-Pollak uncertainty relations.

Different entropic parameters define regimes with distinct informational content.

Entropic uncertainty relations can identify non-trivial minimum uncertainty states.

Abstract

We revisit, in the framework of Mach-Zehnder interferometry, the connection between the complementarity and uncertainty principles of quantum mechanics. Specifically, we show that, for a pair of suitably chosen observables, the trade-off relation between the complementary path information and fringe visibility is equivalent to the uncertainty relation given by Schr\"odinger and Robertson, and to the one provided by Landau and Pollak as well. We also employ entropic uncertainty relations (based on R\'enyi entropic measures) and study their meaning for different values of the entropic parameter. We show that these different values define regimes which yield qualitatively different information concerning the system, in agreement with findings of [A. Luis, Phys. Rev. A 84, 034101 (2011)]. We find that there exists a regime for which the entropic uncertinty relations can be used as criteria to pinpoint non trivial states of minimum uncertainty.