An uncertainty view on complementarity and a complementarity view on uncertainty

TL;DR

This paper explores the deep relationship between quantum uncertainty, complementarity, and quantum correlations, providing new complete relations and measures that unify wave-particle duality and quantum correlations in mixed states.

Contribution

It introduces a comprehensive framework linking quantum and classical uncertainties with quantum correlations and coherence, extending complementarity relations to mixed states and new uncertainty measures.

Findings

Complete complementarity relation for quantum and classical uncertainties.

Quantum uncertainty equals Wigner-Yanase coherence.

Classical uncertainty quantifies quantum correlations.

Abstract

Since the uncertainty about an observable of a system prepared in a quantum state is usually described by its variance, when the state is mixed, the variance is a hybrid of quantum and classical uncertainties. Besides that, complementarity relations are saturated only for pure, single-quanton, quantum states. For mixed states, the wave-particle quantifiers never saturate the complementarity relation and can even reach zero for a maximally mixed state. So, to fully characterize a quanton it is not sufficient to consider its wave-particle aspect; one has also to regard its correlations with other systems. In this paper, we discuss the relation between quantum correlations and local classical uncertainty measures, as well as the relation between quantum coherence and quantum uncertainty quantifiers. We obtain a complete complementarity relation for quantum uncertainty, classical uncertainty, and predictability. The total quantum uncertainty of a d-paths interferometer is shown to be equivalent to the Wigner-Yanase coherence and the corresponding classical uncertainty is shown to be a quantum correlation quantifier. The duality between complementarity and uncertainty is used to derive quantum correlations measures that complete the complementarity relations for $l_1$-norm and $l_2$-norm coherences. Besides, we show that Brukner-Zeilinger's invariant information quantifies both the wave and particle characters of a quanton and we obtain a sum uncertainty relation for the generalized Gell Mann's matrices.