Wave-Particle-Mixedness Complementarity

TL;DR

This paper establishes a ternary complementary relation among wave, particle, and mixedness in quantum systems, revealing how environment-induced mixedness influences quantum duality and entanglement.

Contribution

It introduces a novel ternary complementary relation (TCR) linking wave, particle, and mixedness, extending the understanding of quantum complementarity in mixed states.

Findings

The TCR relates wave, particle, and mixedness measures in multi-state systems.

Wwd can convert entropy into predictability and coherence.

Modified POVM simplifies the TCR to wave-mixedness and particle-mixedness dualities.

Abstract

Wave-particle duality, an important and fundamental concept established upon pure quantum systems, is central to the complementarity principle in quantum mechanics. However, due to the environment effects or even the entanglement between the quanton and the which-way detector (WWD), the quanton should be described by a mixed quantum state but not a pure quantum state. Although there are some attempts to clarify the complementarity principle for mixed quantum systems, it is still unclear how the mixedness affects the complementary relation. Here, we give a ternary complementary relation (TCR) among wave, particle and mixedness aspects for arbitrary multi-state systems, which are respectively quantified by the $l_1$ measure for quantum coherence, the which-path predictability, and the linear entropy. In particular, we show how a WWD can transform entropy into predictability and coherence. Through modifying the POVM (positive-operator valued measure) measurement on WWD, our TCR can be simplified as the wave-mixedness and particle-mixedness duality relations. Beyond enclosing the wave-particle duality relation [PRL 116, 160406 (2016)], our TCR relates to the wave-particle-entanglement complementarity relation [PRL 98, 250501 (2008); Opt. Commun. 283, 827 (2010)].