Wave-Particle-Mixedness Complementarity

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)].

5 + 4 pages, 2 + 2 figures

References in corpus (5)

• Delayed-choice test of complementarity with single photons
• Maximally coherent mixed states: Complementarity between maximal coherence and mixedness
• Equivalence of wave-particle duality to entropic uncertainty
• Quantum Non-Demolition Test of Bipartite Complementarity
• Quantitative complementarity between local and nonlocal character of quantum states in a three-qubit system