Duality of quantum coherence and path distinguishability

Abstract

We derive a generalized wave-particle duality relation for arbitrary multi-path quantum interference phenomena. Beyond the conventional notion of the wave nature of a quantum system, i.e., the interference fringe visibility, we introduce a novel quantifier as the normalized quantum coherence, recently defined in the framework of quantum information theory. To witness the particle nature, we quantify the path distinguishability or the which-path information based on unambiguous quantum state discrimination. Then, the Bohr complementarity principle, for multi-path quantum interference, can be stated as a duality relation between the quantum coherence and the path distinguishability. For two-path interference, the quantum coherence is identical to the interference fringe visibility, and the relation reduces to the well-know complementarity relation. The new duality relation continues to hold in the case where mixedness is introduced due to possible decoherence effects.

Measuring quantum coherence in multislit interference

“…Coherence C had earlier been argued to be a good candidate for quantifying the wave-nature of quantons in interference experiments, and a duality relation between C and the path-distinguishability D Q , based on UQSD, had been proved [23]. Here C has been shown to be related to measured intensities in interference experiments, which puts it firmly as a good measure of wave nature of quantons in multi-slit interference experiments.…”

“…The quantum coherence C, given by (12) or (23), and the path distinguishability, given by (31), are bounded by the inequality (5) [23]. In the light of the demonstration of experimental measurability of C, the inequality (5) can be treated as a general duality relation which works for interference experiments with any number of slits.…”

“…For completeness, we briefly discuss the path distinguishability D Q introduced in [23]. From (8) it is clear that in order to tell which slit the quanton went through, one has to be able to tell which path-detector state, out of the set {|d j } materialized.…”

“…Here ρ i j are the matrix elements of the density operator of the system, in the basis formed by the set of n orthogonal states, which correspond to the quanton passing through the n different slits. Based on this new measure of wave nature, the following duality relation for n-slit interference was obtained [23] …”

“…However, there is no well-defined upper limit to this measure, as it depends on the dimensionality of the Hilbert space of the system. Using this measure, a normalized quantity called coherence was very recently introduced [23] …”