Quantitative wave-particle duality relations from the density matrix properties

TL;DR

This paper establishes quantitative bounds linking quantum coherence and uncertainty measures in multi-level systems, deriving wave-particle duality relations from fundamental density matrix properties.

Contribution

It introduces new bounds for quantum coherence based on density matrix properties and applies them to derive wave-particle duality relations in multi-slit interferometry.

Findings

Derived upper bounds for quantum coherence using entropy measures.

Established coherence-populations trade-off relations.

Formulated quantitative wave-particle duality relations.

Abstract

We derive upper bounds for Hilbert-Schmidt's quantum coherence of general states of a $d$-level quantum system, a qudit, in terms of its incoherent uncertainty, with the latter quantified using the linear and von Neumann's entropies of the corresponding closest incoherent state. Similar bounds are obtained for Wigner-Yanase's coherence. The reported inequalities are also given as coherence-populations trade-off relations. As an application example of these inequalities, we derive quantitative wave-particle duality relations for multi-slit interferometry. Our framework leads to the identification of predictability measures complementary to Hilbert-Schmidt's, Wigner-Yanase's, and $l_{1}$-norm quantum coherences. The quantifiers reported here for the wave and particle aspects of a quanton follow directly from the defining properties of the quantum density matrix (i.e., semi-positivity and unit trace), contrasting thus with most related results from the literature.