Resource theory of quantum coherence with probabilistically non-distinguishable pointers and corresponding wave-particle duality

TL;DR

This paper develops a resource theory of quantum coherence for arbitrary sets of quantum states, including indistinguishable ones, and explores its implications for wave-particle duality and quantum measurements.

Contribution

It introduces a generalized framework for quantum coherence with non-distinguishable states, identifying free states, operations, and measures, and links it to wave-particle duality.

Findings

Identifies free states and operations for coherence with indistinguishable states

Establishes monotonicity of coherence measures in this framework

Derives a complementary relation between coherence and path distinguishability

Abstract

One of the fundamental features of quantum mechanics is the superposition principle, a manifestation of which is embodied in quantum coherence. Coherence of a quantum state is invariably defined with respect to a preferred set of pointer states, and there exist quantum coherence measures with respect to deterministically as well as probabilistically distinguishable sets of quantum state vectors. Here we study the resource theory of quantum coherence with respect to an arbitrary set of quantum state vectors, that may not even be probabilistically distinguishable. Geometrically, a probabilistically indistinguishable set of quantum state vectors forms a linearly dependent set. We find the free states of the resource theory, and analyze the corresponding free operations, obtaining a necessary condition for an arbitrary quantum operation to be free. We identify a class of measures of the quantum coherence, and in particular establish a monotonicity property of the measures. We find a connection of an arbitrary set of quantum state vectors with positive operator valued measurements with respect to the resource theory being considered, which paves the way for an alternate definition of the free states. We notice that the resource theory of magic can be looked upon as a resource theory of quantum coherence with respect to a set of quantum state vectors that are probabilistically indistinguishable. We subsequently examine the wave-particle duality in a double-slit set-up in which superposition of probabilistically indistinguishable quantum state vectors is possible. Specifically, we report a complementary relation between quantum coherence and path distinguishability in such a set-up.