Quantum coherence as asymmetry from complex weak values

TL;DR

This paper introduces a new way to quantify quantum coherence as asymmetry using the imaginary part of weak values, linking it to quantum estimation and providing experimentally accessible bounds.

Contribution

It defines the TC w-coherence measure based on weak values, relates it to standard quantum metrics, and derives bounds and uncertainty relations for this coherence measure.

Findings

The average absolute imaginary part of weak values quantifies coherence as asymmetry.

Upper bounds of TC w-coherence are expressed via quantum Fisher information and standard deviation.

A lower bound and uncertainty relation for TC w-coherence are established.

Abstract

Quantum coherence as an asymmetry relative to a translation group generated by a Hermitian operator, is a necessary resource for the quantum parameter estimation. On the other hand, the sensitivity of the parameter estimation is known to be related to the imaginary part of the weak value of the Hermitian operator generating the unitary imprinting of the parameter being estimated. This naturally suggests a question if one can use the imaginary part of the weak value to characterize the coherence as asymmetry. In this work, we show that the average absolute imaginary part of the weak value of the generator of the translation group, maximized over all possible projective measurement bases, can be used to quantify the coherence as asymmetry relative to the translation group, satisfying certain desirable requirements. We argue that the quantifier of coherence so defined, called TC (translationally-covariant) w-coherence, can be obtained experimentally using a hybrid quantum-classical circuit via the estimation of weak value combined with a classical optimization procedure. We obtain upper bounds of the TC w-coherence in terms of the quantum standard deviation, quantum Fisher information, and the imaginary part of the Kirkwood-Dirac quasiprobability. We further obtain a lower bound and derive a relation between the TC w-coherences relative to two generators of translation group taking a form analogous to the Kennard-Weyl-Robertson uncertainty relation.