Quantum coherence as asymmetry from complex weak values

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.

Quantum coherence from Kirkwood–Dirac nonclassicality, some bounds, and operational interpretation

“…Our results thus strengthen the quantitative link between the quantumness which manifests in the nonclassical values of KD quasiprobability, the associated strange weak value, and quantum contextuality, to the nonclassicality captured by the concept of quantum state subjected to measurement in the forms of asymmetry [64,65], general quantum correlation [66], and uncertainty relation [67]. These results motivate the search for a quantitative connection between KD nonclassicality and strange weak value to quantum entanglement.…”

General quantum correlation from nonreal values of Kirkwood–Dirac quasiprobability over orthonormal product bases

“…Remarkably, the nonreality and/or the negativity of the KD quasiprobability is tighter than noncommutativity [37,38]. Significant works over the past decade showed that KD quasiprobability and its nonclassical values play crucial roles in quantum tomography [39][40][41][42], quantum metrology [43][44][45], quantum thermodynamics [45][46][47], a wide spectrum of quantum fluctuations in condensed matter physics [48], information scrambling in many body quantum chaos [49,50], in quantum foundation to prove quantum contextuality [51,52], and recently in the characterization and quantificiation of coherence and asymmetry [53][54][55].…”