Abstract Just a few years after the inception of quantum mechanics, there has been a research program using the nonclassical values of some quasiprobability distributions to delineate the nonclassical aspects of quantum phenomena. In particular, in Kirkwood–Dirac (KD) quasiprobability distribution, the distinctive quantum mechanical feature of noncommutativity which underlies many nonclassical phenomena, manifests in the nonreal values and/or the negative values of the real part. Here, we develop a faithful quantifier of quantum coherence based on the KD nonclassicality which captures simultaneously the nonreality and the negativity of the KD quasiprobability. The KD-nonclassicality coherence thus defined, is upper bounded by the uncertainty of the outcomes of measurement described by a rank-1 orthogonal projection-valued measure (PVM) corresponding to the incoherent orthonormal basis which is quantified by the Tsallis 1 2 -entropy. Moreover, they are identical for pure states so that the KD-nonclassicality coherence for pure state admits a simple closed expression in terms of measurement probabilities. We then use the Maassen–Uffink uncertainty relation for min-entropy and max-entropy to obtain a lower bound for the KD-nonclassicality coherence of a pure state in terms of optimal guessing probability in measurement described by a PVM noncommuting with the incoherent orthonormal basis. We also derive a trade-off relation for the KD-nonclassicality coherences of a pure state relative to a pair of noncommuting orthonormal bases with a state-independent lower bound. Finally, we sketch a variational scheme for a direct estimation of the KD-nonclassicality coherence based on weak value measurement and thereby discuss its relation with quantum contextuality.Abstract Just a few years after the inception of quantum mechanics, there has been a research program using the nonclassical values of some quasiprobability distributions to delineate the nonclassical aspects of quantum phenomena. In particular, in Kirkwood–Dirac (KD) quasiprobability distribution, the distinctive quantum mechanical feature of noncommutativity which underlies many nonclassical phenomena, manifests in the nonreal values and/or the negative values of the real part. Here, we develop a faithful quantifier of quantum coherence based on the KD nonclassicality which captures simultaneously the nonreality and the negativity of the KD quasiprobability. The KD-nonclassicality coherence thus defined, is upper bounded by the uncertainty of the outcomes of measurement described by a rank-1 orthogonal projection-valued measure (PVM) corresponding to the incoherent orthonormal basis which is quantified by the Tsallis 1 2 -entropy. Moreover, they are identical for pure states so that the KD-nonclassicality coherence for pure state admits a simple closed expression in terms of measurement probabilities. We then use the Maassen–Uffink uncertainty relation for min-entropy and max-entropy to obtain a lower bound for the KD-nonclassicality coherence of a pure state in terms of optimal guessing probability in measurement described by a PVM noncommuting with the incoherent orthonormal basis. We also derive a trade-off relation for the KD-nonclassicality coherences of a pure state relative to a pair of noncommuting orthonormal bases with a state-independent lower bound. Finally, we sketch a variational scheme for a direct estimation of the KD-nonclassicality coherence based on weak value measurement and thereby discuss its relation with quantum contextuality.

arXiv:2309.09162Journal of Physics A Mathematical and TheoreticalMay 22, 202410 citations

A coherence monotone from Kirkwood-Dirac nonclassicality with respect to mutually unbiased bases

Yan Liu, Zhihua Guo, Zhihao Ma +1 more

The Kirkwood-Dirac distribution, serving as an informationally complete representation of a quantum state, has recently garnered increasing attention. We investigate the Kirkwood-Dirac classicality with respect to mutually unbiased bases. For prime dimensional Hilbert spaces, {we demonstrate that a quantum state exhibits Kirkwood-Dirac classicality for two distinct sets of mutually unbiased bases $(A,B)$ and $(A,B')$ if and only if it is incoherent with respect to $A$ }. We subsequently introduce a coherence monotone based on Kirkwood-Dirac nonclassicality with respect to mutually unbiased bases. Additionally, we establish that this coherence monotone can be expressed through weak values, suggesting that quantum coherence can be utilized to detect anomalous weak values.The Kirkwood-Dirac distribution, serving as an informationally complete representation of a quantum state, has recently garnered increasing attention. We investigate the Kirkwood-Dirac classicality with respect to mutually unbiased bases. For prime dimensional Hilbert spaces, {we demonstrate that a quantum state exhibits Kirkwood-Dirac classicality for two distinct sets of mutually unbiased bases $(A,B)$ and $(A,B')$ if and only if it is incoherent with respect to $A$}. We subsequently introduce a coherence monotone based on Kirkwood-Dirac nonclassicality with respect to mutually unbiased bases. Additionally, we establish that this coherence monotone can be expressed through weak values, suggesting that quantum coherence can be utilized to detect anomalous weak values.

arXiv:2411.11666Nov 18, 20240 citations

(A,B)

and

(A,B')

if and only if it is incoherent with respect to

A

}. We subsequently introduce a coherence monotone based on Kirkwood-Dirac nonclassicality with respect to mutually unbiased bases. Additionally, we establish that this coherence monotone can be expressed through weak values, suggesting that quantum coherence can be utilized to detect anomalous weak values.