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Physical Review A

Identifying quantum resources in convex resource theories with Kirkwood-Dirac quasiprobabilities

Kok Chuan Tan, Souradeep Sasmal

Published Jun 10, 2025• 4 citations
Publisher
DOI
10.1103/htcg-9cv2

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Abstract Kirkwood-Dirac (KD) quasiprobability distributions serve as quantum analog representations of the classical phase-space probability, capable of exhibiting nonclassical behavior through negativity or nonreal values. Recent studies have demonstrated that the nonclassicality of KD quasiprobability distributions plays a crucial role in quantum information tasks and provides deep insights into foundational aspects of quantum mechanics. In this work, we introduce an interpolated KD quasiprobability distribution parametrized by a continuous variable α , which smoothly connects different KD quasiprobability representations. Based on the interpolated KD quasiprobability distribution, we propose a quantitative measure of nonclassicality and rigorously analyze its key properties, including unitary covariance, convexity, and monotonicity. Furthermore, we derive an upper bound for this measure and conduct a comparison with other nonclassicality measures. Finally, we evaluate the nonclassicality measures for several prototypical interpolated KD quasiprobability distributions, illustrating their applicability in quantum information theory.
We show that a Kirkwood-Dirac type quasiprobability distribution is sufficient to reveal any arbitrary quantum resource. This is achieved by demonstrating that it is always possible to identify a set of incompatible measurements that distinguishes between resourceful states and nonresourceful states. The quasiprobability reveals a resourceful quantum state by having at least one quasiprobabilty outcome with a strictly negative numerical value. We also show that there always exists a quasiprobabilty distribution where the total negativity can be interpreted as the geometric distance between a resourceful quantum state to the closest nonresourceful state. It can also be shown that Kirkwood-Dirac type quasiprobability distributions, like the Wigner distribution, can be made informationally complete, in the sense that it can provide complete information about the quantum state while simultaneously revealing nonclassicality whenever a quasiprobability outcome is negative. Moreover, we demonstrate the existence of sufficiently strong anomalous weak values whenever the quasiprobability distribution is negative, which suggests a means to experimentally test such quasiprobability distributions. Since incompatible measurements are necessary in order for the quasiprobability to be negative, this result suggests that measurement incompatibility may underlie any quantum advantage gained from utilizing a nonclassical quantum resource
The Kirkwood-Dirac (KD) quasiprobability describes measurement statistics of joint quantum observables, and has generated great interest as prominent indicators of non-classical features in various quantum information processing tasks. It relaxes the Kolmogorov axioms of probability by allowing for negative and even imaginary probabilities, and thus incorporates the classical probability theory as its inner boundary. In this work, we introduce the postquantum quasiprobability under mild assumptions to provide an outer boundary for KD quasiprobability. Specifically, we present qualitative and quantitative evidence to show that the classical, KD, and postquantum quasiprobabilities form a strict hierarchy, in the sense that joint probability distributions are a strict subset of KD quasiprobability distributions that are a strict subset of postquantum ones. Surprisingly, we are able to derive some nontrivial bounds valid for both classical probability and KD quasiprobability, and even valid for the KD quasiprobability generated by an arbitrary number of measurements. Finally, other interesting bounds are obtained, and their implications are noted. Our work solves the fundamental problems of what and how to bound the KD quasiprobability, and hence provides a deeper understanding of utilizing it in quantum information processing.