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
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 is a quantum analog expression of the phase-space probability in classical statistical mechanics, yet it can behave nonclassically by taking negative or nonreal values. The nonclassicality of the KD quasiprobability distribution has lately been shown to provide advantages in some quantum information tasks and shed light on a number of aspects of quantum physics. Characterization and quantification of KD nonclassicality are crucial both theoretically and experimentally. Most of the existing measures of KD nonclassicality concentrate merely on the negative or/and imaginary part(s) of the KD distribution. Yet it is not clear whether these measures can account for all nonclassicality in the distribution. In this paper, we suggest a KD nonclassicality measure based on the two quantum modification terms in the quasiprobability distribution. The present paper is mainly motivated by the following two considerations: First, the two quantum modification terms are actually the sources of the negative and imaginary parts of the KD quasiprobability respectively. Secondly, these quantum modification terms deviate the KD quasiprobability from the classical joint probability distribution function of two successive projective measurements (termed as the Wigner formula), which can be obtained from the multiplication principle in classical probability theory. Our measure refines the quantification of KD nonclassicality in the sense that even for real and nonnegative KD distribution, its quantum modification terms can still exist, resulting in a nonvanishing KD nonclassicality. As an illustration, we analyze the KD nonclassicality of the positive KD distribution in the postselected metrology scheme [S. Das, S. Modak, and M. N. Bera, Phys. Rev. A 107, 042413 (2023)], which provides interpretation on the source of quantum advantages. We also give an upper bound on the KD nonclassicality measure and offer an easily verifiable sufficient condition for nonvanishing KD nonclassicality.
Kirkwood-Dirac (KD) quasiprobability is a quantum analog of classical phase space probability. It offers an informationally complete representation of quantum state wherein the quantumness associated with quantum noncommutativity manifests in its nonclassical values, i.e., the nonreal and/or negative values of the real part. This naturally raises a question: how does such form of quantumness comply with the uncertainty principle which also arises from quantum noncommutativity? Here, first, we obtain sufficient conditions for the KD quasiprobability defined relative to a pair of projection-valued measure (PVM) bases to have nonclassical values. Using these nonclassical values, we then introduce two quantities which capture the amount of KD quantumness in a quantum state relative to a single PVM basis. They are defined, respectively, as the nonreality and the classicality---which captures both the nonreality and negativity---of the associated KD quasiprobability over the PVM basis of interest, and another PVM basis, and maximized over all possible choices of the latter. We obtain their lower bounds, and derive trade-off relations respectively reminiscent of the Robertson and Robertson-Schr\"odinger uncertainty relations but with lower bounds maximized over the convex sets of Hermitian operators whose complete sets of eigenprojectors are given by the PVM bases. We discuss their measurement using weak value measurement and classical optimization. We then suggest an information theoretical interpretation of the KD nonreality relative to a PVM basis as a lower bound to the maximum total root-mean-squared error in an optimal estimation of the PVM basis, and thereby obtain a lower bound and a trade-off relation for the root-mean-squared error. Finally, we suggest an interpretation of the KD nonclassicality relative to a PVM basis as a lower bound to the total state disturbance caused by a nonselective projective binary measurement associated with the PVM basis, and derive a lower bound and a trade-off relation for the disturbance.
Abstract The question of when the Kirkwood-Dirac (KD) quasiprobability serves as the most appropriate description for quantum measurements has remained unresolved, particularly across different measurement strengths. While known to generate anomalous weak values in the weak measurement regime and to reduce to classical probabilities under projective measurement, the physical mechanism governing its continuous transformation has been lacking. Here we demonstrate that the KD quasiprobability provides a general framework for all measurement regimes by identifying pointer-induced decoherence as the universal mechanism controlling this transition. We show that the decoherence factor F ( t ) simultaneously quantifies the loss of quantum coherence and interpolates the measurement strength from weak to strong. Within this framework, the KD quasiprobability naturally deforms from its full complex form—governing weak values—to the real, non-negative Wigner formula describing projective measurements, while maintaining informational completeness throughout the transition. Our work resolves the fundamental question of the KD distribution’s applicability by establishing it as the unified framework that seamlessly connects weak and strong measurement regimes through a physically transparent decoherence pathway.
Abstract Kirkwood discovered in 1933, and Dirac discovered in 1945, a representation of quantum states that has undergone a renaissance recently. The Kirkwood–Dirac (KD) distribution has been employed to study nonclassicality across quantum physics, from metrology to chaos to the foundations of quantum theory. The KD distribution is a quasiprobability distribution, a quantum generalization of a probability distribution, which can behave nonclassically by having negative or nonreal elements. Negative KD elements signify quantum information scrambling and potential metrological quantum advantages. Nonreal elements encode measurement disturbance and thermodynamic nonclassicality. KD distributions’ nonclassicality has been believed to follow necessarily from pairwise noncommutation of operators in the distribution’s definition. We show that noncommutation does not suffice. We prove sufficient conditions for the KD distribution to be nonclassical (equivalently, necessary conditions for it to be classical). We also quantify the KD nonclassicality achievable under various conditions. This work resolves long-standing questions about nonclassicality and may be used to engineer quantum advantages.
The question of when the Kirkwood-Dirac quasiprobability serves as the most appropriate description for quantum measurements has remained unresolved, particularly across different measurement strengths. While known to generate anomalous weak values in the weak measurement regime and to reduce to classical probabilities under projective measurement, the physical mechanism governing its continuous transformation has been lacking. Here we demonstrate that the KD quasiprobability provides a general framework for all measurement regimes by identifying pointer-induced decoherence as the universal mechanism controlling this transition. We show that the decoherence factor F(t) simultaneously quantifies the loss of quantum coherence and interpolates the measurement strength from weak to strong. Within this framework, the KD quasiprobability naturally deforms from its full complex form-governing weak values-to the real, non-negative Wigner formula describing projective measurements, while maintaining informational completeness throughout the transition. Our work resolves the fundamental question of the KD distribution's applicability by establishing it as the universal framework that seamlessly connects all quantum measurement regimes through a physically transparent decoherence pathway.
The Kirkwood-Dirac quasiprobability provides an operational representation of a quantum state, whose negativity serves as a measure of nonclassicality. Despite its fundamental importance, the extremal values of the Kirkwood-Dirac negativity are still unknown in the general case. We investigate the Kirkwood-Dirac quasiprobability of an arbitrary quantum state under Gaussian processes. In this setting, we derive an upper bound on the negativity for any number of modes and measurements. For a single mode and two measurements, we show that the eigenstates of the quadrature operators saturate this upper bound, while a nontrivial minimum is reached by pure Gaussian states. As a consequence, our results indicate that Gaussian states are sufficient to achieve extreme values of nonclassicality.