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 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.
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.
Kirkwood-Dirac (KD) quasiprobability is a quantum analog of phase space probability of classical statistical mechanics, allowing negative or/and nonreal values. It gives an informationally complete representation of a quantum state. Recent works have revealed the important roles played by the KD quasiprobability in the broad fields of quantum science and quantum technology. In the present work, we use the KD quasiprobability to access the quantum coherence in a quantum state. We show that the l1 norm of the imaginary part of the KD quasiprobability over an incoherent reference basis and a second basis, maximized over all possible choices of the latter, can be used to quantify quantum coherence, satisfying certain desirable properties. It is upper bounded by the quantum uncertainty, i.e., the quantum standard deviation, of the incoherent basis in the state. It gives a lower bound to the l1 norm quantum coherence, and for a single qubit, they are identical. We discuss the measurement of the KD coherence based on the measurement of the KD quasiprobability and an optimization procedure in hybrid quantum-classical schemes, and suggest statistical interpretations. We also discuss its relevance in the physics of linear response regime.
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.
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.
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.
Abstract There are several mathematical formulations of quantum mechanics. The Schrödinger picture expresses quantum states in terms of wavefunctions over, e.g. position or momentum. Alternatively, phase-space formulations represent states with quasi-probability distributions over, e.g. position and momentum. A quasi-probability distribution resembles a probability distribution but may have negative and non-real entries. The most famous quasi-probability distribution, the Wigner function, has played a pivotal role in the development of a continuous-variable quantum theory that has clear analogues of position and momentum. However, the Wigner function is ill-suited for much modern quantum-information research, which is focused on finite-dimensional systems and general observables. Instead, recent years have seen the Kirkwood–Dirac (KD) distribution come to the forefront as a powerful quasi-probability distribution for analysing quantum mechanics. The KD distribution allows tools from statistics and probability theory to be applied to problems in quantum-information processing. A notable difference to the Wigner function is that the KD distribution can represent a quantum state in terms of arbitrary observables. This paper reviews the KD distribution, in three parts. First, we present definitions and basic properties of the KD distribution and its generalisations. Second, we summarise the KD distribution’s extensive usage in the study or development of measurement disturbance; quantum metrology; weak values; direct measurements of quantum states; quantum thermodynamics; quantum scrambling and out-of-time-ordered correlators; and the foundations of quantum mechanics, including Leggett–Garg inequalities, the consistent-histories interpretation and contextuality. We emphasise connections between operational quantum advantages and negative or non-real KD quasi-probabilities. Third, we delve into the KD distribution’s mathematical structure. We summarise the current knowledge regarding the geometry of KD-positive states (the states for which the KD distribution is a classical probability distribution), describe how to witness and quantify KD non-positivity, and outline relationships between KD non-positivity, coherence and observables’ incompatibility.