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arXiv:2511.01996 · Symmetry

What is special about the Kirkwood-Dirac distributions? Only they produce natural conditional expectations

Matéo Spriet, Christopher Langrenez, Raymond Brummelhuis, Stephan De Bièvre

quant-phSubmitted Nov 3, 2025• Announced Nov 5, 2025• 0 citations• Revised May 27, 2026 (v2)
v1 · Nov 3, 2025v2 · May 27, 2026
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Abstract

Among the many quasiprobability representations of quantum mechanics, the family of Kirkwood-Dirac (KD) representations has come to the foreground in recent years. Each such KD representation is determined by the choice of two complementary complete sets of commuting observables $\hat A$ and $\hat B$ with respect to which it is Born-compatible, meaning that it correctly reproduces their Born probabilities for every state. We identify in this paper what property uniquely characterizes the KD representations among all such $\hat A$ and $\hat B$ Born-compatible quasiprobability representations. For that purpose, we first define a natural notion of \emph{quantum conditional expectation} of an observable $\hat X$, given an observable $\hat Y$, in a state $\hat ρ$, as a best estimator and we show that it has the basic properties generally expected of a conditional expectation. We then show that only the KD representations provide a notion of conditional expectation, given $\hat B$ (or given $\hat A$) that coincides with the above quantum conditional expectation. As a byproduct of our analysis, we show a state-dependent no-go theorem. We prove that, if the quantum conditional expectation of an observable $\hat X$, given an observable $\hat Y$ in a state $\hat ρ$ admits an anomalous value, then there cannot exist a Born-compatible joint probability distribution $μ(x,y)$ for $\hat X$ and $\hat Y$ in the state $\hat ρ$ for which the associated conditional probability $μ(x|y)$ yields a conditional expectation that coincides with the quantum conditional expectation. We further apply our findings to revisit a standard model for phase estimation in quantum metrology. We show in particular that, within the real sector of a given KD representation, the classical Fisher information of this phase estimation problem vanishes identically.

Journal reference
Symmetry ISSN 2073-8994
DOI
10.3390/sym18061008
Comments
We have strengthened our characterization of the KD distributions. We have also considerably developped the physical interpretation and operational significance of the objects that we introduce. We furthermore apply our results to a metrology problem to give a novel interpretation of the KD real sector. The paper now also serves as an introductory review of the subject

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Quantifying the Nonclassicality of the Kirkwood–Dirac Quasiprobability Distribution Under Discrete-Time Dynamics

Ziheng Ding, Si-Qi Zhou

The Kirkwood–Dirac (KD) quasiprobability distribution describes any quantum state with respect to the eigenbases of two incompatible observables. While the KD quasiprobability distribution behaves similarly to a classical probability distribution, it can take on negative or nonreal values. Recently, the framework of the temporal Kirkwood–Dirac quasiprobability distribution has been proposed, generalizing the KD quasiprobability distribution to arbitrary multi-time quantum processes. In this work, we specifically focus on the temporal KD quasiprobability distribution within the context of two-time dynamics. We begin by constructing a nonclassicality measure derived from the real and imaginary parts of the temporal KD quasiprobability distribution. Next, we establish two uncertainty relations closely linked to this nonclassicality measure, one of which shows that the nonclassicality measure is bounded below by the measurement disturbance caused by the first measurement. Finally, we elucidate the relationships among temporal KD nonclassicality, the spatiotemporal Born rule, and spatiotemporal compatibility.

Entropy•Apr 1, 2026•0 citations

Properties and applications of the Kirkwood–Dirac distribution

David R. M. Arvidsson-Shukur, William F Braasch Jr, Stephan De Bièvre +6 more

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.

arXiv:2403.18899•New Journal of Physics•Dec 1, 2024•57 citations

Conditions tighter than noncommutation needed for nonclassicality

David R. M. Arvidsson-Shukur, Jacob Chevalier Drori, Nicole Yunger Halpern

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.

arXiv:2009.04468•Journal of Physics A Mathematical and Theoretical•Jun 11, 2021•63 citations

Sufficient conditions, lower bounds, and trade-off relations for quantumness in Kirkwood-Dirac quasiprobability

Agung Budiyono

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.

arXiv:2405.08324•Physical Review A•Jun 3, 2024•8 citations

The Structure of the Kirkwood-Dirac Distribution and its Applications in Quantum Foundations, Computation, and Information

Jonathan Thio

Quantum theory is fundamentally different from its classical counterpart. This difference has led to striking new insights in the foundations of physics and the theories of computation and information. Nevertheless, characterising the difference between classical and quantum theory has proven to be a difficult problem. In this thesis, I present my progress towards a general solution for this open problem. My main tool is the Kirkwood-Dirac (KD) quasiprobability distribution. The KD distribution provides an alternative description of quantum states, operations and measurements in terms of quasiprobability distributions. The KD quasiprobability distribution obeys all properties of probability distributions except that its values may, in general, lie in the complex unit disc instead of the interval [0,1]. For some experiments, however, all associated KD distributions are genuine probability distributions. For such `KD-positive’ states, operations and measurements, the KD distribution enables the construction of a classical model. The absence of such a construction can provide a natural criterion for the definition of the quantum-classical boundary. In the first part of my thesis, I present my results on the mathematical structure of the KD distribution. I summarise previous literature, and prove several new results, including a proof that the set of KD-positive states is almost always minimal, which was published in Physical Review A; novel bounds on the entries of the KD distribution and KD nonpositivity, a measure of how far a KD distribution is from being positive; and the construction of exotic states, mixed KD-positive states that cannot be written as the convex combination of pure KD-positive states, for Hilbert spaces of any odd dimension. These results lay the foundation for the objective of the remainder of this thesis: the application of the KD distribution in quantum foundations, computation and information. In the foundational setting, I study contextuality, a notion of nonclassicality often used in quantum foundations. An experiment is contextual if it is impossible to describe its indistinguishable procedures identically classically. I design an experiment that is contextual if and only if the input state of the experiment is not KD-positive. By analysing this connection with respect to the aforementioned exotic states, I obtain a quantum protocol that allows an experimenter to verify contextuality using a noncontextual experiment, thereby uncovering a previously unknown feature of contextuality. This section of the thesis is based on my publication in Physical Review A `Verification of contextuality by noncontextual experiments’ KDContextualityPaper. In the computational setting, I study the Delfosse-Guerin-Bian-Raussendorf (DGBR) model of quantum computation with a restricted set of input states. For certain sets of input states, this model is computationally as powerful as a universal quantum computer. However, for other sets of input states, this model is efficiently simulable on a classical computer and thus cannot be as powerful as a universal quantum computer, unless quantum computers offer no advantage over their classical counterparts. I prove that if all states in the set of input states are KD-positive, then the DGBR model is classically efficiently simulable. In this setting, some exotic states are bound-magic states, magic states that are nevertheless simulable input states in the DGBR model. I use my previous work on exotic states to construct the first bound-magic states on qubits. Furthermore, I show that KD nonpositivity is a resource monotone for quantum computation, establishing KD nonpositivity as a necessary resource for quantum computational advantage. This section of the thesis is based on my article `Kirkwood-Dirac nonpositivity is a necessary resource for quantum computing’, which has been accepted for publication in Physical Review Letters. In the information-theoretical setting, I study the uncertainty principle. In modern quantum information theory, the uncertainty principle is formulated as an `entropic lower bound’ on the sum of the entropies of the outcome distributions of two observables. I extend the definition of the standard entropic functions to the KD distribution and the Margenau-Hill (MH) distribution, which is the real part of the KD distribution. I introduce the MH Shannon entropy and the KD Rényi entropy, and establish their basic properties. For the MH Shannon entropy, I prove a generalised subadditivity relation, which yields a state-dependent entropic uncertainty relation that strengthens the previous best bound in most regimes and implies the finite-dimensional analogue of the Van Herstraeten--Cerf conjecture. For the KD Rényi entropy, I prove an entropic uncertainty relation strengthening the Maassen--Uffink bound, in which the KD nonpositivity appears as an additive correction. This yields a new interpretation of KD nonpositivity as a measure of quantum uncertainty, as well as a new entropic witness of KD positivity. Applying these results to cryptographic games, I show that KD nonpositivity is the key resource governing the optimal strategy in each case. This section is based on my article `Kirkwood-Dirac nonpositivity is sufficient for quantum uncertainty'. These results demonstrate that the KD distribution provides an effective formulation of quantum theory for studying the quantum-classical boundary in various settings. My work establishes explicit connections between the KD distribution and quantum foundations, computation, and information, opening up these fields to be studied using the extensive KD toolkit.

•Jun 5, 2026•0 citations

Almost sure minimality of the set of experiments with classical Kirkwood-Dirac representations

Christopher Langrenez, Wilfred Salmon, Stephan De Bièvre +3 more

A central problem in quantum information is determining quantum-classical boundaries. In the quasiprobability framework, a state is called classical if it is represented by a quasiprobability distribution that is positive, and thus a probability distribution. In recent years, the Kirkwood-Dirac (KD) distributions have gained much interest due to their numerous applications in modern quantum-information research. A particular advantage of the KD distributions is that they can be defined with respect to arbitrary observables. Here, we show that if two $d$-dimensional observables are picked at random, the set of classical (positive) states of the resulting KD distribution is a minimal polytope of dimension $2(d-1)$ with $2d$ explicitly known vertices. This implies minimality of the sets of KD-real observables, of KD-positive measurement elements and of KD-positivity-preserving unitaries. We show how these results have implications on robust observations of nonclassical phenomena, on classical simulations of quantum circuits, and on foundations of quantum theory.

arXiv:2405.17557•Physical Review A•Apr 7, 2026•1 citation
Among the many quasiprobability representations of quantum mechanics, the family of Kirkwood-Dirac (KD) representations has come to the foreground in recent years. Each such KD representation is determined by the choice of two complementary complete sets of commuting observables A^\hat AA^ and B^\hat BB^ with respect to which it is Born-compatible, meaning that it correctly reproduces their Born probabilities for every state. We identify in this paper what property uniquely characterizes the KD representations among all such A^\hat AA^ and B^\hat BB^ Born-compatible quasiprobability representations. For that purpose, we first define a natural notion of \emph{quantum conditional expectation} of an observable X^\hat XX^, given an observable Y^\hat YY^, in a state ρ^\hat ρρ^​, as a best estimator and we show that it has the basic properties generally expected of a conditional expectation. We then show that only the KD representations provide a notion of conditional expectation, given B^\hat BB^ (or given A^\hat AA^) that coincides with the above quantum conditional expectation. As a byproduct of our analysis, we show a state-dependent no-go theorem. We prove that, if the quantum conditional expectation of an observable X^\hat XX^, given an observable Y^\hat YY^ in a state ρ^\hat ρρ^​ admits an anomalous value, then there cannot exist a Born-compatible joint probability distribution μ(x,y)μ(x,y)μ(x,y) for X^\hat XX^ and Y^\hat YY^ in the state ρ^\hat ρρ^​ for which the associated conditional probability μ(x∣y)μ(x|y)μ(x∣y) yields a conditional expectation that coincides with the quantum conditional expectation. We further apply our findings to revisit a standard model for phase estimation in quantum metrology. We show in particular that, within the real sector of a given KD representation, the classical Fisher information of this phase estimation problem vanishes identically.
The Kirkwood–Dirac (KD) quasiprobability distribution describes any quantum state with respect to the eigenbases of two incompatible observables. While the KD quasiprobability distribution behaves similarly to a classical probability distribution, it can take on negative or nonreal values. Recently, the framework of the temporal Kirkwood–Dirac quasiprobability distribution has been proposed, generalizing the KD quasiprobability distribution to arbitrary multi-time quantum processes. In this work, we specifically focus on the temporal KD quasiprobability distribution within the context of two-time dynamics. We begin by constructing a nonclassicality measure derived from the real and imaginary parts of the temporal KD quasiprobability distribution. Next, we establish two uncertainty relations closely linked to this nonclassicality measure, one of which shows that the nonclassicality measure is bounded below by the measurement disturbance caused by the first measurement. Finally, we elucidate the relationships among temporal KD nonclassicality, the spatiotemporal Born rule, and spatiotemporal compatibility.
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.
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.
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.
Quantum theory is fundamentally different from its classical counterpart. This difference has led to striking new insights in the foundations of physics and the theories of computation and information. Nevertheless, characterising the difference between classical and quantum theory has proven to be a difficult problem. In this thesis, I present my progress towards a general solution for this open problem. My main tool is the Kirkwood-Dirac (KD) quasiprobability distribution. The KD distribution provides an alternative description of quantum states, operations and measurements in terms of quasiprobability distributions. The KD quasiprobability distribution obeys all properties of probability distributions except that its values may, in general, lie in the complex unit disc instead of the interval [0,1]. For some experiments, however, all associated KD distributions are genuine probability distributions. For such `KD-positive’ states, operations and measurements, the KD distribution enables the construction of a classical model. The absence of such a construction can provide a natural criterion for the definition of the quantum-classical boundary. In the first part of my thesis, I present my results on the mathematical structure of the KD distribution. I summarise previous literature, and prove several new results, including a proof that the set of KD-positive states is almost always minimal, which was published in Physical Review A; novel bounds on the entries of the KD distribution and KD nonpositivity, a measure of how far a KD distribution is from being positive; and the construction of exotic states, mixed KD-positive states that cannot be written as the convex combination of pure KD-positive states, for Hilbert spaces of any odd dimension. These results lay the foundation for the objective of the remainder of this thesis: the application of the KD distribution in quantum foundations, computation and information. In the foundational setting, I study contextuality, a notion of nonclassicality often used in quantum foundations. An experiment is contextual if it is impossible to describe its indistinguishable procedures identically classically. I design an experiment that is contextual if and only if the input state of the experiment is not KD-positive. By analysing this connection with respect to the aforementioned exotic states, I obtain a quantum protocol that allows an experimenter to verify contextuality using a noncontextual experiment, thereby uncovering a previously unknown feature of contextuality. This section of the thesis is based on my publication in Physical Review A `Verification of contextuality by noncontextual experiments’ KDContextualityPaper. In the computational setting, I study the Delfosse-Guerin-Bian-Raussendorf (DGBR) model of quantum computation with a restricted set of input states. For certain sets of input states, this model is computationally as powerful as a universal quantum computer. However, for other sets of input states, this model is efficiently simulable on a classical computer and thus cannot be as powerful as a universal quantum computer, unless quantum computers offer no advantage over their classical counterparts. I prove that if all states in the set of input states are KD-positive, then the DGBR model is classically efficiently simulable. In this setting, some exotic states are bound-magic states, magic states that are nevertheless simulable input states in the DGBR model. I use my previous work on exotic states to construct the first bound-magic states on qubits. Furthermore, I show that KD nonpositivity is a resource monotone for quantum computation, establishing KD nonpositivity as a necessary resource for quantum computational advantage. This section of the thesis is based on my article `Kirkwood-Dirac nonpositivity is a necessary resource for quantum computing’, which has been accepted for publication in Physical Review Letters. In the information-theoretical setting, I study the uncertainty principle. In modern quantum information theory, the uncertainty principle is formulated as an `entropic lower bound’ on the sum of the entropies of the outcome distributions of two observables. I extend the definition of the standard entropic functions to the KD distribution and the Margenau-Hill (MH) distribution, which is the real part of the KD distribution. I introduce the MH Shannon entropy and the KD Rényi entropy, and establish their basic properties. For the MH Shannon entropy, I prove a generalised subadditivity relation, which yields a state-dependent entropic uncertainty relation that strengthens the previous best bound in most regimes and implies the finite-dimensional analogue of the Van Herstraeten--Cerf conjecture. For the KD Rényi entropy, I prove an entropic uncertainty relation strengthening the Maassen--Uffink bound, in which the KD nonpositivity appears as an additive correction. This yields a new interpretation of KD nonpositivity as a measure of quantum uncertainty, as well as a new entropic witness of KD positivity. Applying these results to cryptographic games, I show that KD nonpositivity is the key resource governing the optimal strategy in each case. This section is based on my article `Kirkwood-Dirac nonpositivity is sufficient for quantum uncertainty'. These results demonstrate that the KD distribution provides an effective formulation of quantum theory for studying the quantum-classical boundary in various settings. My work establishes explicit connections between the KD distribution and quantum foundations, computation, and information, opening up these fields to be studied using the extensive KD toolkit.
A central problem in quantum information is determining quantum-classical boundaries. In the quasiprobability framework, a state is called classical if it is represented by a quasiprobability distribution that is positive, and thus a probability distribution. In recent years, the Kirkwood-Dirac (KD) distributions have gained much interest due to their numerous applications in modern quantum-information research. A particular advantage of the KD distributions is that they can be defined with respect to arbitrary observables. Here, we show that if two ddd-dimensional observables are picked at random, the set of classical (positive) states of the resulting KD distribution is a minimal polytope of dimension 2(d−1)2(d-1)2(d−1) with 2d2d2d explicitly known vertices. This implies minimality of the sets of KD-real observables, of KD-positive measurement elements and of KD-positivity-preserving unitaries. We show how these results have implications on robust observations of nonclassical phenomena, on classical simulations of quantum circuits, and on foundations of quantum theory.