The Kirkwood–Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables A and B. KD distributions behave similarly to classical joint probability distributions but can assume negative and nonreal values. In recent years, KD distributions have proven instrumental in mapping out nonclassical phenomena and quantum advantages. These quantum features have been connected to nonpositive entries of KD distributions. Consequently, it is important to understand the geometry of the KD-positive and -nonpositive states. Until now, there has been no thorough analysis of the KD positivity of mixed states. Here, we investigate the dependence of the full convex set of states with positive KD distributions on the eigenbases of A and B and on the dimension d of the Hilbert space. In particular, we identify three regimes where convex combinations of the eigenprojectors of A and B constitute the only KD-positive states: (i) any system in dimension 2; (ii) an open and dense probability one set of bases in dimension d = 3; and (iii) the discrete-Fourier-transform bases in prime dimension. Finally, we show that, if for example d = 2m, there exist, for suitable choices of A and B, mixed KD-positive states that cannot be written as convex combinations of pure KD-positive states. We further explicitly construct such states for a spin-1 system.
The Kirkwood–Dirac (KD) quasiprobability distribution can describe any quantum state with respect to the eigenbases of two observables A and B. KD distributions behave similarly to classical joint probability distributions but can assume negative and nonreal values. In recent years, KD distributions have proven instrumental in mapping out nonclassical phenomena and quantum advantages. These quantum features have been connected to nonpositive entries of KD distributions. Consequently, it is important to understand the geometry of the KD-positive and -nonpositive states. Until now, there has been no thorough analysis of the KD positivity of mixed states. Here, we investigate the dependence of the full convex set of states with positive KD distributions on the eigenbases of A and B and on the dimension d of the Hilbert space. In particular, we identify three regimes where convex combinations of the eigenprojectors of A and B constitute the only KD-positive states: (i) any system in dimension 2; (ii) an open and dense probability one set of bases in dimension d = 3; and (iii) the discrete-Fourier-transform bases in prime dimension. Finally, we show that, if for example d = 2m, there exist, for suitable choices of A and B, mixed KD-positive states that cannot be written as convex combinations of pure KD-positive states. We further explicitly construct such states for a spin-1 system.
Abstract Given two orthonormal bases ( | a i ⟩ ) i and ( | b j ⟩ ) j , one can associate to every quantum state a Kirkwood–Dirac (KD) quasiprobability distribution. KD distributions are like joint classical probabilities except that they can have negative or nonreal values, which are associated to nonclassical features of the state. In the last decade, KD distributions have proven to be versatile tools to investigate and construct quantum advantages and nonclassical phenomena. KD distributions are also used to determine quantum–classical boundaries, for example in the theory of (non)contextuality. Optimal use of the KD distributions relies on useful witnesses of KD nonpositivity. Previous works have established such a witness based on the observation that KD-nonpositive pure states have large support uncertainty with respect to the bases ( | a i ⟩ ) i and ( | b j ⟩ ) j . Here, we extend this witness to mixed states. We introduce two convex roof constructions: the convex roof of the support uncertainty and the convex roof of the KD nonpositivity. These constructions both witness the set of quantum states that cannot be written as a convex combination of KD-positive pure states. Moreover, the first one extends to mixed states the relation between two phenomena: large support uncertainty and the fact that the state belongs to the convex hull of the KD-positive pure states. The convex roof of the KD nonpositivity together with the KD nonpositivity itself give a necessary and sufficient condition for a state to be KD positive but not to belong to the convex hull of the KD-positive pure states.
The Kirkwood-Dirac (KD) distribution is a quantum state representation that relies on two chosen fixed orthonormal bases, or alternatively, on the transition matrix of these two bases. In recent years, it has been discovered that the KD distribution has numerous applications in quantum information science. The presence of negative or nonreal KD distributions may indicate certain quantum features or advantages. If the KD distribution of a quantum state consists solely of positive or zero elements, the state is called a KD-positive state. Consequently, a crucial inquiry arises regarding the determination of whether a quantum state is KD-positive when subjected to various physically relevant transition matrices. When the transition matrix is discrete Fourier transform (DFT) matrix of dimension p [Langrenez et al., J. Math. Phys. 65, 072201 (2024)] or p2 [Yang et al., J. Phys. A: Math. Theor. 57, 435303 (2024)] with p being prime, it is proved that any KD-positive state can be expressed as a convex combination of pure KD-positive states. In this work, we prove that when the transition matrix is the DFT matrix of any finite dimension, any KD-positive state can be expressed as a real linear combination of pure KD-positive states.
Kirkwood-Dirac (KD) quasiprobability distributions are increasingly used across quantum information science, yet their computational significance remains unclear. We classify unitary dynamics that preserve KD positivity and connect this structure to classical simulation. In contrast with the discrete Wigner setting, we show that KD positivity preservation, stochastic evolution of quasiprobabilities, and preservation of total nonpositivity do not coincide. We identify three classes of positivity-preserving unitaries: type I gates, which are exactly the KD-stochastic unitaries; type II gates, which are nonstochastic yet preserve total nonpositivity by permuting and conjugating KD entries; and, for Fourier-conjugate bases in dimension d = p q , type III gates, which preserve KD positivity but can change the total nonpositivity of nonreal distributions. The classification is complete for Haar-random bases and for Fourier-conjugate dimensions d = p k and d = p q , with p , q distinct primes. Adapting the sampling algorithm of Pashayan [], we simulate all of these positivity-preserving circuits efficiently on KD-positive inputs; type III gates, however, create a sharp distinction between real inputs, which remain efficiently simulable, and nonreal inputs, whose sampling overhead can grow exponentially. Consequently, for d = p q no resource theory can both treat the KD total nonpositivity as a monotone and admit every efficiently simulable positivity-preserving unitary as a free operation.
The Kirkwood-Dirac (KD) quasiprobability distribution is a fundamental representation for quantum states and has been widely applied in quantum metrology, quantum chaos, weak values in recent years. A quantum state is KD-classical if its KD-quasiprobability distribution forms a valid classical probability distribution with respect to two given bases, and KD-nonclassical otherwise, with the latter being closely associated with quantum advantages in various quantum processes. In this work, we investigate the structural characteristics of the KD-classical state set when the transition matrix between two orthonormal bases takes the form of a discrete Fourier transform (DFT) matrix. First, we adopt an alternative analytical approach to prove that the set of KD-classical states in a pr-dimensional Hilbert space is the convex hull of KD-classical pure states--a conclusion that was recently established by De Bi{è}vre et al [Annales Henri Poincar{é}, 1-20, 2025]. Furthermore, we define a directed graph and use it to characterize KD-classical pure states in a Hilbert space of arbitrary dimension d. That is, the convex hull of KD-classical pure states along any path from the start vertex to the end vertex in this directed graph is exactly the intersection of the KD-classical state set and the linear space spanned by these path-associated KD-classical pure states. This general result not only yields the pr-dimensional conclusion in a straightforward manner but also encompasses Theorem 2 in the existing work [J. Phys. A, 57, 435303, 2024], demonstrating its generality and inclusiveness.
Abstract The characterization of Kirkwood–Dirac (KD) classicality or non-classicality is very important in quantum information processing. In general, the set of KD classical states with respect to two bases is not a convex polytope (Langrene et al 2024 J. Math. Phys. 65 072201), which makes us interested in finding out in which circumstances they do form a polytope. In this paper, we focus on the characterization of KD classicality of mixed states for the case where the transition matrix between two bases is a discrete Fourier transform (DFT) matrix in Hilbert space with dimensions p 2 and pq , respectively, where p , q are prime. For the two particular cases we investigate, the sets of extremal points are finite, implying that the set of KD classical states we characterize forms a convex polytope. We show that for p 2 dimensional system, the set KD A , B + is a convex hull of the set pure ( KD A , B + ) based on DFT, where KD A , B + is the set of KD classical states with respect to two bases and pure ( KD A , B + ) is the set of all the rank-one projectors of KD classical pure states with respect to two bases. In pq dimensional system, we believe that this result also holds. Unfortunately, we do not completely prove it, but some meaningful conclusions are obtained about the characterization of KD classicality.
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.