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 $p^r$ -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 $p^r$ -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.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 $p^r$-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 $p^r$-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.

quant-pharXiv:2603.13863Mar 17, 20260 citations

From pure quantum states to quantum classical systems

Itai Carmeli

Jan 1, 20250 citations

Geometry of Kirkwood–Dirac classical states: a case study based on discrete Fourier transform

Ying‐Hui Yang, Shuang Yao, Shi-Jiao Geng +2 more

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.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.

arXiv:2404.09399Journal of Physics A Mathematical and TheoreticalSep 30, 20244 citations

Convex roofs witnessing Kirkwood–Dirac nonpositivity

Christopher Langrenez, David R. M. Arvidsson-Shukur, Stephan De Bièvre

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.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.

arXiv:2407.04558Journal of Physics A Mathematical and TheoreticalMay 7, 20257 citations

p^r

-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

p^r

-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.