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arXiv:2404.09399 · Journal of Physics A Mathematical and Theoretical

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

Ying‐Hui Yang, Shuang Yao, Shi-Jiao Geng, Xiaoli Wang, Pei-Ying Chen

Published Sep 30, 2024• 4 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1751-8121/ad819a

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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.
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 prp^rpr-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 ddd. 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 prp^rpr-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) 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.
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.
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.
In this paper, we investigate an uncertainty diagram and Kirkwood–Dirac (KD) nonclassicality based on discrete Fourier transform (DFT) in a d-dimensional system. We first consider the uncertainty diagram of the DFT matrix, which is a transition matrix from basis A to basis B. Here, the bases A, B are not necessarily completely incompatible. We show that for the uncertainty diagram of the DFT matrix, there is no “hole” in the region of the (nA,nB) plane above and on the line nA+nB=d+1. Then, we present where the holes are in the region strictly below the line and above the hyperbola nAnB=d. Finally, we provide an alternative proof of the conjecture about KD nonclassicality based on DFT.