Characterizing an Uncertainty Diagram and Kirkwood–Dirac Nonclassicality Based on Discrete Fourier Transform

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.


I. INTRODUCTION
In quantum mechanics, there exist many nonclassical properties such as entanglement, discord, coherence, nonlocality, contextuality, noncommutativity of two operators, uncertainty principles, and negativity or nonreality of quasiprobability distributions. By studying these nonclassical properties one can not only obtain a better understanding of quantum mechanics but also explore their applications in quantum information processing. Kirkwood-Dirac (KD) distribution is a quasiprobability distribution that is independently developed by Kirkwood [1] and Dirac [2]. It is a finite dimensional analog of the well-known Wigner distribution [3,4]. A quasiprobability distribution behaves like a probability distribution, but negative or nonreal values are allowed to appear in the distribution. For a quantum state and some observables, the KD distribution of this state can be obtained. A quantum state is called KD classical if KD distribution of the state is real nonnegative everywhere, i.e., a probability distribution. Otherwise, it is called KD nonclassical. Recently, KD nonclassicality has come to the forefront due to the application in quantum tomography [5][6][7] and weak measurements [8,9].
The noncommutativity of observables cannot guarantee the KD nonclassicality of a state. The KD nonclassicality of a state depends not only on the state but also on the eigenbases of observables. Given a state |ψ and an eigenbasis A of observable A and an eigenbasis B of observable B, authors in Ref. [10] gave a sufficient condition on the KD nonclassicality of a state, that is, |ψ is KD nonclassical if n A (ψ) + n B (ψ) > ⌊ 3d 2 ⌋, where n A (ψ) * Electronic address: yangyinghui4149@163.com counts the number of nonvanishing coefficients in the basis A representation, similar for n B (ψ). In 2021, De Bièvre [11] introduce the concept of complete incompatibility on eigenbases A, B of two observables A, B, and presented the relations among complete incompatibility, support uncertainty, and KD nonclassicality, also showed that |ψ is KD nonclassical if n A (ψ) + n B (ψ) > d + 1 and a i |b j = 0, where |a i and |b j are the eigenvectors of A, B, respectively. Xu [12] generalized the concept of complete incompatibility to s-order incompatibility and established a link between s-order incompatibility and the minimal support uncertainty. Recently, De Bièvre [13] provided an in-depth study of the links of complete incompatibility to support uncertainty and to KD nonclassicality.
Discrete Fourier transform (DFT) is an important linear transform in quantum information theory. The uncertainty diagram is a practical and visual tool to study the uncertainty of a state with respect to bases A, B. De Bièvre [13] characterized the uncertainty diagram of complete incompatibility bases. However, for the DFT matrix with nonprime order, the bases A, B are not completely incompatible bases. The uncertainty diagram of the DFT matrix with nonprime order is still unclear. In addition, in d-dimensional system, all the states are nonclassical except for basis vectors if d is prime [11]. However, for nonprime d, it is still an open problem. In this paper, we consider these two questions. Firstly, for the uncertainty diagram of DFT, we show that for any dimension d there is no "hole" in the region of the (n A , n B )plane above and on the line n A + n B ≥ d + 1, i.e., there is no absence of states with (n A (ψ), n B (ψ)) in the region. We also show some positions of holes in the uncertainty diagram of DFT. Secondly, we present the KD nonclassicality of a state based on the DFT matrix that is a transition matrix of bases A, B. The KD nonclassicality of a state based on the DFT matrix can be completely characterized by using the support uncertainty relation n A (ψ)n B (ψ) ≥ d. That is, a state |ψ is KD nonclassical if and only if n A (ψ)n B (ψ) > d, whenever d is prime or not. In other words, the lower bound, n A (ψ)n B (ψ) = d, is just attained for the KD classical states. The DFT is an example of a transition matrix for mutually unbiased bases (MUBs). For general MUBs, we also give a sufficient condition on the KD nonclassicality.
The rest of this paper is organized as follows. In Sec. II, we recall some relevant notions and notations. In Sec. III we study the uncertainty diagram of DFT. In Sec. IV, we characterize the KD classicality of a state based on DFT and give a sufficient condition of KD nonclassicality for general MUBs. Conclusions are given in Sec. V.

Consider a Hilbert space H with dimension d. Let an orthonormal basis
j=0 , be the eigenbasis of observable A, respectively of observable B. Let U be the unitary transition matrix with entries U ij = a i |b j from basis A to basis B. In terms of these two bases, the Kirkwood-Dirac (KD) distribution of a state |ψ ∈ H can be written as It is a quasi-probability distribution and satisfies A state |ψ is called to be classical if the KD distribution of |ψ is a probability distribution, i.e., Q ij ≥ 0 for all i, j ∈ Z d . Otherwise, |ψ is called to be nonclassical. Obviously, all of the basis vectors |a i and |b j are classical.
Given a state |ψ ∈ H, let n A (ψ), respectively n B (ψ), be the number of nonzero components of |ψ on A, respectively on B. That is, n A (ψ) = |S ψ | and n B (ψ) = |T ψ |, where and | · | denotes the cardinality of a set. Two bases A and B are called completely incompatible [11] if all index set S, T ∈ Z d for which |S| + |T | ≤ d have the property that where Π A (S) is an orthogonal projector i∈S |a i a i | and Π A (S)H is a |S|-dimensional subspace. Notice that any |ψ ∈ Π A (S)H implies S ψ ⊆ S. If A and B are completely incompatible, the only classical states are the basis states [13].
The uncertainty diagram for orthonormal bases A, B, denoted by UNCD (A, B), is a set of points (n A , n B ) ∈ Z * d+1 × Z * d+1 in the n A n B -plane for which there exists a state |ψ such that n A (ψ) = n A and n B [11,13,14]. It is called the support uncertainty relation. If A, B are mutually unbiased bases (MUBs) [15,16] It means that all the points (n A , n B ) ∈ UNCD(A, B) are above or on the hyperbola n A n B = d.
The following lemma was introduced in Ref. [13]. It can be employed to determine whether a point (n A , n B ) belongs to UNCD(A, B).
(4) It follows that H(S, T ) is the null space of the matrix ( a i |b j ) (d−k)×l . In this paper, a submatrix of a matrix U is denoted by where i k and j l are the i k -th row and j l -th column of U and i k , j l ∈ Z d . Now an improved lemma is given to show the exitance of point (n A , n B ) in UNDC (A, B).
which satisfies the following three conditions: and The proof of Lemma 2 is given in Appendix A. Note that M ′ in Eq. (7) is a submatrix of U and condition (ii) means the rank will increase by one if a new row is added to the submatrix M . And condition (iii) means the rank is invariant if a column of M is removed. Now we introduce the discrete Fourier transform Obviously, F is a symmetric and reversible Vandermonde matrix. The DFT matrix F has the following property.
where t ≤ d m and j l = j k mod d m for l = k. Then Rank(M )= min{s, t}.
The proof of Lemma 3 is given in Appendix B. Since F is symmetric, a similar property can be obtained if one interchanges indices of the rows with that of columns of M in Eq. (8). Lemma 3 means that M in Eq.(8) is a row full rank matrix or a column full rank matrix.

III. UNCERTAINTY DIAGRAM OF DFT
De Bièvre [11] has shown that the points on the hyperbola n A (ψ)n B (ψ) = d belong to UNDC(A, B) of the DFT matrix F . He [13] also showed that where i ∈ Z n m and k ∈ Z nB . Obviously, N be a n m × n B submatrix of F . Since n m < n B , N is a Vandermonde Matrix that is a row full rank matrix by Lemma 3. The matrix N has the following two properties.
(i) If row i 1 ∈ { n m , n m + 1, . . . , d m − 1} of F is added to N to obtain submatrix N ′ , then Rank(N ′ )=Rank(N )+1= n m + 1 by Lemma 3. It is because N ′ is still a Vandermonde matrix and n m + 1 ≤ n B and ω im Secondly, consider the following n × n B submatirx The equality in Eq.(9) holds due to ω  Taking m = 1 and n A = d − n, we have the following result by Theorem 1 and discussions above.
Note that in Corollary 1, n A can run over set Z * d+1 due to the above discussion of Corollary 1. Corollary 1 means that all the points above and on the line segment n A +n B = d+1 do exist whether A and B are completely incompatible or not. It implies that there is no "hole" in the region of the (n A , n B )-plane above and on the line n A + n B ≥ d + 1 for any d, that is, there is no absence of states with (n A (ψ), n B (ψ)) in the region. The absence of states lies strictly above the hyperbola of n A (ψ)n B (ψ) = d and strictly below the line n A + n B = d + 1. This is illustrated in Fig.1. The following theorems will show where the holes are. Proof. Sufficiency can be obtained by taking n = m in Theorem 1 and by discussion above Corollary 1 for n = 0. We now show the necessary. Since (d, 2) and (d − 1, 2) belong to UNDC (A, B), we have n = 0, 1, respectively. Then we consider n ≥ 2. A point (d − n, 2) belongs to UNDC(A, B) of F . It means there exists a n × 2 submatrix satisfying Lemma 2. Then we have Rank(M )=1. It means that ω That is, i s is in the congruence class of i t modulo d p . Notice that the cardinality of the congruence class of i t modulo d p is p. It implies that there are at most p rows in submatrix M by the arbitrariness of s, t ∈ Z n , i.e., n ≤ p. In fact, the submatrix M has only p rows, i.e., n = p. Otherwise, M cannot satisfy the second condition of Lemma 2. Thus, n|d since p|d. However, n = p = d since j = k mod d.

IV. KD NONCLASSICALITY ON MUB
In this section, we focus on the KD nonclassicality of a state based on MUBs. In Ref. [11], De Bièvre gave a conjecture, that is, whether it is true that the only KD classical states for the DFT are the ones on the hyperbola n A (ψ)n B (ψ) = d ? So we first consider the transition matrix of a pair of MUBs A and B is the DFT matrix F and try to answer this question.  Proof. The necessity has been proved by De Bièvre in Ref. [11]. Here we only need to show the sufficiency, i.e., |ψ ∈ H is KD nonclassical if n A (ψ)n B (ψ) > d.
We proceed by contradiction. Suppose that |ψ is KD classical, i.e., a i |ψ ψ|b j b j |a i ≥ 0 for any i, j ∈ Z d . Since KD distribution is insensitive to global phase rotations, we perform global phase rotations |a i → e √ −1φi |a i and |b j | → e √ −1φj |b j | such that a i |ψ and ψ|b j are nonnegative for i, j ∈ Z d . Reordering the basis vectors, we can suppose that a im |ψ > 0 and ψ|b js > 0 for m ∈ Z nA(ψ) and s ∈ Z nB (ψ) , where i m , j s are initial indices of basis vectors |a i and |b j , respectively. Thus, for the same range of i m and j s , It follows a im |b js = 1 √ d for m ∈ Z nA(ψ) and s ∈ Z nB (ψ) . It means that the top left-hand block V = (v ij ) in the new transition matrix after reordering the basis vectors is a n A ×n B submatrix with all entries of 1 √ d .
Let us first consider the trivial case. If n A (ψ) = 1 (or n B (ψ) = 1), then n B (ψ) = d (or n A (ψ) = d) since F is the DFT matrix. Thus n A (ψ)n B (ψ) = d. It is a contradiction with n A (ψ)n B (ψ) > d.
Next we will consider the case n A (ψ) ≥ 2 and n B (ψ) ≥ 2. For any m ∈ Z nA(ψ) , any s, t ∈ Z nB(ψ) and s < t, calculate the product of two numbers √ dv * ms and √ dv mt in V . We have (10) where α s,t := d 2π (φ jt − φ js ). It implies that α s,t + i m (j t − j s ) ≡ 0 mod d. Notice that α s,t is independent of m. Thus, for any m, n ∈ Z nA(ψ) and m < n, we have It follows that Suppose gcd(j t − j s , d) = p and q := d p . If p = 1, then i n ≡ i m mod d. It is impossible since m, n ∈ Z nA(ψ) and m < n. If p = 1, then we have It implies that i n is in the congruence class of i m modulo q and the cardinality of the congruence class of i m is p.
Similarly, j t is in the congruence class of j s modulo p, and the cardinality of the congruence class of j s is q. Because of the arbitrariness of i n , i m ∈ Z nA(ψ) and j t , j s ∈ Z nB(ψ) , we obtain n A ≤ p and n B ≤ q. Therefore, n A (ψ)n B (ψ) ≤ d. It is a contradiction with n A (ψ)n B (ψ) > d.
From the above proof we find that n A (ψ)n B (ψ) ≤ d if |ψ is KD classical. It follows that n A (ψ)n B (ψ) = d since n A (ψ)n B (ψ) ≥ d for the DFT matrix. It implies that only the KD classical states lie on the hyperbola of n A (ψ)n B (ψ) = d. This result gives a positive answer to the conjecture in Ref. [11].
When d is prime, the bases A, B are completely incompatible. Then all the states are nonclassical except for basis vectors [11]. In Theorem 4, the KD nonclassicality of a state based on the DFT matrix, whenever d is prime or not, is completely characterized by the support uncertainty relation. See Fig.1.
For two mutually unbiased observables, the author in Ref. [13] showed that all states, except for the eigenstates of the two observables, are KD nonclassical when two observables are completely incompatible. Next, we will consider the general case, that is, two observables are not necessarily completely incompatible.
Theorem 5. Let U be the unitary transition operator between A and B in a d dimensional Hilbert space H and suppose that |U ij | = | a i |b j | = 1 √ d for i, j ∈ Z d . |ψ ∈ H but not a basis vector is KD nonclassical if n A (ψ) > d 2 or n B (ψ) > d 2 . Proof. Suppose that |ψ is KD classical. Similar to the proof of Theorem 4, we perform global phase rotations and reorder and relabel the basis vectors such that a i |ψ > 0 and ψ|b j > 0 for i ∈ Z nA(ψ) and j ∈ Z nB(ψ) . Thus, b j |a i ≥ 0 for the same range of i and j. Since |U ij | = | a i |b j | = 1 √ d , we have a i |b j = 1 √ d for i ∈ Z nA(ψ) and j ∈ Z nB (ψ) . Note that it is impossible that n A (ψ) = 1 or n B (ψ) = 1 since |ψ is not a basis vector. For any 0 ≤ j < j ′ ≤ Z nB(ψ) , we have where P := d d−1 i=nA b j |a i a i |b j ′ . It follows that P = −n A (ψ).
If n A (ψ) > d 2 , we have |P | > d 2 . While It is a contradiction. Note that the first inequality follows from Cauchy-Schwarz inequality. This contradiction implies that |ψ ∈ H is KD nonclassical. The case when n B (ψ) > d 2 can be proved similarly by analyzing a i |a i ′ for any 0 ≤ i < i ′ ≤ Z nA(ψ) .
Theorem 5 gives a sufficient condition of KD nonclassicality for two mutually unbiased observables, whether they are completely incompatible or not. However, it is still unclear that |ψ ∈ H is KD nonclassical or not if n A (ψ) ≤ d 2 and n B (ψ) ≤ d 2 .

V. CONCLUSION
We have studied the uncertainty diagram and the Kirkwood-Dirac nonclassicality based on DFT in a d dimensional system. We show that for the uncertainty diagram of the DFT matrix, there is no "hole" in the region of the (n A , n B )-plane above and on the line n A + n B ≥ d + 1, whether the bases A, B are not complete incompatible bases or not. The absence of states lies strictly above the hyperbola of n A (ψ)n B (ψ) = d and strictly below the line n A + n B = d + 1. We also show where the holes are when n B = 2, 3. Then we present that the KD nonclassicality of a state based on the DFT matrix can be completely characterized by using the support uncertainty relation n A (ψ)n B (ψ) ≥ d. That is, a state |ψ is KD nonclassical if and only if n A (ψ)n B (ψ) > d, whenever d is prime or not. This result gives a positive answer to the conjecture in Ref. [11]. There are still some questions left. For example, a state is nonclassical or not when A, B are MUBs and n A (ψ) ≤ d 2 and n B (ψ) ≤ d 2 . Furthermore, how is the strength of the KD nonclassicality established ?