1. Introduction
In quantum mechanics, there exist many nonclassical properties, such as entanglement, discord, coherence, nonlocality, contextuality 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. The Kirkwood–Dirac (KD) distribution is a quasiprobability distribution that was 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 the KD distribution of the state is real non-negative everywhere, i.e., a probability distribution. Otherwise, it is called KD nonclassical. Recently, KD nonclassicality has come to the forefront due to its application in quantum tomography [
5,
6,
7,
8,
9,
10,
11,
12,
13,
14] and weak measurements [
15,
16,
17].
The noncommutativity of observables cannot guarantee the nonclassicality of a state [
18]. The KD nonclassicality of a state depends not only on the state but also on the eigenbases of observables. Given a state
and an eigenbasis
of observable
A and an eigenbasis
of observable
B, authors in Ref. [
19] gave a sufficient condition on the KD nonclassicality of a state; that is,
is KD is nonclassical if
, where
counts the number of nonvanishing coefficients in the basis
representation, and it is similar for
. In 2021, De Bièvre [
20] introduced the concept of complete incompatibility on the eigenbases
of two observables
and presented the relations among complete incompatibility, support uncertainty and KD nonclassicality, also showing that
is KD nonclassical if
and
, where
and
are the eigenvectors of
, respectively. Xu [
21] generalized the concept of complete incompatibility to
s-order incompatibility and established a link between
s-order incompatibility and the minimal support uncertainty. Fiorentino et al. [
22] generalized Tao’s uncertainty relation [
23] to complete sets of mutually unbiased bases in spaces of prime dimensions. Recently, De Bièvre [
24] provided an in-depth study of the links of complete incompatibility to support uncertainty and to KD nonclassicality. Xu [
25] gave some characterizations for the general structure of KD classical pure states and showed that when
are mutually unbiased bases,
is KD classical if and only if
. This answered the conjecture in Ref. [
20]. Langrenez et al. characterized how the full convex set of states with positive KD distributions depends on the eigenbases of
and
[
26].
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
. De Bièvre [
24] characterized the uncertainty diagram of complete incompatibility bases. However, for the DFT matrix with nonprime order, the bases
are not completely incompatible bases. The uncertainty diagram of the DFT matrix with nonprime order is still unclear. In this paper, we consider this question. Firstly, for the uncertainty diagram of DFT, we show that for any dimension
d, there is no “hole” in the region of the
plane above or on the line defined by Tao’s uncertainty relation [
23],
, i.e., there is no absence of states with
in the region. Secondly, we provide some holes in the region strictly below the line and above the hyperbola defined by Donoho and Stark’s product uncertainty relation [
27],
. Finally, we give a new method to prove the conjecture [
20] about KD nonclassicality of a state based on DFT. Our method avoids analyzing the phases and only uses the concept of congruence class.
The rest of this paper is organized as follows. In
Section 2, we recall some relevant notions and notations. In
Section 3, we study the uncertainty diagram of DFT. In
Section 4, we give an alternative method to characterize the KD nonclassicality of a state based on DFT. Conclusions are given in
Section 5.
2. Preliminaries
Consider a Hilbert space
with dimension
d. Let an orthonormal basis
, respectively,
, be the eigenbasis of observable
A, respectively, of observable
B. Let
U be the unitary transition matrix with entries
from basis
to basis
. In terms of these two bases, the Kirkwood–Dirac (KD) distribution of a state
can be written as
It is a quasiprobability distribution and satisfies
, with conditional probabilities
and
. A state
is called classical if the KD distribution of
is a probability distribution, i.e.,
for all
. Otherwise,
is called nonclassical. Obviously, all of the basis vectors
and
are classical.
Given a state
, let
, respectively,
, be the number of nonzero components of
on
, respectively, on
. That is,
and
, where
and
denotes the cardinality of a set.
Two bases
and
are called completely incompatible [
20] if all index sets
for which
have the property that
where
is an orthogonal projector
and
is a
-dimensional subspace. Notice that any
implies
. If
and
are completely incompatible and mutually unbiased (or close to mutually unbiased), the only classical states are the basis states [
24].
The physical interpretation of completely incompatible bases is based on the theory of selective projective measurements [
24]. Suppose, on a state
, successive measurements in basis
then basis
yields the outcomes
S then
T. If the outcome
S occurs with probability one when measuring again in
after having obtained
T, then it implies that the measurement
is not disturbed by the first outcome
S. The completely incompatible bases implies that such repeated compatible selective measurements cannot occur.
The uncertainty diagram for orthonormal bases
, denoted by UNCD
, is a set of points
in the
plane, for which there exists a state
such that
and
, where
. For any state
, we have
max
, where
[
20,
24,
27]. If
are mutually unbiased bases (MUBs) [
28,
29], i.e.,
for all
, we have
. This is Donoho and Stark’s product uncertainty relation [
27]. This means that all the points
UNCD
are above or on the hyperbola
. The inequality
is Tao’s uncertainty relation [
23].
The following lemma was introduced in Ref. [
24]. It can be employed to determine whether a point
belongs to UNCD
.
Lemma 1. Let be two subsets of and suppose dim . Suppose that for all for which , one has dim , and that for all for which , one has dim . Then, the set of for which , is an open and dense set in . The opposite implication is also true.
By Lemma 1, the point
belongs to the UNCD
, since
and
. In order to better employ Lemma 1, let us first consider subspace
. Without loss of generality, let
,
. If
, then
. If
,
is isomorphic to the null space of a
matrix. Notice that [
20]
where
, and ≅ denotes two sets are isomorphic. It follows that
is isomorphic to the null space of the matrix
. It implies that
− Rank
. In this paper, a submatrix of a matrix
U is denoted by
where
and
are the
-th row and
-th column of
U and
.
Now, an improved lemma is given to show the existence of point in UNCD.
Lemma 2. In UNCD, suppose . Then, a point UNCD if and only if there exists a submatrix M of the transition matrix U,which satisfies the following three conditions: (i) Rank;
(ii) Rank Rank, whereand ; (iii) Rank Rank, where is a new submatrix of U that is obtained by removing one column of M.
Proof. Let us first consider the sufficiency. Without loss of generality, suppose that a submatrix with and satisfies the three conditions in Lemma 2. Let and .
Note that
is equal to the dimension of the null space of
M, that is,
Rank
. Then,
, since Rank
. For any
for which
, assume
. Then,
=
Rank
, where
is a
submatrix of
U that is obtained by adding row
k, i.e.,
to
M. By condition (ii), we have Rank
Rank
. Thus,
For any for which , assume . Then, is equal to the dimension of the null space of , i.e., Rank, where is a submatrix of U that is obtained by removing column l of M, i.e., . By condition (iii), we have Rank Rank. It follows Rank Rank. Therefore, UNCD by Lemma 1.
Now, we turn to showing the necessity. We proceed by contradiction. If condition (i) does not hold, i.e., Rank, it implies . Thus, UNCD by Lemma 1. It is a contradiction. If condition (ii) cannot be satisfied, i.e., for any submatrix M of U, there exists a row, called row k, that is added to M such that Rank Rank. Hence, Rank, where . It means UNCD by Lemma 1. Similarly, we can also obtain the desired result when condition (iii) cannot be satisfied. □
Note that
in Equation (
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. Lemma 2 provides a more efficient method to determine whether a point
belongs to UNCD
or not.
Now, we introduce the discrete Fourier transform (DFT). Suppose that F is the DFT matrix with , where and . Obviously, F is a symmetric and reversible Vandermonde matrix. The DFT matrix F has the following property.
Lemma 3. Suppose but . Letwhere and for . Then, Rank(M). The proof of Lemma 3 is given in
Appendix A. Since
F is symmetric, a similar property can be obtained if one interchanges indices of the rows with that of columns of
M in Equation (
8). Lemma 3 means that
M in Equation (
8) is a row full-rank matrix or a column full-rank matrix.
3. Uncertainty Diagram of DFT
De Bièvre [
20] has shown that the points on the hyperbola
belong to UNCD
of the DFT matrix
F. He [
24] also showed that
and
are completely incompatible if and only if
However, it is unclear if
and
are not completely incompatible. In this section, we continue to explore UNCD
of
F.
The UNCD
of
F is symmetric, since
F is symmetric [
24]. That is,
UNCD
of
F if and only if
UNCD
of
F.
Theorem 1. Suppose and . A point belongs to the UNCD of F if and .
Proof. In order to show
UNCD
, we only need to find an
submatrix that satisfies Lemma 2. First of all, let
where
and
. Obviously,
N is a
submatrix of
F, since
,
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 of F is added to N to obtain submatrix , then Rank() = Rank(N) + 1 = by Lemma 3. It is because is still a Vandermonde matrix, and and , where , .
(ii) If a column of N is removed to obtain submatrix , then Rank() = Rank(N) by Lemma 3 and .
Secondly, consider the following
submatirx
The equality in Equation (
9) holds due to
. It follows Rank(
M) = Rank(
N) =
.
Since
N has the above two properties and
M has the form in Equation (
9), we have Rank
Rank
and Rank
Rank
, where
is obtained by adding a new row
to
M,
, and
is obtained by removing a column of
M. Here, we employ the condition
. So,
M is the required submatrix. By Lemma 2, the required result is obtained. □
Notice that . This means Theorem 1 cannot work for . However, taking and in Theorem 1, we obtain UNCD, where . By the symmetry of UNCD of F, we have UNCD, where . In addition, UNCD by Lemma 1.
Taking and , we have the following result by Theorem 1 and the discussions above.
Corollary 1. A point UNCD of F if .
Note that in Corollary 1,
can run over set
due to the above discussion of Corollary 1. Corollary 1 means that all the points above and on the line segment
do exist, whether
and
are completely incompatible or not. It implies that there is no “hole” in the region of the
plane above and on the line
for any
d; that is, there is no absence of states with
in the region. The absence of states lies strictly above the hyperbola of
and strictly below the line
. This is illustrated in
Figure 1. The following theorems show where the holes are.
Theorem 2. A point belongs to the UNCD of F if and only if or but .
Proof. Sufficiency can be obtained by taking
in Theorem 1 and by the discussion above of Corollary 1 for
. We now show the necessity. Since
and
belong to UNCD
, we have
, respectively. Then, we consider
. A point
belongs to UNCD
of
F. It means there exists an
submatrix
satisfying Lemma 2. Then, we have Rank
=1. This means that
for any
and
. It follows that
. Assume
. Then,
. That is,
is in the congruence class of
modulo
. Notice that the cardinality of the congruence class of
modulo
is
p. It implies that there are at most
p rows in submatrix
M by the arbitrariness of
, i.e.,
. In fact, the submatrix
M has only
p rows, i.e.,
. Otherwise,
M cannot satisfy the second condition of Lemma 2. Thus,
since
. However,
, since
. □
Note that for the UNCD
of
F,
for any state
. Thus,
is meaningless in Theorem 2. This means that a point
the UNCD
of
F if and only if
. In fact, we only consider the case
since
. For example, if
, there is no hole for
, since
are all the divisors of 6. See panel (a) of
Figure 1. If
, there is a hole
, since
. See panel (b) of
Figure 1.
Theorem 3. Suppose d has only nontrivial prime divisors. Then, a point belongs to the UNCD of F if and only if or there exists a divisor m of d such that and or .
The proof of Theorem 3 is given in
Appendix B. Theorem 3 presents where the holes are when
. For instance, if
,
. Then,
. It implies that
, but
. Thus, there is two holes,
and
, for
. See panel (c) of
Figure 1. Panel (d) is for
. It should be noted that although 8 has a nontrivial nonprime divisor, one can check that the result of Theorem 3 for
still holds, since the proof can be followed similarly for the proof of Theorem 3.
4. KD Nonclassicality on DFT
In
Section 3, the existence of states in UNCD
has been shown. In this section, we focus on the KD nonclassicality of a state based on DFT. In Ref. [
20], 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
. From a different perspective than Xu [
25], we give an alternative method to prove this conjecture. Our method avoids analyzing the phases and only uses the concept of congruence class.
Theorem 4. Suppose that the bases and are related by the DFT matrix F. A state is KD nonclassical if and only if . In other words, is KD classical if and only if .
Proof. The necessity has been proved by De Bièvre in Ref. [
20]. Here, we only need to show the sufficiency, i.e.,
is KD nonclassical if
.
We proceed by contradiction. Suppose that
is KD classical, i.e.,
for any
. Since the KD distribution is insensitive to global phase rotations, we perform global phase rotations
and
such that
and
are non-negative for
. Reordering the basis vectors, we can suppose that
and
for
and
, where
are initial indices of basis vectors
and
, respectively. Thus, for the same range of
and
, we have
. Since
F is the DFT matrix, we have
. It follows
for
, and
. It means that the top left-hand block
in the new transition matrix after reordering the basis vectors is an
submatrix with all entries of
.
Let us first consider the trivial case. If (or ), then (or ), since F is the DFT matrix. Thus, . It is a contradiction with .
Next, we consider the case
and
. For any
, any
with
, calculate the product of two numbers,
and
, in
V. We have
where
. This implies that
. Notice that
is independent of
m. Thus, for any
and
, we have
It follows that
Suppose gcd
and
. If
, then
. This is impossible, since
and
. If
, then we have
It implies that
is in the congruence class of
modulo
q and the cardinality of the congruence class of
is
p. Similarly,
is in the congruence class of
modulo
p, and the cardinality of the congruence class of
is
q. Because of the arbitrariness of
and
, we obtain
and
. Therefore,
. This is a contradiction with
. □
From the above proof, we find that
if
is KD classical. It follows that
, since
for the DFT matrix. This implies that only the KD classical states lie on the hyperbola of
. This result gives a positive answer to the conjecture in Ref. [
20].
When
d is prime, the bases
are completely incompatible. Then, all the states are nonclassical except for basis vectors [
20]. In Theorem 4, the KD nonclassicality of a state based on the DFT matrix, whenever
d is prime or not, is completely characterized by Donoho and Stark’s product uncertainty relation [
27],
. See
Figure 1. It should be noted that this method is completely different from the method in Ref. [
25]. Here, we only use the concept of congruence class.
In the following example, we analyze the positions of two states in UNCD and their KD nonclassicality.
Example 1. Consider the computational basis and in , where . Obviously, the DFT matrix F is the transition matrix from to . Take a state . Then, and . The KD distribution, presented in Table 1, is nonclassical. Take , then and . The KD distribution, presented in Table 2, is real and non-negative. It should be noted that the notions of nonclassicality based on the different contexts are different. In Ref. [
30], Ferraro et al. focus on two celebrated criteria for defining the nonclassicality of
bipartite bosonic quantum systems. One stems from physical constraints on the quantum phase space. The other stems from information theoretic concepts. They showed that the two defining criteria are maximally inequivalent. Notice that these two types of definitions of the nonclassicality are for bipartite quantum systems. The quasiprobability formalism provides a useful alternative to describe the nonclassicality of quantum states. The Wigner function is the most famous quasiprobability distribution which deals with continuous variable systems [
3,
4]. However, it is ill-suited for finite-dimensional systems and observables. The KD distribution can deal with finite-dimensional systems. It is a versatile tool in studying quantum information processing. Recently, Budiyono et al. quantified quantum coherence via KD quasiprobability [
31]. Since the terms “classicality” and “nonclassicality” lack unique definition, Langrenez et al. changed the terminology from KD-classical to KD-positive [
26]. Notice that the KD distribution can be used for a single qudit system. It implies that a finite-dimensional system is not always nonclassical. For instance, the KD classical states only lie on the hyperbola of
for the DFT matrix. See
Figure 1.