1. Introduction
Quantum entanglement is the phenomenon of particles interacting in a system consisting of two or more particles, although the particles may be separated by a distant space [
1]. It is an important physical resource and plays an important role in quantum information processing, such as quantum computation [
2], cryptographic protocols [
3,
4], quantum state tomography [
5,
6], and modern quantum technologies [
7,
8]. Specifically, the maximally entangled states play a central role in quantum mechanics and quantum information processing [
9,
10].
In 1960, mutually unbiased bases (MUBs), first introduced by Schwinger in [
11], also have useful applications in quantum information processing, and Ivanovic applied the mutually unbiased bases to the problem of quantum state determination in [
12]. Two orthonormal bases
and
of
are called mutually unbiased if and only if:
A set of orthonormal bases
is called mutually unbiased if any two bases of them are mutually unbiased. Denote
the maximum number of any set of MUBs in
. An open problem with mutually unbiased bases is to determine the value of
when
d is not a power of a prime number. It is already known that
for any dimension
d and
if
d is a prime power [
13,
14,
15]. However, if
d has at least two different prime divisors, the result for
is known very little, even for
. We refer the readers to [
16] and the references cited in that paper.
The unextendible maximally entangled basis (UMEB) in
was introduced in [
17]. It was shown that the UMEB is constructed explicitly when
and
[
18]. In Reference [
18], Chen and Fei studied the UMEB in any bipartite system
. They presented a method to construct
-member UMEBs in
when
and gave two examples of mutually unbiased unextendible maximally entangled bases (MUUMEBs) in
. In recent years, UMEBs have been studied more extensively in arbitrary bipartite systems. In [
19], Nan et al. presented the construction of the
-member UMEBs in
when
and gave two complete MUUMEBs in
. In [
20], Tao et al. studied the mutually unbiased maximally entangled bases in bipartite systems
. They presented five MEBs in
and three MEBs in
that are mutually unbiased. In [
21], Zhang et al. provided two constructions of UMEBs in
based on the constructions of UMEBs in
and in
. In [
22], Han et al. presented an easy way of constructing a mutually unbiased entangled basis with a fixed Schmidt number of two (MUSEB2) in
when
. In [
23], Tang et al. constructed two complete UMEBs in bipartite system
and obtained the sufficient and necessary conditions of these two UMEBs to develop MUBs. In [
24], Xu constructed new types of mutually unbiased maximally entangled bases (MUMEBs) in
by using Galois rings.
In this paper, we provide a new construction of MEBs in bipartite system
when
. We first review some basic concepts, then present a systematic way of constructing MEBs, which is a different approach from [
18,
20] and Theorem 1 in [
19]. To briefly summarize their methods, these three articles obtained a new set of MEBs by using all the operators of the Weyl–Heisenberg group of the low-dimensional system to act on the standard maximally entangled state. None of them applied the Weyl–Heisenberg group of the high-dimensional system to construct MEBs. In this paper, we shall consider this type of construction of MEBs. The difficult problem we need to overcome is choosing a part of the operators in the Weyl–Heisenberg group and finding a suitable state. Since the operators of the Weyl–Heisenberg group are invertible, the MEBs we construct are certainly different from those of [
18,
19,
20]. Furthermore, we give explicit examples of MEBs in
, which are mutually unbiased, and eight-member UMEBs in
. Finally, we give some conclusions and discussions of this paper.
2. Preliminaries
For the sake of convenience, we review some basic definitions and notations for quantum entanglement states in the following.
Suppose the Hilbert space associated with some isolated physical bipartite system is
. Let
and
be the computational bases in
and
, respectively. A state
in
is called a product state (or separable state) if it can be written as
, where
and
are any two quantum states of the corresponding subsystems. Otherwise, the state
is called an entangled state. For any given orthonormal complete basis
of subsystem
A, if there exists an orthonormal basis
of subsystem
B such that
can be expressed as
, then
is said to be a maximally entangled state [
18].
One can also describe the maximally entangled state by the so-called Schmidt decomposition. For any vector
, one has the corresponding Schmidt decomposition [
25]:
where
are positive real numbers and
,
are orthonormal bases in
and
, respectively. If
is a pure state of a composite system, then
are called its Schmidt coefficients and
the Schmidt number. The pure state
is maximally entangled if and only if its Schmidt number is
d and all Schmidt coefficients are equal to
.
We denote by
the vector space of all
complex matrices.
is a Hilbert space under the Hilbert–Schmidt inner product defined by
for any two
matrices
. If
is a pure state in
, then there is a corresponding
matrix
, and the Schmidt number of
and the rank of matrix
M are equal. Moreover, if
and
are two pure states in
, then
[
26,
27]. It is easy to see that
is a maximally entangled state if and only if all the singular values of the matrix
equal one.
A collection of states
is called an unextendible maximally entangled basis (UMEB) [
18] if and only if:
- (i)
are all maximally entangled states;
- (ii)
, for ;
- (iii)
All the states in the orthogonal complement space of cannot be maximally entangled.
Next, let us consider a set of
unitary matrices:
where
,
is any primitive
dth root of unity, and
denotes
. These
matrices constitute a basis of the vector space
(or equivalently, the operator space on
) and:
The above
linear transformations
correspond to the Weyl–Heisenberg group. We use some of these operators to construct maximally entangled bases in
, which is a different method compared to that in [
18,
20].
3. MEBs in ℂd ⊗ ℂqd
In this section, we present a new method of constructing a maximally entangled basis (MEB) in when .
Consider the following pure state:
If we assume
, then the state in (
4) happens to be the standard maximally entangled state in
. In [
18,
19,
20], they all used this type of state to construct mutually unbiased MEBs and unextendible MEBs by making a transformation to the bases of subsystems. In this case, our constructions coincide with theirs. However, when
, we did obtain a different kind of maximally entangled states. In our case, it is not difficult to check that all the singular values of the complex matrix
(the definition of this complex matrix is in the
Section 2) equal one. That is to say,
is a maximally entangled state.
Now, let
be the
operators on
. We apply these operators on the above state as follows:
Obviously, we obtain
maximally entangled states in
. These states cannot form a basis of
since
. However, we can always choose part of the states in (
5). Next, we briefly list some simple properties of the Weyl–Heisenberg group. Based on this, we found a family of maximally entangled bases of form (
5). Then, in particular, we illustrate our method with two examples of low-dimensional systems.
We first considered the
unitary matrices
defined in (
2) of the vector space
. Each of these
matrices has only
d nonzero entries. These nonzero entries are all the
dth root of unity. The form of these matrices
is determined by
m, which is independent of the value of
n. When
,
is a diagonal matrix, and the entries on the diagonal are the
dth root of unity. When
, the nonzero entry of the first row is in the last column, and the other
nonzero entries are directly below the main diagonal, i.e., they are in the lower left subdiagonal of the matrix. When
, the
d entries of the bottom left corner and the superdiagonal of the matrix are nonzero elements. The nonzero entries of these matrices move parallel along the main diagonal. There are exactly
d different forms of matrices. There are exactly
d matrices in each form, which are determined by the value of
n. For example, we have
d diagonal matrices in the Weyl–Heisenberg group. According to group theory, the form of the product of two matrices of the Weyl–Heisenberg group is also one of these
d different types.
Now, let us discuss the maximally entangled states defined in (
5). Let
and
be two operators on
. Denote the matrix
by
. If
, then by (
3), we have:
If
, after a simple calculation, we have:
In order to make the right-hand side of the above formula equal to zero, we need to find out how many forms of the matrix in the Weyl–Heisenberg group satisfy that the corresponding elements are all zero. Since are parallel to the main diagonal and all the other ’s appearing in the above formula are closer to the main diagonal, we have forms of matrices in the Weyl–Heisenberg that make these entries not equal to zero. According to our analysis in the previous paragraph, if we choose m or from the set , then the sum of the ’s of the right-hand side of the above equation is equal to zero. Because there are matrices in each form (that is, n can be chosen from all these numbers ), thus we find matrices to make satisfy the orthogonal property. At this point, we have proven that we obtained a set of maximally entangled bases. In order to make our construction clearer, we illustrate this method with two examples below by writing the states in detail.
Let us first construct MEBs in
. According to the above construction, we have:
We chose
and
, such that:
Since:
where we denote the matrix
. We derived that the sum of
’s in Equation (
8) is equal to zero when
. It is easy now to check that the above 12 states exactly form an orthonormal maximally entangled basis in
.
Next, we deal with the maximally entangled bases in
. Similarly, we have:
In this case, we chose
and
, such that:
Since:
where we denote again the matrix
. Similar to the previous case, the 18 states constitute an orthonormal maximally entangled basis in
.
4. MUMEBs in ℂ2 ⊗ ℂ4
In this section, we shall study the mutually unbiased maximally entangled bases (MUMEBs) in
. Let
and
be the computational bases in
and
, respectively. Throughout this section,
defined in the
Section 2 is the set of 16 operators on
.
Let:
and:
where
denotes
for
.
In this case, we have
and
. Thus, the value of
m =
. According to the previous discussion of our construction of the maximally entangled basis and after a simple calculation, we obtained that:
is a maximally entangled basis in
.
Suppose
is another basis in
and the transition matrix from
to
is the Hadamard matrix:
Then we have
After a simple calculation, we obtain another maximally entangled basis in
:
where
.
It is easy to check that the bases
and
in
are not mutually unbiased. In order to obtain a mutually unbiased basis, we just need to match the above states with some coefficients. That is, let:
where
. As for the coefficients, we have the following four inequivalent choices:
Now, it is not difficult to prove that the basis
and the basis
are mutually unbiased. In other words, we have:
Let us choose another basis
in
as:
Thus, we obtain the following maximally entangled states:
where
. If we let the coefficients
be any four of the following:
then we obtain that the basis
and the basis
are mutually unbiased. That is to say,
To summarize this section, we need to explain that is not a mutually unbiased maximally entangled basis in . One can check this straightforwardly by the definition of mutually unbiased bases that and are not unbiased. We found that this is related to the selection of m. We hope that this will be helpful to the research of mutually unbiased bases.
5. UMEBs in ℂ2 ⊗ ℂ5
In this section, we discuss a new construction of unextendible maximally entangled bases (UMEBs) in
. Let
and
be the computational bases in
and
, respectively. We prove the following maximally entangled states form of the UMEB in
:
where
.
We used reductio ad absurdum to prove the above statement. Suppose there exists an extended maximally entangled state:
where
and
are two unitary operators on
and
with respect to the above bases. Then, for all
, we have:
Let us compute these two equations in two steps.
At first, by a direct computation, we have:
Substitute
into the above formula, we have the following four equations:
Combining the above equations, that is (
26) ± (
28) and (
27) ± (
29), respectively, after a simplification, we obtain the following four equivalent equations:
Secondly, according to
, we have:
Then, the four equations (
30), (
32), (
38), and (
40) can be expressed as:
The other four equations (31), (33), (39), and (41) can be expressed as:
Based on the construction of the matrix
M and
X being a unitary matrix, we derived that
M is invertible, and we obtain:
Thus, we know the determinant of matrix
Y is equal to zero, which is a contradiction to
Y being a unitary matrix. Therefore, we proved that the eight states (
22), (
23) form an unextendible maximally entangled basis.