Mutually unbiased maximally entangled bases (MUMEBs) are an interesting topic combining mutually unbiased bases (MUBs) and maximally entangled states. Mutually unbiased bases play an central role in quantum kinematics [1] , quantum state tomography [2] - [4] and many tasks in quantum information processing, such as quantum key distribution [5] , cryptographic protocols [6] [7] , mean king problem [8] , quantum teleportation and superdense coding [9] - [11] . Maximally entangled state is central both to the foundations of quantum mechanics and to quantum information and computation [12] - [24] .

A state is said to be a () maximally entangled state if and only if for an arbitrary given orthonormal complete basis of subsystem A, there exists an orthonormal basis of subsystem B such that can be written as [24] . Two orthonormal bases and of are mutually unbiased if and only if. A set of

orthonormal bases in are said to be a set of mutually unbiased bases if every pair of bases in the set is mutually unbiased.

Mutually unbiased bases are recently combined with other bases, such as product basis (PB) [25] , unextendible product basis (UPB) [26] , unextendible maximally entangled basis (UMEB) [27] - [32] and maximally entangled basis (MEB) [33] - [35] . The MEB is a set of orthonormally maximally entangled states in consisting of vectors. In [33] - [35] , by systematically constructing MEBs, the concrete construction of pairs of

MUMEBs in bipartite systems is studied.

In this note, we study the problem of constructing more than two mutually unbiased maximally entangled bases in bipartite spaces. Through the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased, we find the special matrices and present steps of using special matrix to construct five mutually unbiased maximally entangled bases in.

We first recall the sufficient and necessary conditions of two maximally entangled bases to be mutually unbiased in.

Let be the orthonormal basis in, and be two othonormal bases in, A denotes the transition matrix between them, that is, i.e., , are entries of the matrix A.

We first consider two MEBs in [33] as follows:

where are Pauli matrices and.

From [33] , the above two MEBs (1) and (2) in are mutually unbiased if and only if the matrices A satisfy the following relations:

where and denotes mod 2.

To visualize the conditions (3), we divide the transition matrix A into 4 submatrices of 2 × 2 from left to right, then the conditions (3) hold if and only if each 2 × 2 submatrix satisfying the similar conditions as follows (we might take the upper left submatrix as a representative):

From [33] , it is easy to find matrices satisfying the above conditions (4) such as

In this note, we want to find more than two MUMEBs, so how to find the third MEB mutually unbiased with the above two MEBs (1) and (2), it depends on the property transit matrix satisfied. Suppose that

be the third orthonormal basis in, and B denotes the transition matrix between and, that is, i.e., are entries of the matrix B. Then ac-

cording to [33] , we have the third MEB as follows

Then, the above three MEBs in are mutually unbiased if and only if the matrices A, B and BA all satisfy the conditions (4) simultaneously.

Since the transit matrix A is easy to choose, we really want to know the way to construct matrix B. Assume that

where P is a 2 × 2 matrix, if A is known, how can we choose the matrix P to assure B and BA all satisfy the conditions (4)? For simplicity, we can first assume that P be a diagonal block matrix

then we have

Since B satisfy the conditions (4), then we have

thus we must have

It follows from the unitarity of matrix P that

Similarly, we can have

so there are many choices about the values of. To avoid the trivial diagonal case of matrix P, we may take, then the values of can be divided into the following two cases:

We first discuss the case I. Obviously, there are many forms of P satisfying the above property, such as

No loss of generality, we first choose

then we have

It is direct to verify that the transformation matrix B and BA both satisfy the conditions (4), then the MEBs (1), (2) and (5) in are mutually unbiased.

Let, then

Denoting be the fourth orthonormal basis in, and C denotes the transition matrix between and, that is, then the fourth MEB in can be con-

structed as follows:

Obviously, , and

It is easy to check the above matrices C, and all satisfy the conditions (4), so the fourth MEB (13) is mutually unbiased with the former three bases (1), (2) and (5) in.

Moreover, let, then

Denoting be the fifth orthonormal basis in, and denotes the transition matrix between and, that is, then the fourth MEB in can be constructed as follows:

Obviously, , , and

One can directly check that the above matrices, , and all satisfy the conditions (4), so the fifth MEB (14) is mutually unbiased with the former four bases (1), (2), (5) and (13) in.

Furthermore, let, then

Denoting be the fifth orthonormal basis in, and be the transition matrix between and, that is, then and

Since is exactly equal to A, the sixth orthonormal basis is equal to, thus using matrix p, we can only get five MUMEBs (1), (2), (5), (13), (14) and no the sixth one.

Next, we discuss Case II of. Now there are many forms of P satisfying the property, such as

If we take the same A in (12) and choose the following form of P:

similar to the above analysis, we can get the five MUMEBs in in [33] .

In this note, we have constructed five mutually unbiased maximally entangled bases in bipartite spaces using special matrices. Thus, we have presented a method to construct more than two mutually unbiased maximally entangled bases in. Similar problems can be discussed in arbitrary bipartite spaces.

Jun Zhang,Qiang Yang,Hua Nan,Yuanhong Tao, (2015) Special Matrices in Constructing Mutually Unbiased Maximally Entangled Bases in C^{C4. Open Access Library Journal,02,1-7. doi: 10.4236/oalib.1101620}