^{1}

^{1}

^{1}

^{1}

^{*}

Some special matrices can really help us to construct more than two mutually unbiased maximally entangled bases in

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 [

A state

orthonormal bases

Mutually unbiased bases are recently combined with other bases, such as product basis (PB) [

MUMEBs in bipartite systems

In this note, we study the problem of constructing more than two mutually unbiased maximally entangled bases in bipartite spaces

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

Let

We first consider two MEBs in

where

From [

where

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 [

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

cording to [

Then, the above three MEBs in

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

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

Let

Denoting

structed as follows:

Obviously,

It is easy to check the above matrices C,

Moreover, let

Denoting

Obviously,

One can directly check that the above matrices

Furthermore, let

Denoting

Since

Next, we discuss Case II of

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 this note, we have constructed five mutually unbiased maximally entangled bases in 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}