The classical zero-error channel and coding method were discussed in the previous section. This section deals with the coding problem in the quantum zero-error channel using the linear transformation of matrix.
3.1. Quantum n-Symbol Obfuscation Model
In quantum channels, probabilities between inputs and outputs can be represented by coefficients.
Figure 2 illustrates the quantum channel.
Figure 3 shows the pure states of n inputs and outputs as |0>, |1>, |2> … |n − 1>. The pure state
(i is an integer and
) becomes the superposition state
via the quantum channel (ground state
is in the outputs; variable j is an integer and
).
is the coefficient of the relation between ground states
and
and satisfies the condition of
. Analogous to classical channels, the probability that ground state
becomes ground state
is
.
In the quantum channel, as is shown in
Figure 3, the input set can be expressed as
, while the output set is
. Any input can be represented as
Variable i is an integer and . Ground state is in the outputs. Variable j is an integer and . is a complex number which satisfies .
According to quantum channel theory,
3.2. Preliminary Theorems for Zero-Error Coding in Quantum Cases
In a zero-error channel, outputs should be distinguished through coding. However, quantum states can be distinguished while they are orthogonal to each other. Considering the matrices theory, linearly independent vectors can be used to construct a set of orthogonal vectors.
Firstly, we introduce certain theorems of linear zero-error coding based on linear transformation, which is a lemma in matrices theory. Two theorems are then provided, following the lemma.
Lemma 1. The rank of a matrix is equal to the rank of its column vectors [27]. Proof. Let
be an n × n matrix satisfying rank (A) = m. Suppose that A has an m × m submatrix with . The necessary and sufficient condition for the linear independence of a vector group is that matrix , which consists of m vectors, should be of m rank. We determine that the m column in is linearly independent because . In addition, each m + 1 column vector in A is linearly dependent. Therefore, m column vectors compose the maximum linearly independent group among all column vector groups in A. Hence, the column vector groups in A are ranked m. □
Theorem 1. Letbe an n× n matrix (rank(A) = k). Suppose that A has an n× n similarity matrixconsisting of k linearly independent column vectors in A, such asand n–k zero column vectors. Then, an n× n matrixoccurs, which makes.
Proof. Matrix A can be transformed into matrix D through the elementary transformation of the matrix, because D is a similarity matrix of A.
An elementary matrix E differs from the identity matrix via a single elementary row operation. Right multiplication (post-multiplication) by an elementary matrix represents elementary column operations.
For example, right multiplication by is equal to the additional transformation of the column operation on matrix A, which adds column i multiplied by scalar k to column j.
Here, .
In the elimination process, the matrix can be reduced step by step to matrix D. The first step is to reduce the value of to 0. That is to say, the first column multiplying is added to the second column (i.e., ) while the elements of the first row and the second column are 0s. Therefore, .
Matrix A can be transformed into matrix D through finite right multiplication by the elementary matrices .
In other words, it can be expressed as . Here, . Therefore, a matrix B always exists and can produce AB = D. □
Theorem 2. Supposing the channel coefficient matrix A with rank (A) = k and the set of k inputs {},. Here, the value ofrefers to the value of the corresponding position in matrix B. Matrix B is equivalent to matrix B in Theorem 1. The set of k outputs {} can then be distinguished.
Proof. Let .
Considering the isomorphism between quantum states and vectors, the normalization of quantum states and Theorem 1, k linearly independent outputs and inputs can be found, which can be expressed in the following form.
In accordance with
Section 3.1, the input
turns into the output
after the signal passes through the channel. Here,
In Theorem 1,
are linearly independent. Thus, the condition is satisfied by
, and therefore
It can be seen from the above equation that the inner product of any two different outputs is zero. This makes the discrimination of two quantum states possible. Therefore, we can say that the set of k outputs {} can be distinguished. □
3.3. Quantum Method
The quantum zero-error communication coding method based on linear transformation is shown in
Figure 4.
Linear transformation here plays a central role in realizing the quantum zero-error coding. The following steps present the details of how it works.
A. Establishing the coefficient matrix
In the first step, the information of the quantum zero-error channel can be expressed in the form of a graph, function, or matrix depicting the relation between the ground states of n inputs and outputs. Channel matrix P and coefficient matrix A can be determined by the information of the channel, and can be represented by the following matrix:
In channel matrix P, row and column coordinates are input and output, respectively (i.e., the probability of input to output is ). The corresponding coefficient matrix is also input to the column and row coordinates, and the expression is the probability coefficient of input to output . Moreover, the column vectors in the coefficient matrix can be expressed with .
B. The rank of the coefficient matrix and the set of linearly independent vectors
The second step is to calculate the rank k of the coefficient matrix and find the k linearly independent vectors in the coefficient matrix via matrix transformation. The set of these vectors can be expressed as
, with any
denoted by
The rank of the coefficient matrix is k, and, in accordance with Lemma 1, the rank of the column of the matrix is also known as k. Therefore, knowing that the coefficient matrix can find at most k linearly independent column vectors is possible. These column vectors and n-k zero vectors form n × n matrix D as
C The relationship between linearly independent vectors and matrix column vectors
The third step is to determine the relation between the k linearly independent and column vectors of the coefficient matrix.
In accordance with Theorem 1, matrix B creates the following relationship between coefficient matrices A and D:
Matrix B can be represented as
By substituting Equations (4), (6) and (8) into Equation (7),
By substituting Equation (9) into Equation (5),
Therefore, the relation between the k linearly independent and column vectors of the coefficient matrix is Equation (10).
D Outputs and inputs as well as channel capacity
The last step is to find the outputs and inputs and the channel capacity after encoding. The encoded outputs and inputs are determined by using the isomorphism between the outputs and the linearly independent vectors. Moreover, k distinguishable outputs and their corresponding inputs can be obtained by finding the linearly independent vectors and normalization of quantum states. The outputs of the zero-error coding using the linear transformation method are distinguished by Theorem 2. Therefore, this method is appropriate.