Pauli Matrices

Here is a summary of useful properties of Pauli matrices in quantum computation.

I. Definition

Pauli matrices are $2\times 2$ matrices which are defined as below,

\[X = \sigma_x = \begin{pmatrix} 0 & 1\\ 1 &0 \end{pmatrix}, Y = \sigma_y =\begin{pmatrix} 0 & -i\\ i &0 \end{pmatrix}, Z = \sigma_z =\begin{pmatrix} 1 & 0\\ 0 &-1 \end{pmatrix}. \nonumber\]

From the definition above, we could easily check the following properties of Pauli matrices:

Pauli matrices are Hermitian, $X = X^{\dagger},Y = Y^{\dagger},Z = Z^{\dagger}$.
Pauli matrices are unitary, $X^{\dagger}X = I,Y^{\dagger}Y = I,Z^{\dagger}Z = I$.

II. Eigenvalues, Eigenstates, Diagonalized Form

Here are eigenvalues, eigenstates and diagonalized form of Pauli matrices.

For $X$, the eigenvalues $\lambda_1=1$ and $\lambda_2=-1$, and their corresponding eigenvectors are

\[|v_1\rangle = |+\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 1 \\ 1\end{pmatrix}, |v_2\rangle = |-\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\end{pmatrix}. \nonumber\]

For $Y$, the eigenvalues $\lambda_1=1$ and $\lambda_2=-1$, and their corresponding eigenvectors are

\[|v_1\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \\ i\end{pmatrix}, |v_2\rangle = \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -i\end{pmatrix}. \nonumber\]

For $Z$, the eigenvalues $\lambda_1=1$ and $\lambda_2=-1$, and their corresponding eigenvectors are

\[|v_1\rangle = |0\rangle = \begin{pmatrix}1 \\ 0\end{pmatrix}, |v_2\rangle = |1\rangle= \begin{pmatrix}0\\ 1\end{pmatrix}. \nonumber\]

III. Operation on Basis

For $Z-$basis state, the operation of Pauli matrices are given by

For $X-$basis state, the operation of Pauli matrices are given by

IV. Commutation Rules

The commutation of two matrices/operators are defined as $[A, B] = AB-BA$, and the anti-commutation of two matrices/operators are defined as ${A, B} = AB+BA$. Pauli matrices obey the following commutation and anti-commutation rules.

Commutation. Pauli matrix commutes with themselves but does not commute with another Pauli matrix,

\[[X, X] = [Y, Y] = [Z, Z] = 0. \nonumber\\ [X, Y] = 2iZ, [Y, Z] = 2iX, [Z, X] = 2iY.\nonumber\]

Anti-commucation. Pauli matrix does not commute with themselves but anticommute with another Pauli matrix,

\[\{X, X\} = \{Y, Y\} = \{Z, Z\} = 2I.\nonumber\\ \{X, Y\} = \{Y, X\} = \{Y, Z\} = \{Z, Y\} = \{Z, X\} = \{X, Z\} =0.\nonumber\]

V. Useful Identities

Here are several useful identities of Pauli matrices.

For the multiplication $\sigma_i\sigma_j$, we have

\[XX = YY = ZZ = I\nonumber\\ XY = - YX = iZ \nonumber\\ YZ = -ZY = iX \nonumber\\ ZX = -XZ = iY \nonumber\]

For the trace ${\rm Tr}{\sigma_i\sigma_j}$, we have ${\rm Tr}{\sigma_i\sigma_j} = 2\delta_{ij}$, or

\[{\rm Tr}\{XX\} = {\rm Tr}\{YY\} = {\rm Tr}\{ZZ\} = 2\nonumber \\ {\rm Tr}\{XY\} = {\rm Tr}\{YX\} = {\rm Tr}\{XZ\} = {\rm Tr}\{ZX\} = {\rm Tr}\{YZ\} = {\rm Tr}\{ZY\} = 0 \nonumber\]

For the multiplication in the form $\sigma_{i}\sigma_j\sigma_i$, we have

\[XXX = X, XYX = -Y, XZX = -Z \nonumber\\ YXY = -X, YYY = Y, YZY = -Z \nonumber\\ ZXZ = -X, ZYZ = -Y, ZZZ = Z \nonumber\]

For the multiplication in the form $H\sigma_i H$, we have

\[HXH = Z, HYH = -Y , HZH = X \nonumber\]

VI. Pauli matrices as basis

For any $2\times 2$ complex matrix and $a,b,c,d\in\mathbb{C}$

\[\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \frac{a+d}{2}I + \frac{b+c}{2}X - \frac{b-c}{2i}Y+ \frac{a-d}{2}Z \nonumber\]

Specially, for any $2\times 2$ Hermitian matrix and $a,b,c,d\in\mathbb{R}$,

\[\begin{pmatrix} a & b \\ c & d \end{pmatrix} = \frac{a+d}{2}I + bX - cY+ \frac{a-d}{2}Z \nonumber\]

Dawei Zhong