Pauli Basis Matrices and Pauli Vector

1. The Pauli Basis Matrices denoted \(\sigma_x\), \(\sigma_y\) and \(\sigma_z\) (or \(\sigma_1\), \(\sigma_2\) and \(\sigma_3\)) are a Set of 3 Unitary Traceless, Hermitian \(2\times 2\) Matrices given as follows
   
   \(\sigma_x=\sigma_1=\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix}\),   \(\sigma_y=\sigma_2=\begin{bmatrix}0 & -i \\ i & 0\end{bmatrix}\),   \(\sigma_z=\sigma_3=\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}\)
2. Following are Properties of Pauli Basis Matrices
   1. They are Hermitian
   2. They are Unitary
   3. They are Involutary
   4. Their Trace is 0
   5. They have Determinant Value of -1
   6. Product of Any 2 Pauli Basis Matrices is given as
      
      \(\sigma_j\sigma_k = i\hspace{1mm}\epsilon_{jkl}\hspace{1mm}\sigma_l\)
      
      This means
      
      \(\sigma_x\sigma_y=i\sigma_z\),   \(\sigma_y\sigma_z=i\sigma_x\),   \(\sigma_z\sigma_x=i\sigma_y\)
      
      \(\sigma_y\sigma_x=-i\sigma_z\),   \(\sigma_z\sigma_y=-i\sigma_x\),   \(\sigma_x\sigma_z=-i\sigma_y\)
      
      This means that the Product of Any 2 Pauli Basis Matrices is Anti-Commutative. That is
      
      \(\sigma_{i}\sigma_{j}=-\sigma_{j}\sigma_{i}\)
   7. The Commutator of Any 2 Pauli Basis Matrices is given as
      
      \([\sigma_j,\sigma_k] = \sigma_j\sigma_k-\sigma_k\sigma_j = 2i\hspace{1mm}\epsilon_{jkl}\hspace{1mm}\sigma_l\)
   8. The Anti Commutator of Any 2 Pauli Basis Matrices is given as
      
      \(\{\sigma_j,\sigma_k\} = \sigma_j\sigma_k+\sigma_k\sigma_j = 2\hspace{1mm}\delta_{jk}\hspace{1mm}I\)
   9. Product of 3 Pauli Basis Matrices is given as
      
      \(\sigma_j\sigma_k\sigma_l=i\hspace{1mm}\epsilon_{jkl}\hspace{1mm}I\)
3. Using the Pauli Basis Matrices, any 3 Dimensional Real Vector can be represented as a Traceless, Hermitian, \(2\times 2\) Complex Matrix. Any 3 Dimensional Real Vector represented as such is called a Pauli Vector. For example, the 3D Vector \(V=\begin{bmatrix}x,y,z\end{bmatrix}^T\) can be represented as a Pauli Vector \(P\) as follows
   
   \(V=\begin{bmatrix}x\\y\\z\end{bmatrix}=x\mathbf{\hat{i}}+y\mathbf{\hat{j}}+z\mathbf{\hat{k}}=x\mathbf{\sigma_x}+y\mathbf{\sigma_y}+z\mathbf{\sigma_z}= x\begin{bmatrix}0 & 1 \\ 1 & 0\end{bmatrix} + y\begin{bmatrix}0 & -i \\ i & 0\end{bmatrix} + z\begin{bmatrix}1 & 0 \\ 0 & -1\end{bmatrix}= \begin{bmatrix}z & x-yi \\ x+yi & -z\end{bmatrix}=V\cdot\sigma=P\)
   
   Similarly, Any Pauli Vector \(P\) can be converted to a 3D Vector as follows
   
   \(P=\begin{bmatrix}z & x-yi \\ x+yi & -z\end{bmatrix}=\begin{bmatrix}a_{00} & a_{01} \\ a_{10} & a_{11}\end{bmatrix}=\begin{bmatrix}{\Large\frac{a_{10}+a_{01}}{2}}\\{\Large\frac{a_{10}-a_{01}}{2i}}\\a_{00}\end{bmatrix}=\begin{bmatrix}x\\y\\z\end{bmatrix}=V\)
4. Any Property/Operation of/on a 3D Vector has a corresponding analog for it's Pauli Vector representation as follows
   1. The Square of a Pauli Vector is the Identity Matrix Scaled by the Square of the Magnitude of the Corresponding 3D Vector. For example, if \(|V|\) is the Magnitude of 3D Vector \(V\), then the Square of a Corresponding Pauli Vector \(P\), \(P^2=|V|^2I\).
   2. The Determinant of a Pauli Vector is the Negative of Square of the Magnitude of the Corresponding 3D Vector. For example, if \(|V|\) is the Magnitude of 3D Vector \(V\), then the Determinant of Corresponding Pauli Vector \(P\), \(|P|=-|V|^2\).
   3. The Anti-Commutator of any 2 Pauli Vectors is the Identity Matrix Scaled by Twice the Value of Dot Product of the Corresponding 3D Vectors. For example, if \(V_1\cdot V_2\) is the Dot Product of 2 3D Vectors \(V_1\) and \(V_2\), then the Anti-Commutator of their Corresponding Pauli Vectors \(P_1\) and \(P_2\), \(\{P_1,P_2\} = P_1P_2+P_2P_1 = 2(V_1\cdot V_2)I\).
   4. The Commutator of any 2 Pauli Vectors is the Pauli Vector Corresponding to Cross Product of the Corresponding 3D Vectors Scaled by \(2i\). For example, if \(V_1\times V_2\) is the Cross Product of 2 3D Vectors \(V_1\) and \(V_2\), then the Commutator of their Corresponding Pauli Vectors \(P_1\) and \(P_2\), \([P_1,P_2] = P_1P_2-P_2P_1 = 2iP_R\), where \(P_R\) is the Pauli Vector Corresponding to the Vector \(V_1\times V_2\).
5. The Product of 2 Pauli Vectors corresponding to 2 Parallel 3D Vectors is Commutative. For example if \(P_1\) and \(P_2\) are 2 Pauli Vectors corresponding to 2 3D Parallel Vectors \(V_1\) and \(V_2\), then \(P_1P_2 = P_2P_1\), that is, \([P_1,P_2] = P_1P_2-P_2P_1 = 0\).
6. The Product of 2 Pauli Vectors corresponding to 2 Perpendicular 3D Vectors is Anti-Commutative. For example if \(P_1\) and \(P_2\) are 2 Pauli Vectors corresponding to 2 Perpendicular 3D Vectors \(V_1\) and \(V_2\), then \(P_1P_2 = -P_2P_1\).
7. Any Pauli Vector \(P\) can be rotated by using \(SU(2)\) Rotation Matrix \(R\) as follows
   
   \(P_R = RPR^\dagger = (-R)P(-R)^\dagger\)
   
   where
   
   \(P =\) Given Pauli Vector
   
   \(P_R =\) Rotated Pauli Vector
   
   \(R,-R =\) \(SU(2)\) Rotation Matrices
   
   \(R^\dagger, -R^\dagger =\) Conjugate Transpose of Matrices \(R\) and \(-R\) respectively
   
   \(SU(2)\) Rotation Matrix \(R\) for an Axis of Rotation Vector \(A\) and Rotation Angle \(\theta\) Radians is calculated as follows
   
   \(R=\cos({\Large\frac{\theta}{2}})I - i\sin({\Large\frac{\theta}{2}}) (A\cdot\sigma)\)
   
   \(SU(2)\) Rotation Matrix \(-R\) can be calculated either by negating the above equation or as follows
   
   \(-R=\cos({\Large\frac{2\pi-\theta}{2}})I - i\sin({\Large\frac{2\pi-\theta}{2}}) (-A\cdot\sigma)\)
   
   Please note that Both \(SU(2)\) Rotation Matrices \(R\) and \(-R\) can be resolved into Counter Clockwise Rotation Angle \(\theta\) with respect to Axis \(A\) and Counter Clockwise Rotation Angle \(2\pi-\theta\) with respect to Axis \(-A\).
8. Any Pauli Vector \(P\) can be Reflected Across a Plane whose Unit Normal is given by a Pauli Vector \(A\) as follows
   
   \(P_{Ref}=-APA\)
   
   where
   
   \(P =\) Given Pauli Vector
   
   \(P_{Ref} =\) Reflected Pauli Vector
   
   \(A =\) Pauli Vector Corresponding to Unit Vector Normal to a Plane