%%What Pauli matrices are besides simply matrices that look a particular way needs to be clarified right in the definition.%% The _Pauli Matrices_ are a set of three $2\times2$ matrices expressed as $\sigma_1 = \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix} \;\;\;\; \sigma_2 = \begin{pmatrix} 0 & -i \\ i & 0\end{pmatrix} \;\;\;\; \sigma_3 = \begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix}$^3583c2 where alternatively we may also write $\sigma_x, \sigma_y, \sigma_z.$ In this context we may also denote the identity matrix as $\sigma_0,$ such that $\sigma_0 = \begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix}$^4ef47a We include the identity matrix here since taken together, it and the Pauli matrices are among the elements of the [single-qubit Pauli group.](Pauli%20groups.md#Single-qubit%20Pauli%20group) # Vector of Pauli Matrices A [vector](Linear%20Algebra%20and%20Matrix%20Theory%20(index).md#Vectors) containing the [Pauli matrices](Pauli%20Matrices.md) is defined as follows: $\pmb{\sigma}^{T} = \begin{pmatrix} \begin{pmatrix} 0 & 1 \\ 1 & 0\end{pmatrix}, & \begin{pmatrix} 0 & -i \\ i & 0\end{pmatrix}, & \begin{pmatrix} 1 & 0 \\ 0 & -1\end{pmatrix} \end{pmatrix}$ # Basic Properties The [Pauli matrices,](Pauli%20Matrices.md) $\sigma_i,$ have the following properties: 1) $\sigma_i^\dagger=\sigma_i$ ([Hermiticity](Hermitian%20operators.md)). 2) $\sigma_i^\dagger=\sigma_i^{-1}$ ([unitarity](Unitary%20operators.md)). 3) $\sigma_i^{-1}=\sigma_i$ ([Involutivity](Involutive%20operators.md#Involutory%20matrices)) 4) [$\mathrm{det}$](Determinants.md)$(\sigma_i)=1$ [Property 3.](Pauli%20Matrices.md#Basic%20Properties) follows directly from [properties 1. and 2.](Pauli%20Matrices.md#Basic%20Properties) and equivalently property 3. may be expressed as $\sigma_i^2=\sigma_0,$ where $\sigma_0$ is how we express the $2\times 2$ identity matrix [in this context.](Pauli%20Matrices.md) # Eigenvalues and eigenvectors The [eigenvalues and eigenvectors](Eigenvalues%20and%20eigenvectors.md) are given as follows where we start with the simplest eigenvectors: ## Normalized eigenvectors The [normalized eigenvectors](Eigenvalues%20and%20eigenvectors.md#Normalized%20eigenvectors) are given as follows where we start with the simplest eigenvectors: * Every normalized $1\times 2$ vector is an eigenvector of $\sigma_0$ * Eigenvalues and normalized eigenvectors for $\sigma_3:$ $\lambda_+ = 1, \;\;\; \chi_+ = \begin{pmatrix}1\\ 0 \end{pmatrix}; \;\;\;\;\;\; \lambda_- = -1, \;\;\; \chi_- = \begin{pmatrix}0\\ 1 \end{pmatrix}$ * Eigenvalues and normalized eigenvectors for $\sigma_2:$ $\lambda_+^y = 1, \;\;\; \chi_+^y = \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\ i \end{pmatrix}; \;\;\;\;\;\; \lambda_-^y = -1, \;\;\; \chi_-^y = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \\ -i \end{pmatrix}$ * Eigenvalues and normalized eigenvectors for $\sigma_1:$ $\lambda_+^x = 1, \;\;\; \chi_+^x = \frac{1}{\sqrt{2}}\begin{pmatrix} 1\\ 1 \end{pmatrix}; \;\;\;\;\;\; \lambda_-^x = -1, \;\;\; \chi_-^x = \frac{1}{\sqrt{2}}\begin{pmatrix}1 \\ -1 \end{pmatrix}$ # Pauli matrix [[Commutators]] $[\sigma_1,\sigma_2] =2i\sigma_3$, $\;\;\;\;[\sigma_2,\sigma_1] =-2i\sigma_3$, $\;\;\;\;[\sigma_3,\sigma_1] =2i\sigma_2$, $\;\;\;\;[\sigma_1,\sigma_3] = -2i\sigma_2$, $\;\;\;\;[\sigma_2,\sigma_3] =2i\sigma_1$, $\;\;\;\;[\sigma_3,\sigma_2] = -2i\sigma_1$ Where these relations relations are condensed to: $[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k.$ Here $\epsilon_{ijk}$ is the [[Levi-Civita Symbol]]. # Pauli matrix [[Anti-commutator]]s $\{\sigma_x,\sigma_y\} =0$, $\;\;\;\;\{\sigma_x,\sigma_z\} =0$, $\;\;\;\;\{\sigma_y,\sigma_z\} =0$\{\sigma_x,\sigma_x\} =2\sigma_0$, $\;\;\;\;\{\sigma_y,\sigma_y\} =2\sigma_0$, $\;\;\;\;\{\sigma_z,\sigma_z\} =2\sigma_0$ where these relations are condensed to: $\{\sigma_i,\sigma_j\} =2\delta_{ij}.$ Here $\delta_{ij}$ is the [Kronecker delta.](Kronecker%20delta.md) # Pauli matrices as elements of $\mathfrak{su}(2)$ The matrices $i\sigma$ form a basis for the [algebra](SU(2).md#Group%20Algebra) of $\mbox{SU}(2)$. Putting the [vector of pauli matrices](Pauli%20Matrices.md#Vector%20of%20Pauli%20Matrices) (along with the identity matrix) in a [complex matrix exponential,](Complex%20matrix%20exponentials.md), $e^{i\pmb{\sigma}\cdot\pmb{a}}$ yields a vector containing the elements of the group [$\mathrm{SU}(2)$](SU(2).md), which is written as follows: $e^{\pm i\pmb{\sigma}\cdot\pmb{a}} = I\cos(|\pmb{a}|^2) \pm i(\pmb{\sigma}\cdot\pmb{a})\sin(|\pmb{a}|^2)$ This expression is derived from taking the [power series representation](Complex%20matrix%20exponentials.md#Power%20series%20representation%20of%20complex%20matrix%20exponentials) of the complex matrix exponential and separating out the even and odd terms as shown [here.](Pauli%20Matrices.md#Derivation%20of%20SU%202%20Group%20elements%20from%20Pauli%20Matrices) # Pauli Groups ([... see more](Pauli%20groups.md)) --- # Proofs and Examples ## Derivation of eigenvalues and eigenvectors of the Pauli matrices ### Normalizing the eigenvectors of Pauli matrices ## Derivation of SU(2) Group elements from Pauli Matrices Here we show how [Pauli matrices](Pauli%20Matrices.md) act as [as generators of $\mathfrak{su}(2)$](Pauli%20Matrices.md#Pauli%20matrices%20as%20elements%20of%20$%20mathfrak{su}(2)$) by plugging them into [complex matrix exponential.](Complex%20matrix%20exponentials.md) This will give a set of matrices in [$\mathrm{SU}(2)$.](SU(2).md) We start with [power series representation of $e^{iX}$,](Complex%20matrix%20exponentials.md#Power%20series%20representation) and separate the even from the odd terms. Even terms have exponents $2n$, odd terms have exponents $2n+1$: $e^{i\pmb{\sigma}\cdot\pmb{a}} = \sum_{n=0}^{\infty} \frac{(i\pmb{\sigma}\cdot\pmb{a})^n}{n!} = \sum_{n=0}^{\infty} \frac{(i\pmb{\sigma}\cdot\pmb{a})^{2n}}{(2n)!} + \sum_{n=0}^{\infty} \frac{(i\pmb{\sigma}\cdot\pmb{a})^{2n+1}}{(2n+1)!}$ $= \sum_{n=0}^{\infty} \frac{(-1)^n(\pmb{\sigma}\cdot\pmb{a})^{2n}}{(2n)!} + \sum_{n=0}^{\infty} \frac{i(-1)^n(\pmb{\sigma}\cdot\pmb{a})^{2n+1}}{(2n+1)!}$ Notice that $(\pmb{\sigma}\cdot\pmb{a})^2 = I(a_1^2+a_2^2+a_3^2) = I|\pmb{a}|^2$. Therefore $(\pmb{\sigma}\cdot\pmb{a})^{2n} = I|\pmb{a}|^2$, $(\pmb{\sigma}\cdot\pmb{a})^{2n+1} = I|\pmb{a}|^2(\pmb{\sigma}\cdot\pmb{a})$, and $e^{i\pmb{\sigma}\cdot\pmb{a}}= \sum_{n=0}^{\infty} \frac{(-1)^n I|\pmb{a}|^{2}}{(2n)!} + \sum_{n=0}^{\infty} \frac{i (-1)^n I|\pmb{a}|^2}{(2n+1)!}(\pmb{\sigma}\cdot\pmb{a}) = I\cos(|\pmb{a}|^2) + i(\pmb{\sigma}\cdot\pmb{a})\sin(|\pmb{a}|^2)$ The resulting vector contains the elements of $\mathrm{SU}(2).$ %%I think "contains" might be the wrong word. It may that that it spans the group%% ## Proof that $e^{i\pmb{\sigma}\cdot\pmb{a}}$ contains the matrices in $\mathrm{SU}(2)$ %%A few comments on this note: - This note is math note instead of a physics note, but due to the connection between Pauli matrices and quantum physics it will be index as both. - Since it is a math note, it follows the notation conventions of math notes, and thus the Pauli matrices won't have hats like they do for quantum physics. - Currently the role of Pauli matrices as observables in quantum physics is placed in a note called "Observables of two level systems" A similar note for "dynamics of two level systems" will be used t denote their role in quantum dynamics of two level systems - This note will connect directly to the note on qubits. By convention, we treat qubits as eigenvectors of the z Pauli matrix.%% #QuantumMechanics/QuantumMeasurement/QuantumObservables #QuantumMechanics/TwoLevelSystems #QuantumMechanics/MathematicalFoundations #MathematicalFoundations/Algebra/AbstractAlgebra/LinearAlgebra/Matrices