Pauli matrices

The Pauli matrices ${\sigma }_x,{\sigma }_y,{\sigma }_z$ are traceless Hermitian $2\times 2$ matrices that, together with the identity, form a basis of the complex $2\times 2$ matrices.

The identity and the three Pauli matrices form an orthogonal basis for the complex vector space of two-by-two matrices, with Hilbert-Schmidt inner product $\mathrm{Tr}(P^{†}Q)=2{\delta }_{PQ}$. Thus every operator has the expansion $A=\frac{1}{2}{\sum }_{P\in \{I,X,Y,Z\}}\mathrm{Tr}(PA)P$. For a density matrix, the identity coefficient is $1/2$ and the remaining real coefficients are half the components of its Bloch vector. This converts unitary conjugation into transformations of three real components.

Direct multiplication of the three Pauli matrices gives $X^2=Y^2=Z^2=I$, $XY=iZ$ and their cyclic permutations, with reversed products changing sign. Consequently $(\mathbf{a}\cdot \mathbf{\sigma })(\mathbf{b}\cdot \mathbf{\sigma })=(\mathbf{a}\cdot \mathbf{b})I+i(\mathbf{a}\times \mathbf{b})\cdot \mathbf{\sigma }$. In particular $(\mathbf{s}\cdot \mathbf{\sigma }{)}^2=|\mathbf{s}{|}^{2}I$, which gives the eigenvalues $(1\pm |\mathbf{s}|)/2$ of a qubit density matrix.

For $H=\mathbf{d}\cdot \mathbf{\sigma }$ with $\mathbf{d}\ne 0$, the eigenvalues are $\pm |\mathbf{d}|$ and the projectors are $P_s=(I+s\hat{\mathbf{d}}\cdot \mathbf{\sigma })/2$. A normalized eigenstate has $⟨\mathbf{\sigma }⟩=s\hat{\mathbf{d}}$. At $\mathbf{d}=0$ the spectrum is degenerate and that direction is undefined.

Every Hermitian matrix of size two can be written uniquely as $M=dI+b_x{\sigma }_x+b_y{\sigma }_y+b_z{\sigma }_z$ with real coefficients. For $M=\left(\begin{array}{cc}u& w\\ w^{\ast }& z\end{array}\right)$, take $d=(u+z)/2$, $b_z=(u-z)/2$, $b_x=\mathrm{Re}w$ and $b_y=-\mathrm{Im}w$. Alternatively, $d=\mathrm{tr}M/2$ and $b_i=\mathrm{tr}(M{\sigma }_i)/2$ follow from the trace orthogonality of the Pauli matrices and the identity matrix.

The Pauli matrix multiplication law gives $(\mathbf{b}\cdot \mathbf{\sigma }{)}^2=|\mathbf{b}{|}^{2}I$, hence the two eigenvalues are $d\pm |\mathbf{b}|$. For $\mathbf{b}\ne 0$, the corresponding pure-state Bloch vectors are $\pm \mathbf{b}/|\mathbf{b}|$.

For a real unit vector $\mathbf{n}$, the Pauli matrix multiplication law gives $(\mathbf{n}\cdot \mathbf{\sigma }{)}^2=I$. Separating the even and odd powers in the matrix exponential proves the displayed identity. For spin one-half, the physical rotation through angle $\theta$ has $a=\theta /2$.

For $A=\Omega T/2$, $B=\Phi /2$, the ordered product of a $z$ rotation after an $x$ rotation has $C=\cos A\cos B$ and $\mathbf{v}=(\cos A\sin B,\sin A\sin B,\sin A\cos B)$. Since $C^2+|\mathbf{v}{|}^2=1$, its angle is $2atan2(|\mathbf{v}|,C)$ and its axis is $\mathbf{v}/|\mathbf{v}|$. At $\mathbf{v}=0$ the group element is central and its axis is undetermined.

A traceless Hermitian matrix in a Hilbert-Schmidt inner product orthonormal basis of the real vector space of traceless Hermitian $N$-by-$N$ matrices. One choice uses symmetric and imaginary antisymmetric off-diagonal matrix units, normalized by $1/\sqrt{2}$, and traceless diagonal generators. Together with $I/\sqrt{N}$ they form an orthonormal basis of all Hermitian matrices. The coordinates $s_k=\mathrm{Tr}(\rho {\sigma }_k)$ give the Generalized Bloch representation.

The Pauli matrices satisfyConsequently each squares to the identity, distinct Pauli matrices anticommute, and their products span $M_2(\mathbb{C})$.

The identity follows because the identity matrix and the three Pauli matrices form an orthogonal basis of two-by-two matrices under the trace inner product. It rewrites a spin-vector bilinear square as a scalar four-fermion expression, with the Grassmann signs supplied when fields are reordered. This is the algebra behind a spin-vector Hubbard–Stratonovich decoupling.

For vectors $\mathbf{a},\mathbf{b}\in {\mathbb{R}}^3$, the Pauli matrix multiplication law gives