Pauli Matrices

The Pauli matrices form a trace-orthogonal basis for complex $2\times2$ matrices. They encode spin-$1/2$ generators, qubit observables, two-level Hamiltonians, Bloch vectors, and the smallest nontrivial representation of the angular-momentum algebra.

This page is a convention-fixed calculation table. For derivations and geometric interpretation, use the canonical Pauli Matrices in Spin-1/2 Systems page or the linear-algebra treatment of Pauli Matrices.

Scope

Included here:

• the identity and three Pauli matrices;
• products, commutators, anticommutators, and traces;
• eigenvectors and spectral projectors;
• vector, exponential, and rotation identities;
• decomposition of arbitrary $2\times2$ matrices;
• qubit density matrices and two-level Hamiltonians;
• conversion between Pauli and physical spin operators.

General spin-$j$ representations belong in the Spin Matrices table. Gate conventions and controlled operations belong in Quantum Gates.

Convention

Use the ordered orthonormal basis

$$\lbrace\lvert 0\rangle,\lvert 1\rangle\rbrace, \qquad \lvert 0\rangle= \begin{pmatrix}1\\0\end{pmatrix}, \qquad \lvert 1\rangle= \begin{pmatrix}0\\1\end{pmatrix},$$

with

$$\sigma_z\lvert 0\rangle=+\lvert 0\rangle, \qquad \sigma_z\lvert 1\rangle=-\lvert 1\rangle.$$

In spin language, these states are also written $\lvert {+z}\rangle$ and $\lvert {-z}\rangle$. Cartesian indices run over $x,y,z$, the orientation is $\epsilon_{xyz}=+1$, and repeated Cartesian indices are summed when doing so is unambiguous.

Physical spin operators carry angular-momentum units:

$$S_i=\frac{\hbar}{2}\sigma_i.$$

Active rotations of states use

$$R_{\hat{\boldsymbol n}}(\theta) =\exp\left( -\frac{i\theta}{2}\, \hat{\boldsymbol n}\cdot\boldsymbol{\sigma} \right),$$

where $\hat{\boldsymbol n}$ is a real unit vector and positive $\theta$ follows the right-hand rule.

Matrices

Symbol	Matrix in the stated basis	Trace	Determinant	Eigenvalues
$I$	$\begin{pmatrix}1&0\\0&1\end{pmatrix}$	$2$	$1$	$1,1$
$\sigma_x$	$\begin{pmatrix}0&1\\1&0\end{pmatrix}$	$0$	$-1$	$+1,-1$
$\sigma_y$	$\begin{pmatrix}0&-i\\i&0\end{pmatrix}$	$0$	$-1$	$+1,-1$
$\sigma_z$	$\begin{pmatrix}1&0\\0&-1\end{pmatrix}$	$0$	$-1$	$+1,-1$

The names $X,Y,Z$ are common in quantum information:

$$X\equiv\sigma_x, \qquad Y\equiv\sigma_y, \qquad Z\equiv\sigma_z.$$

Elementary properties

Property	$I$	$\sigma_x$	$\sigma_y$	$\sigma_z$
Hermitian conjugate	$I$	$\sigma_x$	$\sigma_y$	$\sigma_z$
Square	$I$	$I$	$I$	$I$
Inverse	$I$	$\sigma_x$	$\sigma_y$	$\sigma_z$
Transpose	$I$	$\sigma_x$	$-\sigma_y$	$\sigma_z$
Complex conjugate	$I$	$\sigma_x$	$-\sigma_y$	$\sigma_z$

Thus each $\sigma_i$ is Hermitian and unitary:

$$\sigma_i^\dagger=\sigma_i, \qquad \sigma_i^{-1}=\sigma_i, \qquad \sigma_i^2=I.$$

Multiplication Table

The row factor multiplies the column factor in the displayed order.

Row $\times$ column	$\sigma_x$	$\sigma_y$	$\sigma_z$
$\sigma_x$	$I$	$i\sigma_z$	$-i\sigma_y$
$\sigma_y$	$-i\sigma_z$	$I$	$i\sigma_x$
$\sigma_z$	$i\sigma_y$	$-i\sigma_x$	$I$

Equivalently,

$$\sigma_i\sigma_j =\delta_{ij}I+i\epsilon_{ijk}\sigma_k.$$

The order matters whenever $i\ne j$. Reversing two distinct Pauli matrices changes the sign:

$$\sigma_i\sigma_j=-\sigma_j\sigma_i \qquad (i\ne j).$$

Useful consequences are

Identity	Formula
Commutator	$[\sigma_i,\sigma_j]=2i\epsilon_{ijk}\sigma_k$
Anticommutator	$\{\sigma_i,\sigma_j\}=2\delta_{ij}I$
Single trace	$\operatorname{tr}\sigma_i=0$
Pair trace	$\operatorname{tr}(\sigma_i\sigma_j)=2\delta_{ij}$
Triple trace	$\operatorname{tr}(\sigma_i\sigma_j\sigma_k)=2i\epsilon_{ijk}$
Four-factor trace	$\operatorname{tr}(\sigma_i\sigma_j\sigma_k\sigma_l)=2(\delta_{ij}\delta_{kl}-\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk})$

Eigenvectors and Projectors

The following normalized eigenvectors use the basis and phases fixed above.

Matrix	Eigenvalue $+1$	Eigenvalue $-1$
$\sigma_x$	$\frac{1}{\sqrt2}\begin{pmatrix}1\\1\end{pmatrix}$	$\frac{1}{\sqrt2}\begin{pmatrix}1\\-1\end{pmatrix}$
$\sigma_y$	$\frac{1}{\sqrt2}\begin{pmatrix}1\\i\end{pmatrix}$	$\frac{1}{\sqrt2}\begin{pmatrix}1\\-i\end{pmatrix}$
$\sigma_z$	$\begin{pmatrix}1\\0\end{pmatrix}$	$\begin{pmatrix}0\\1\end{pmatrix}$

Multiplying either eigenvector by an overall phase gives the same ray. The relative phase between its components is fixed by the eigenvalue equation.

For any real unit vector $\hat{\boldsymbol n}$, $\hat{\boldsymbol n}\cdot\boldsymbol{\sigma}$ has eigenvalues $\pm1$ and spectral projectors

$$P_\pm(\hat{\boldsymbol n}) =\frac{1}{2} \left( I\pm\hat{\boldsymbol n}\cdot\boldsymbol{\sigma} \right).$$

They obey

$$\begin{gathered} P_\pm^2=P_\pm, \qquad P_+P_-=0, \qquad P_++P_-=I, \\ (\hat{\boldsymbol n}\cdot\boldsymbol{\sigma})P_\pm =\pm P_\pm. \end{gathered}$$

The probability of outcome $\pm1$ in state $\rho$ is $p_\pm=\operatorname{tr}(\rho P_\pm)$.

Vector Identities

For real or complex three-component vectors $\boldsymbol a$ and $\boldsymbol b$,

$$(\boldsymbol a\cdot\boldsymbol{\sigma}) (\boldsymbol b\cdot\boldsymbol{\sigma}) = (\boldsymbol a\cdot\boldsymbol b)I +i(\boldsymbol a\times\boldsymbol b) \cdot\boldsymbol{\sigma}.$$

Consequently,

$$\begin{aligned} \lbrace \boldsymbol a\cdot\boldsymbol{\sigma}, \boldsymbol b\cdot\boldsymbol{\sigma} \rbrace &=2(\boldsymbol a\cdot\boldsymbol b)I, \\ [ \boldsymbol a\cdot\boldsymbol{\sigma}, \boldsymbol b\cdot\boldsymbol{\sigma} ] &=2i(\boldsymbol a\times\boldsymbol b) \cdot\boldsymbol{\sigma}. \end{aligned}$$

For a real unit vector $\hat{\boldsymbol n}$,

$$(\hat{\boldsymbol n}\cdot\boldsymbol{\sigma})^2=I.$$

This reduces analytic functions to their even and odd parts. Two especially useful cases are

$$\begin{aligned} \exp\left( \alpha\hat{\boldsymbol n}\cdot\boldsymbol{\sigma} \right) &= \cosh\alpha\,I +\sinh\alpha\, \hat{\boldsymbol n}\cdot\boldsymbol{\sigma}, \\ \exp\left( -\frac{i\theta}{2} \hat{\boldsymbol n}\cdot\boldsymbol{\sigma} \right) &= \cos\frac{\theta}{2}\,I -i\sin\frac{\theta}{2}\, \hat{\boldsymbol n}\cdot\boldsymbol{\sigma}. \end{aligned}$$

For the active-rotation convention stated above,

$$R_{\hat{\boldsymbol n}}(\theta) (\boldsymbol a\cdot\boldsymbol{\sigma}) R_{\hat{\boldsymbol n}}^\dagger(\theta) = \left[ \mathcal R_{\hat{\boldsymbol n}}(\theta) \boldsymbol a \right]\cdot\boldsymbol{\sigma},$$

where $\mathcal R_{\hat{\boldsymbol n}}(\theta)$ is the ordinary three-dimensional right-handed rotation.

Pauli-Basis Decomposition

Every complex $2\times2$ matrix $A$ has the unique expansion

$$A=a_0I+\boldsymbol a\cdot\boldsymbol{\sigma},$$

with

$$a_0=\frac{1}{2}\operatorname{tr}A, \qquad a_i=\frac{1}{2}\operatorname{tr}(\sigma_iA).$$

The trace orthogonality of $\lbrace I,\sigma_x,\sigma_y,\sigma_z\rbrace$ makes these coefficients immediate. Useful invariants are

$$\operatorname{tr}A=2a_0, \qquad \det A=a_0^2-\boldsymbol a\cdot\boldsymbol a.$$

Here $\boldsymbol a\cdot\boldsymbol a=\sum_i a_i^2$, without complex conjugation. The matrix $A$ is Hermitian exactly when $a_0$ and all $a_i$ are real. For a Hermitian matrix its eigenvalues are

$$\lambda_\pm=a_0\pm\lVert\boldsymbol a\rVert.$$

Qubit States and Observables

A qubit density operator has the Bloch form

$$\rho =\frac{1}{2} \left( I+\boldsymbol r\cdot\boldsymbol{\sigma} \right), \qquad \boldsymbol r\in\mathbb R^3, \qquad \lVert\boldsymbol r\rVert\le1.$$

The Bloch components are observable expectation values:

$$r_i=\operatorname{tr}(\rho\sigma_i) =\langle\sigma_i\rangle.$$

Quantity	Expression
Eigenvalues of $\rho$	$(1\pm\lVert\boldsymbol r\rVert)/2$
Determinant	$(1-\lVert\boldsymbol r\rVert^2)/4$
Purity	$\operatorname{tr}(\rho^2)=(1+\lVert\boldsymbol r\rVert^2)/2$
Pure-state condition	$\lVert\boldsymbol r\rVert=1$
Maximally mixed state	$\boldsymbol r=\boldsymbol 0$, so $\rho=I/2$
Directional probability	$p_\pm=(1\pm\boldsymbol r\cdot\hat{\boldsymbol n})/2$

For an observable $A=a_0I+\boldsymbol a\cdot\boldsymbol{\sigma}$ with real coefficients,

$$\langle A\rangle_\rho =a_0+\boldsymbol a\cdot\boldsymbol r.$$

The Bloch Sphere page develops the geometry and state interpretation.

Two-Level Hamiltonians

Every Hermitian two-level Hamiltonian can be written

$$H=c_0I+\boldsymbol c\cdot\boldsymbol{\sigma}, \qquad c_0\in\mathbb R, \qquad \boldsymbol c\in\mathbb R^3.$$

For $\boldsymbol c\ne\boldsymbol 0$:

Quantity	Expression
Energies	$E_\pm=c_0\pm\lVert\boldsymbol c\rVert$
Gap	$\Delta E=2\lVert\boldsymbol c\rVert$
Energy projectors	$P_\pm=(I\pm\hat{\boldsymbol c}\cdot\boldsymbol{\sigma})/2$
Propagator	$e^{-ic_0t/\hbar}\left[\cos(\lVert\boldsymbol c\rVert t/\hbar)I-i\sin(\lVert\boldsymbol c\rVert t/\hbar)\hat{\boldsymbol c}\cdot\boldsymbol{\sigma}\right]$

The scalar term $c_0I$ changes only the global dynamical phase in a closed two-level system. The vector $\boldsymbol c$ fixes both the energy axis and the Bloch-vector precession axis. See the Two-Level System Hamiltonian for assumptions, spectrum, dynamics, and realizations.

Spin Operators and Ladder Matrices

Object	Definition	Matrix or action
Physical spin	$S_i=\hbar\sigma_i/2$	Eigenvalues of $S_z$ are $\pm\hbar/2$
Dimensionless raising matrix	$\sigma_+=(\sigma_x+i\sigma_y)/2$	$\begin{pmatrix}0&1\\0&0\end{pmatrix}=\lvert0\rangle\langle1\rvert$
Dimensionless lowering matrix	$\sigma_-=(\sigma_x-i\sigma_y)/2$	$\begin{pmatrix}0&0\\1&0\end{pmatrix}=\lvert1\rangle\langle0\rvert$
Spin raising operator	$S_+=S_x+iS_y$	$S_+=\hbar\sigma_+$
Spin lowering operator	$S_-=S_x-iS_y$	$S_-=\hbar\sigma_-$

The most-used ladder identities are

$$\begin{gathered} \sigma_+\sigma_-=\frac{I+\sigma_z}{2}, \qquad \sigma_-\sigma_+=\frac{I-\sigma_z}{2}, \\ [\sigma_z,\sigma_\pm]=\pm2\sigma_\pm, \qquad [\sigma_+,\sigma_-]=\sigma_z. \end{gathered}$$

Pauli Strings

For $n$ qubits, tensor products of $I,X,Y,Z$ form an orthogonal operator basis. If

$$P=P_1\otimes\cdots\otimes P_n, \qquad P_j\in\lbrace I,X,Y,Z\rbrace,$$

then the $4^n$ Pauli strings satisfy

$$\operatorname{tr}(P^\dagger Q) =2^n\delta_{P,Q}.$$

Two Pauli strings either commute or anticommute. They anticommute exactly when the number of tensor positions at which their nonidentity factors differ is odd. This criterion underlies stabilizer calculations, but stabilizer codes and circuit conventions have their own canonical treatments.

How to Use This Table

1. Write the ordered basis next to every imported matrix.
2. Decide whether symbols denote dimensionless Pauli matrices or physical spin operators.
3. Reduce products with the multiplication table or vector identity.
4. Use spectral projectors for functions of one Pauli direction.
5. For a general two-state matrix, extract Pauli coefficients with trace orthogonality.
6. Check the result through Hermiticity, trace, determinant, or a direct action on basis vectors.

Common Mistakes

• Reversing the sign of the upper-right entry of $\sigma_y$.
• Forgetting that $\sigma_x\sigma_y=-\sigma_y\sigma_x$.
• Dropping the factor $\hbar/2$ when converting $\sigma_i$ to $S_i$.
• Using $\sigma_\pm$ and $S_\pm$ interchangeably even though their dimensions differ.
• Treating $\boldsymbol a\cdot\boldsymbol a$ as $\boldsymbol a^\ast\cdot\boldsymbol a$ in the determinant of a general complex matrix.
• Assuming every vector $\boldsymbol r$ defines a density matrix; positivity requires $\lVert\boldsymbol r\rVert\le1$.
• Confusing $R\sigma R^\dagger$ with $R^\dagger\sigma R$; the associated three-dimensional rotations are inverse to one another.
• Treating a $2\pi$ spinor sign change as a change in the associated Bloch vector.
• Copying a qubit matrix without checking the computational-basis order.

Verification

The entries are exact and editorially maintained. They can be checked directly from the four displayed matrices. A compact independent audit is:

1. square each $\sigma_i$ and obtain $I$;
2. verify $\sigma_x\sigma_y=i\sigma_z$ and cyclic permutations;
3. check $\operatorname{tr}(\sigma_i\sigma_j)=2\delta_{ij}$;
4. diagonalize each matrix and compare with the stated eigenspaces;
5. insert the vector identity into the exponential power series.

Last reviewed: 2026-08-19.

Exercises

Exercise 1: Squaring a Pauli combination

Use the vector product identity to compute $(\sigma_x+\sigma_z)^2$ without multiplying matrices entry by entry.

Solution

Set $\boldsymbol a=(1,0,1)$. Then

$$(\sigma_x+\sigma_z)^2 =(\boldsymbol a\cdot\boldsymbol{\sigma})^2 =(\boldsymbol a\cdot\boldsymbol a)I +i(\boldsymbol a\times\boldsymbol a) \cdot\boldsymbol{\sigma} =2I.$$

The cross product vanishes because any vector crossed with itself is zero.

Exercise 2: Exponentiating a two-level Hamiltonian

Let $H=(\hbar\Omega/2)\sigma_x$. Find $U(t)=e^{-iHt/\hbar}$ and the probability of measuring $\lvert1\rangle$ after evolving from $\lvert0\rangle$ for time $t$.

Solution

Using $\sigma_x^2=I$,

$$U(t) =\cos\frac{\Omega t}{2}\,I -i\sin\frac{\Omega t}{2}\,\sigma_x.$$

Since $\sigma_x\lvert0\rangle=\lvert1\rangle$,

$$U(t)\lvert0\rangle = \cos\frac{\Omega t}{2}\lvert0\rangle -i\sin\frac{\Omega t}{2}\lvert1\rangle.$$

Therefore

$$\Pr(1;t)=\sin^2\frac{\Omega t}{2}.$$

Exercise 3: Testing a proposed Bloch vector

An experiment reports $\langle\sigma_x\rangle=0.6$, $\langle\sigma_y\rangle=0.3$, and $\langle\sigma_z\rangle=0.8$. Does this triple define a physical qubit density matrix?

Solution

The proposed Bloch-vector length satisfies

$$\lVert\boldsymbol r\rVert^2 =0.6^2+0.3^2+0.8^2 =1.09.$$

Because $\lVert\boldsymbol r\rVert>1$, the smaller eigenvalue $(1-\lVert\boldsymbol r\rVert)/2$ is negative. The reconstructed matrix is Hermitian and has unit trace, but it is not positive semidefinite and therefore is not a physical density operator. In practice, such a result can arise from finite-sample noise or unconstrained state reconstruction.

Canonical Links

• Pauli Matrices in Spin-1/2 Systems
• Pauli Matrices in Linear Algebra
• Pauli Identity Index
• Bloch Sphere
• Angular Momentum Algebra
• Two-Level System Hamiltonian
• Quantum Gates
• Symbol: Sigma

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th anniversary ed., Cambridge University Press, 2010.
• L. E. Ballentine, Quantum Mechanics: A Modern Development, World Scientific, 1998.