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Pauli matrix algebra and trace identities

Statement

For the matrices $X, Y, Z$ of The Pauli matrices, each is self-adjoint and unitary, and $X^{2} = Y^{2} = Z^{2} = I$ . Distinct Pauli matrices anticommute. For $j, k \in {x, y, z}$ , $σ_{j} σ_{k} = δ_{j k} I + i \sum_{ℓ \in {x, y, z}} ϵ_{j k ℓ} σ_{ℓ}, tr (σ_{j} σ_{k}) = 2 δ_{j k} .$ Here $σ_{x} = X, σ_{y} = Y, σ_{z} = Z$ , $δ_{j k} = 1$ when $j = k$ and $0$ otherwise, and $ϵ$ is the alternating symbol with $ϵ_{x y z} = 1$ . Also $tr (I) = 2$ and $tr (σ_{j}) = 0$ .

Facts & Assumptions

Given: The ordered computational basis of $Q_{2}$ and the four matrices $I, X, Y, Z$ displayed in The Pauli matrices.

[F1]

The Pauli matrices: $X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ , $Y = (\begin{array}{cc} 0 & - i \\ i & 0 \end{array})$ , and $Z = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array})$ .

[F2]

In orthonormal bases, the matrix of the adjoint is the conjugate transpose of the matrix: in an orthonormal basis, the adjoint matrix is the conjugate transpose.

[F3]

The trace $tr (A)$ as the sum of the diagonal entries: the trace of a square matrix is the sum of its diagonal entries.

[F4]

For an endomorphism in finite dimension, preserving lengths, preserving inner products, carrying orthonormal bases to orthonormal bases, and $T^{*} T = I$ are equivalent: on a finite-dimensional inner-product space, $T^{*} T = I$ implies that $T$ is unitary and $T T^{*} = I$ .

[F5]

Linear isometries, and orthogonal or unitary operators on finite-dimensional inner product spaces: a unitary operator on a complex inner-product space is an invertible linear isometry from the space to itself.

Proof

technique · direct

1.1F1algebra

Multiplying the entries of the matrices in [F1] gives the nine row-by-column products $X^{2} = (\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}) = I$ , $X Y = (\begin{array}{cc} i & 0 \\ 0 & - i \end{array}) = i Z$ , $X Z = (\begin{array}{cc} 0 & - 1 \\ 1 & 0 \end{array}) = - i Y$ , $Y X = (\begin{array}{cc} - i & 0 \\ 0 & i \end{array}) = - i Z$ , $Y^{2} = (\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}) = I$ , $Y Z = (\begin{array}{cc} 0 & i \\ i & 0 \end{array}) = i X$ , $Z X = (\begin{array}{cc} 0 & 1 \\ - 1 & 0 \end{array}) = i Y$ , $Z Y = (\begin{array}{cc} 0 & - i \\ - i & 0 \end{array}) = - i X$ , and $Z^{2} = (\begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array}) = I$ .

2.1F1F2F4F5step 1.1algebra

By [F1] and [F2], $X^{*} = X$ , $Y^{*} = Y$ , and $Z^{*} = Z$ ; the diagonal entries of each product $X^{*} X, Y^{*} Y, Z^{*} Z$ are $1$ and its off-diagonal entries are $0$ , so each product is $I$ . By [F4] each is a unitary operator, in the sense of [F5].

3.1F1F3step 1.1∎

The six off-diagonal products in step 1.1 have opposite signs when their order is reversed, so the matrices anticommute pairwise and the table is exactly the $δ, ϵ$ product formula in the Statement. The same matrices have $tr (X) = tr (Y) = tr (Z) = 0$ and $tr (I) = 2$ by [F3]. Taking traces in step 1.1 gives $tr (σ_{j} σ_{k}) = 2$ when $j = k$ and $0$ otherwise.

Remarks

The sign $X Y = i Z$ is fixed by the displayed $Y$ matrix; changing the sign of $Y$ would change every orientation formula that follows.

Depends on

Used by

Phase aware n qubit pauli group Definition
Bloch coordinates of |0> and H|0> Example
Pauli commutator and binary symplectic sign Example
CNOT Pauli action Lemma
Computational basis states have an extended stabilizer tableau Lemma
Hadamard pauli action Lemma
Minimum-Weight Recovery Failure Yields a Half-Error Crossing Path Lemma
Phase gate pauli action Lemma
Binary Pauli quotient isomorphism Theorem
Existence and uniqueness of Bloch coordinates Theorem
Hadamard conjugation of the Pauli matrices Theorem
Pauli formula for an axis rotation Theorem
Pauli product and adjoint Theorem
Purity is the boundary of the Bloch ball Theorem
The Bloch ball is exactly the qubit density-state space Theorem

Dependency tree · two levels

25 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (updated July 2015) (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (2026-08-27) (standard reference, not scraped)
Aram Harrow, 6.443/8.371/18.436 Quantum Information Science II, Lecture 1: Quantum states and operations (2018) (standard reference, not scraped)