Commutator

The commutator of two operators $A$ and $B$ is defined as $[A, B] = AB - BA$. When the commutator is zero, the operators commute, meaning the order of application does not matter. In quantum mechanics, commuting Hermitian operators share a common eigenbasis and represent observables that can be measured simultaneously with arbitrary precision. Non-commuting operators are subject to uncertainty relations that limit simultaneous knowledge of their values.

Mathematical definition

For two linear operators $A$ and $B$ acting on the same Hilbert space:

$[A, B] = AB - BA$

The anticommutator is also frequently used:

$\{A, B\} = AB + BA$

Key algebraic properties of the commutator:

• Antisymmetry: $[A, B] = -[B, A]$
• Linearity: $[\alpha A + \beta B, C] = \alpha[A, C] + \beta[B, C]$
• Product rule (Leibniz): $[A, BC] = [A, B]C + B[A, C]$
• Jacobi identity: $[A, [B, C]] + [B, [C, A]] + [C, [A, B]] = 0$

Pauli commutation relations

The Pauli operators satisfy the fundamental commutation relations:

$[X, Y] = 2iZ, \quad [Y, Z] = 2iX, \quad [Z, X] = 2iY$

These are cyclic: $[P_a, P_b] = 2i\epsilon_{abc}P_c$ where $\epsilon_{abc}$ is the Levi-Civita symbol. The anticommutation relations are:

$\{X, Y\} = \{Y, Z\} = \{Z, X\} = 0$

Meaning $XY = -YX$, and similarly for the other pairs. The Pauli operators anticommute but do not commute (except with themselves and the identity).

Role in quantum mechanics

Uncertainty principle: For two observables $A$ and $B$, the Robertson uncertainty relation states:

$\Delta A \cdot \Delta B \geq \frac{1}{2}|\langle [A, B] \rangle|$

where $\Delta A = \sqrt{\langle A^2 \rangle - \langle A \rangle^2}$ is the standard deviation. Non-zero commutator implies a fundamental limit on simultaneous precision, not merely a measurement disturbance effect.

Simultaneous measurability: Two observables can be simultaneously measured (are “compatible”) if and only if they commute. For qubits, $Z$ and $X$ do not commute, so they cannot be simultaneously known with arbitrary precision. This is why measuring in the computational basis ($Z$ basis) disturbs information about the $X$ basis.

Time evolution: The Heisenberg equation of motion relates the time derivative of an observable to its commutator with the Hamiltonian:

$\frac{dA}{dt} = \frac{i}{\hbar}[H, A]$

An observable that commutes with the Hamiltonian is a constant of motion (conserved quantity).

Applications in quantum computing

Commutation relations matter practically in several ways:

• Gate ordering: If two gates $U_A$ and $U_B$ commute, they can be applied in either order, which the transpiler can exploit for circuit optimization
• Pauli grouping: When measuring a Hamiltonian $H = \sum_i c_i P_i$, Pauli terms that mutually commute can be measured simultaneously, reducing the total number of circuits needed
• Trotterization: The Trotter-Suzuki decomposition error depends on commutators between Hamiltonian terms: $e^{(A+B)t} \approx (e^{At/n}e^{Bt/n})^n$ with error proportional to $[A, B]t^2/n$

Why it matters for learners

Commutators are the algebraic language of quantum mechanics. They determine what can be simultaneously known, how observables evolve, how errors in Hamiltonian simulation scale, and which quantum gates can be reordered. Comfort with commutator algebra is essential for working with quantum algorithms at a level deeper than circuit diagrams.

See also

• Hermitian Operator
• Pauli Gates
• Measurement
• Trotterization
• Unitary Operator