Pauli operator

n

+ 1

and

- 1

Pauli decomposition of a qubit-environment isometry ( $V ∣ ψ ⟩ = \sum_{k = 0}^{3} σ_{k} ∣ ψ ⟩ \otimes ∣ e_{k} ⟩$ )

Every linear map from a qubit into the tensor product of that qubit and an environment can be expanded in the Pauli matrix orthonormal basis of operators. If

V ∣0 ⟩ = ∣0 ⟩ ∣ e_{00} ⟩ + ∣1 ⟩ ∣ e_{01} ⟩

and

V ∣1 ⟩ = ∣0 ⟩ ∣ e_{10} ⟩ + ∣1 ⟩ ∣ e_{11} ⟩

, then the four coefficient vectors are

∣ e_{0} ⟩ = \frac{∣ e _{00} ⟩ + ∣ e _{11} ⟩}{2}, ∣ e_{1} ⟩ = \frac{∣ e _{01} ⟩ + ∣ e _{10} ⟩}{2}, ∣ e_{2} ⟩ = \frac{i ( ∣ e _{10} ⟩ - ∣ e _{01} ⟩)}{2}, ∣ e_{3} ⟩ = \frac{∣ e _{00} ⟩ - ∣ e _{11} ⟩}{2} .

These environment vectors need not be orthogonal, so this is an operator expansion rather than a claim that the quantum channel is a classical Pauli channel. Physical isometries additionally preserve the inner products of the two input basis states.

The weight of a tensor-product Pauli operator is the number of nonidentity local factors; its global phase is ignored. Products satisfy

wt (P^{†} Q) \leq wt (P) + wt (Q)

because their support is contained in the union of the two supports. The quantum code distance uses this weight to measure the smallest undetectable nontrivial action on logical information.

n

qubits, so every linear error operator has a Pauli expansion. Using

X^{a} Z^{b}

, with scalar phases absorbed into the coefficients, its computational basis action is

E ∣ x ⟩ = \sum_{a, b} c_{a, b} (- 1)^{b \cdot x} ∣ x + a ⟩

, with binary arithmetic. This describes coherent errors as well as individual Pauli errors; it does not assume that the noise is a probabilistic Pauli channel. A physical noise map has Kraus operators of this form when the usual quantum channel description applies, for instance when the initial system and environment are uncorrelated.