Pauli operator

An $n$-qubit Pauli operator is a tensor product of single-qubit Pauli matrices, possibly multiplied by a phase. Hermitian Pauli operators have eigenvalues $+1$ and $-1$.

One of the six pure single-qubit eigenstates of $X,Y,Z$. Their Bloch vectors are the six coordinate unit vectors with both signs. Their convex hull is the stabilizer octahedron.

The two eigenstate rays of the Pauli Y gate have the displayed normalized representatives and respective eigenvalues $+1$ and $-1$. They form an orthonormal basis. Multiplication by $Y$ changes only the global phase of each of these states; it does not exchange their eigenvalue labels.

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 areThese 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 $\mathrm{wt}(P^{†}Q)\le \mathrm{wt}(P)+\mathrm{wt}(Q)$ 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.

The tensor products of single-qubit Pauli matrices form an orthogonal basis of the operator space on $n$ 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.

A Pauli gate applies a Pauli matrix as a unitary operator to a qubit. The Pauli X gate, Pauli Y gate and Pauli Z gate are common correction operations in quantum teleportation.

A Pauli correlator is the expectation value of a tensor product of Pauli matrices, such as $⟨X\otimes Z⟩$. It measures a specified correlation between qubit observables.