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Two-site Pauli observables commute at disjoint sites

Statement

Take $d = 1$ , spin dimension $q = 2$ , and distinct sites $x, y \in Z$ . On $C^{2}$ define the Pauli matrices

$X = (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array}), Z = (\begin{array}{cc} 1 & 0 \\ 0 & - 1 \end{array}) .$

Let $X_{x}$ and $Z_{y}$ be their one-site observables embedded in the two-site algebra $A_{{x, y}}$ by the factor inclusions of Local spin matrix algebras and their factor inclusions. Then $X_{x}$ and $Z_{y}$ are self-adjoint unitaries of norm one and commute. This is a finite example of the locality clause in Quasi-local algebra completion and locality.

Facts & Assumptions

Given: The two-site qubit algebra and its isometric factor inclusions.

[F1]

A finite region has algebra $B (H_{Λ})$ , and inclusion into a larger region tensors with the identity (Local spin matrix algebras and their factor inclusions).

[F2]

Those factor inclusions preserve products, adjoints and operator norm (Local spin factor inclusions preserve the operator norm).

[F3]

Observables supported on disjoint finite regions commute in their common finite tensor algebra (Quasi-local algebra completion and locality).

Proof

technique · direct

Given: $x \neq y$ , $q = 2$ , and facts [F1]–[F3].

1.1givenF1F2algebra

Direct matrix multiplication gives $X^{*} = X$ , $Z^{*} = Z$ , $X^{2} = Z^{2} = I$ , and $∥ X ∥ = ∥ Z ∥ = 1$ because each preserves the Euclidean norm and is nonzero. By [F2], the embedded $X_{x}$ and $Z_{y}$ have the same adjoint, square and norm properties.

1.2givenF1algebra

If $x < y$ , lexicographic factor order gives $X_{x} = X \otimes I$ and $Z_{y} = I \otimes Z$ , so $(X \otimes I) (I \otimes Z) = X \otimes Z = (I \otimes Z) (X \otimes I)$ . If $y < x$ , the factors are interchanged and both products are $Z \otimes X$ . Thus in either ordering $X_{x} Z_{y} = Z_{y} X_{x}$ .

2.1F3step 1.2∎

The supports ${x}$ and ${y}$ are disjoint; the computed commutation is therefore the stated instance of the general disjoint-locality clause.

Two-site Pauli observables commute at disjoint sites

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources