LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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An anticommuting stabilizer measurement is fair

Statement

Let $ρ = ∣ ψ ⟩ ⟨ ψ ∣$ be a pure stabilizer state density operator, and let $M$ be a Hermitian Pauli observable that anticommutes with one of its stabilizer generators. Then $Tr (P_{0} ρ) = Tr (P_{1} ρ) = 1 / 2$ .

Facts & Assumptions

Given: A rank- $n$ stabilizer subgroup $S$ fixing the ray of $ψ$ , a normalized representative $ψ$ , and a Hermitian Pauli $M$ with $g M = - M g$ for some $g \in S$ .

[F1]

Stabilizer state: every $g \in S$ fixes $ψ$ .

[F2]

Hermitian Pauli phase criterion: a Hermitian Pauli is self-adjoint and squares to $I$ .

[F3]

Pauli measurement branch semantics: $P_{b} = (I + (- 1)^{b} M) / 2$ , and its probability on a pure state is $p_{b} = Tr (∣ ψ ⟩ ⟨ ψ ∣ P_{b})$ .

[F4]

Pure-state projective branch probability is a squared norm: for a unit vector $ψ$ and orthogonal projection $P$ , $Tr (∣ ψ ⟩ ⟨ ψ ∣ P) = ∥ P ψ ∥^{2}$ .

[F5]

Ray-to-projector is well-defined and bijective: the projector of a unit representative of a pure-state ray is a positive trace-one density operator.

[F6]

Pauli measurement projectors and total probability: $P_{0}, P_{1}$ are orthogonal complementary projections.

Proof

Given: The data in the Statement.

1.1givenF1F2F3algebra

Since $g = g^{†}$ and $g^{2} = I$ by [F2], it is unitary. From $g ψ = ψ$ and $g M g = - M$ , we get $g P_{b} g = P_{1 - b}$ and $g P_{b} ψ = P_{1 - b} ψ$ . Hence $∥ P_{b} ψ ∥ = ∥ P_{1 - b} ψ ∥$ .

2.1F3F4F5F6step 1.1algebra

By [F5], the unit ray projector is a density operator, so its branch law [F3] and the norm identity [F4] give $p_{b} = ∥ P_{b} ψ ∥^{2} = ∥ P_{1 - b} ψ ∥^{2} = p_{1 - b}$ . The complementary orthogonal projections in [F6] satisfy $∥ P_{0} ψ ∥^{2} + ∥ P_{1} ψ ∥^{2} = ∥ ψ ∥^{2} = 1$ , so equal probabilities must both be $1 / 2$ .

3.1step 2.1∎

This proves that each outcome bit has probability $1 / 2$ .

Sources

De Wolf §20.3, PDF p. 188, proves equality of the two outcome probabilities by conjugating with an anticommuting stabilizer. Aaronson–Gottesman §III, PDF pp. 4–5, gives the corresponding tableau measurement case.

Depends on

Used by

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Sources

Ronald de Wolf, Quantum Computing: Lecture Notes, §20.3 (standard reference, not scraped)
Scott Aaronson and Daniel Gottesman, Improved Simulation of Stabilizer Circuits, §III (standard reference, not scraped)