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

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The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

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.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Positive-probability Pauli branches are density operators

Statement

Let $H$ be a nonzero finite-dimensional complex inner-product space, let $ρ$ be a density operator on $H$ , and let $M$ be a Hermitian Pauli observable on $H$ with $M^{2} = I_{H}$ . For $b \in {0, 1}$ put $P_{b} = (I_{H} + (- 1)^{b} M) / 2$ and $p_{b} = Tr (ρ P_{b})$ . If $p_{b} > 0$ , then the normalized branch

$ρ_{b} : = \frac{P_{b} ρ P_{b}}{p_{b}}$

is a density operator.

Facts & Assumptions

Given: The data in the Statement.

[F1]

The setup defines $P_{b} = (I_{H} + (- 1)^{b} M) / 2$ , $p_{b} = Tr (ρ P_{b})$ , and $ρ_{b} = P_{b} ρ P_{b} / p_{b}$ when $p_{b} > 0$ .

[F2]

Pauli measurement projectors and total probability: $Q_{b} = P_{b} ρ P_{b}$ is nonnegative and $Tr (Q_{b}) = p_{b}$ .

[F3]

Trace is a linear functional on $M_{n} (F)$ : trace respects multiplication by a scalar.

[F4]

Density operator: a density operator is a nonnegative endomorphism of trace one on a nonzero finite-dimensional complex-inner-product space.

Proof

Given: The data in the Statement.

1.1F2givenalgebra

Set $Q_{b} = P_{b} ρ P_{b}$ . By [F2], $Q_{b}$ is nonnegative and has trace $p_{b}$ . Since $p_{b} > 0$ , the scalar $1 / p_{b}$ is positive. Hence, for every $v$ , $⟨ (Q_{b} / p_{b}) v, v ⟩ = (1 / p_{b}) ⟨ Q_{b} v, v ⟩ \geq 0$ .

2.1F2F3F4step 1.1algebra

By [F3] and [F2], $Tr (Q_{b} / p_{b}) = p_{b}^{- 1} Tr (Q_{b}) = 1$ . Thus $Q_{b} / p_{b}$ is a density operator by [F4].

3.1F1step 2.1∎

The conditional state in [F1] is exactly $Q_{b} / p_{b}$ , so it is the density operator established in step 2.1.

Sources

Watrous Lesson 14 §14.1, printed pp. 405–407 (PDF pp. 411–413), defines the projective conditional state; de Wolf §20.2, PDF pp. 186–188, gives the Pauli branch update. The positivity and trace-one check is proved locally.

Depends on

Used by

Phase-aware tableau measurement agrees with the projective instrument Theorem

Cited to discharge well-definedness by Pauli measurement branch semantics.

Dependency tree · two levels

22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

John Watrous, Understanding Quantum Information and Computation, Lesson 14 §14.1 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §20.2 (standard reference, not scraped)