TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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Rank-one density operators are pure-state projectors

Statement

Every pure-state ray projector on a nonzero finite-dimensional complex-inner-product space is a density operator. Conversely, every rank-one density operator is the projector of a unique pure-state ray.

Facts & Assumptions

Given: A nonzero finite-dimensional complex-inner-product space $H$ .

[F1]

Pure state and ray: a pure state is a ray represented by a unit vector's rank-one projector.

[F2]

Density operator: a density operator is non-negative with trace one.

[F3]

Ray-to-projector is well-defined and bijective: unit-vector rays correspond bijectively to rank-one orthogonal projectors, each non-negative and trace one.

[F4]

Density operators have an orthogonal spectral ensemble: a density operator is a positive-weight sum of orthogonal unit-ray projectors, with one term per positive eigenvalue and number of terms equal to its rank.

[F5]

Rank and nullity of a linear map with finite-dimensional domain: rank is the dimension of the image of a finite-dimensional linear map.

[F6]

The basis-independent trace of an endomorphism of a finite-dimensional vector space and Trace is a linear functional on $M_{n} (F)$ : endomorphism trace is matrix trace in a basis, and trace is linear under scalar multiplication.

Proof

1.1F1F2F3given

For any pure-state ray, [F1] gives a unit representative and [F3] makes its projector non-negative with trace one. By [F2] it is a density operator.

2.1F2F3F4F5F6algebra∎

Let $ρ$ be a density operator whose image has dimension one by [F5]. By [F4] its spectral ensemble has exactly one positive term, so $ρ = λ ∣ u ⟩ ⟨ u ∣$ for a unit $u$ and $λ > 0$ . The trace-one condition in [F2] gives $1 = tr (ρ) = λ tr (∣ u ⟩ ⟨ u ∣) = λ$ , by [F3] and linearity [F6]. Hence $ρ$ is the projector of $[u]$ , and the bijection in [F3] makes that ray unique.

Remarks

The spectral-ensemble route uses a finite-dimensional eigenbasis and a finite positive-eigenvalue list; no arbitrary-index choice is used.

Depends on

Used by

Fusion measurement branches are normalized exactly when nonzero Theorem

Dependency tree · two levels

25 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, The Theory of Quantum Information, §2.1.2 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, §15.1 (standard reference, not scraped)