TheoremStatement: Literature-sourcedProof: AI-generatedPipeline-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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Every nonzero finite-dimensional complex inner-product space admits a density operator

Statement

Every nonzero finite-dimensional complex inner-product space $H$ admits a density operator; specifically, if $d = \dim_{C} H$ , then $I_{H} / d$ is a density operator on $H$ .

Facts & Assumptions

Given: A nonzero finite-dimensional complex inner-product space $H$ , $d = \dim_{C} H$ , and the identity endomorphism $I_{H} (x) = x$ .

[F1]

Density operator: on nonzero finite-dimensional $H$ , a density operator is a non-negative endomorphism with trace one.

[F2]

Non-negative and positive operators: non-negative means self-adjoint and having nonnegative quadratic form on every vector.

[F3]

Real and complex inner-product spaces and their induced length: the inner product is linear in its first argument and positive definite.

[F4]

The basis-independent trace of an endomorphism of a finite-dimensional vector space: trace of an endomorphism is the matrix trace in any ordered basis.

[F5]

The trace $tr (A)$ as the sum of the diagonal entries: matrix trace is the sum of diagonal entries.

[F6]

Trace is a linear functional on $M_{n} (F)$ : matrix trace is linear under scalar multiplication and addition.

[F7]

Finite-dimensional vector space, and its dimension $\dim_{F} V$ ; infinite-dimensional means having no finite basis: $d$ is the finite size of a basis of $H$ ; since $H \neq {0}$ , $d \geq 1$ .

Proof

1.1givenF2F3F7algebra

Put $ρ = I_{H} / d$ . The scalar $1 / d$ is positive real. The identity is self-adjoint, and for every $x \in H$ , $⟨ ρ x, x ⟩ = (1 / d) ⟨ x, x ⟩ \geq 0$ by [F3] and [F7]. Thus $ρ$ is non-negative by [F2].

2.1F1F4F5F6F7step 1.1algebra∎

In a basis of $H$ with $d$ elements, the matrix of $ρ$ is $(1 / d) I_{d}$ . By [F4]–[F6], its trace is $(1 / d) \sum_{j = 1}^{d} 1 = 1$ . Therefore $ρ$ is a density operator by [F1].

Depends on

Used by

Cited to discharge well-definedness by Density operator.

Dependency tree · two levels

31 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)