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

How statement and proof provenance work

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.

Dirac Adjoint Amplitude Translation

Statement

Let $H, K$ be finite-dimensional complex inner-product spaces, let $T : H \to K$ be linear, and let $x \in K$ , $y \in H$ . With the Dirac conversion Dirac Notation Conversion,

$⟨ x ∣ T ∣ y ⟩_{D} : = ⟨ x ∣ T y ⟩_{D} = ⟨ T y, x ⟩_{K, r o o t} = ⟨ y, T^{*} x ⟩_{H, r o o t} = ⟨ T^{*} x ∣ y ⟩_{D} .$

Facts & Assumptions

Given: Finite-dimensional complex inner-product spaces $H, K$ , a linear map $T : H \to K$ , and vectors $x \in K$ , $y \in H$ .

[F1]

The Dirac pairing reverses the root pairing: $⟨ a ∣ b ⟩_{D} = ⟨ b, a ⟩_{r o o t}$ (Dirac Notation Conversion).

[F2]

The adjoint satisfies $⟨ T v, w ⟩_{K} = ⟨ v, T^{*} w ⟩_{H}$ (The adjoint $T^{*} : W \to V$ is characterised by $⟨ T v, w ⟩_{W} = ⟨ v, T^{*} w ⟩_{V}$ ).

[F3]

Every linear map between finite-dimensional inner-product spaces has a unique adjoint (Every linear map between finite-dimensional inner product spaces has a unique adjoint).

Proof

1.1givenF1F3

The adjoint $T^{*}$ exists by [F3]. The definition [F1] gives $⟨ x ∣ T ∣ y ⟩_{D} = ⟨ T y, x ⟩_{K, r o o t}$ .

2.1F1F2step 1.1∎

Apply [F2] with $v = y$ and $w = x$ , then apply [F1] again. This gives $⟨ T y, x ⟩_{K, r o o t} = ⟨ y, T^{*} x ⟩_{H, r o o t} = ⟨ T^{*} x ∣ y ⟩_{D}$ , proving the stated chain.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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 (2018 author-hosted draft), Chapters 1–2 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (2026-08-27), Appendices A.5 and A.10 (standard reference, not scraped)