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.

The measurement law for a finite continuous-time walk

Statement

Let $V$ be a nonzero finite-dimensional complex inner-product space, let $(e_{y})_{y \in Y}$ be a finite orthonormal basis, and let $ϕ \in V$ be a unit vector. Measuring the pure state $[ϕ]$ by the projective measurement onto the basis lines outputs $y \in Y$ with probability $p_{ϕ} (y) = ∣ ⟨ ϕ, e_{y} ⟩ ∣^{2}, \sum_{y \in Y} p_{ϕ} (y) = 1.$ In particular, for self-adjoint $H : V \to V$ , real $t$ , and unit $ψ$ , the walk output $ϕ = U_{H} (t) ψ$ is unit and $p_{t} (y) = ∣ ⟨ U_{H} (t) ψ, e_{y} ⟩ ∣^{2}$ . For an initial vertex $e_{x}$ , the probability of observing vertex $y$ is $∣ ⟨ U_{H} (t) e_{x}, e_{y} ⟩ ∣^{2}$ .

Facts & Assumptions

Given: The nonzero space, orthonormal basis, and unit vector $ϕ$ in the Statement. For its walk specialization, also take self-adjoint $H$ , real $t$ , and a unit input $ψ$ with $ϕ = U_{H} (t) ψ$ .

[F1]

Unitary and differential identities for finite matrix dynamics: $U_{H} (t)$ is unitary, so it preserves the norm and $U_{H} (t) ψ$ is unit.

[F2]

Pure state and ray: a pure state is a ray represented by any unit vector.

[F3]

Ray-to-projector is well-defined and bijective: for a unit representative $u$ , $P_{u} = ∣ u ⟩ ⟨ u ∣$ is a nonnegative rank-one projection with trace one.

[F4]

Finite projective measurements: orthogonal projections summing to the identity form a projective measurement, and its weight on $ρ$ is $tr (ρ P_{y})$ .

[F5]

Projective measurement outcome weights are probabilities: for a density operator and a finite projective measurement, these weights lie in $[0, 1]$ and sum to one.

[F6]

Dirac Notation Conversion: $∣ x ⟩ ⟨ y ∣$ acts by $z \mapsto ⟨ z, y ⟩_{r o o t} x$ .

[F7]

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

[F8]

Bessel's inequality for a finite orthonormal list and Parseval's identity for an orthonormal basis: if $(e_{y})$ is an orthonormal basis, then $v = \sum_{y} ⟨ v, e_{y} ⟩ e_{y}$ and $\sum_{y} ∣ ⟨ v, e_{y} ⟩ ∣^{2} = ∥ v ∥^{2}$ .

[F9]

Real and complex inner-product spaces and their induced length: the root inner product is linear in its first argument and conjugate-symmetric.

Proof

technique · direct

1.1givenF1

The vector $ϕ$ is unit by hypothesis. In the walk specialization, [F1] also shows $U_{H} (t) ψ$ is unit.

1.2givenF4F6F8F9algebra

For each $y$ , define $P_{y} z : = ⟨ z, e_{y} ⟩ e_{y}$ . Orthonormality gives $P_{y}^{2} = P_{y}$ and $P_{y} P_{y^{'}} = 0$ for $y \neq y^{'}$ . By [F6] and [F9], $⟨ P_{y} z, w ⟩ = ⟨ z, e_{y} ⟩ ⟨ e_{y}, w ⟩ = ⟨ z, P_{y} w ⟩$ , so $P_{y}^{*} = P_{y}$ . The basis expansion in [F8] gives $\sum_{y} P_{y} = I_{V}$ . Hence $(P_{y})_{y \in Y}$ is the basis projective measurement in [F4].

1.3givenF2F3F4F5F6F7algebra

The ray $[ϕ]$ has density operator $P_{ϕ}$ by [F2]–[F3]. By [F4]–[F5] its probability at $y$ is $tr (P_{ϕ} P_{y}) \in [0, 1]$ . In the basis $(e_{z})$ , $P_{y} e_{z} = 0$ for $z \neq y$ and $P_{y} e_{y} = e_{y}$ ; [F6] gives $P_{ϕ} e_{y} = ⟨ e_{y}, ϕ ⟩ ϕ$ . Therefore [F7] yields $tr (P_{ϕ} P_{y}) = ⟨ e_{y}, ϕ ⟩ ⟨ ϕ, e_{y} ⟩ = ∣ ⟨ ϕ, e_{y} ⟩ ∣^{2}$ .

2.1step 1.1step 1.3F8algebra∎

Summing the formula in step 1.3 and applying [F8] gives $\sum_{y} p_{ϕ} (y) = ∥ ϕ ∥^{2} = 1$ . Substituting $ϕ = U_{H} (t) ψ$ in the walk specialization gives the stated walk law and vertex formula.

Depends on

Used by

Generator convention matters Counterexample
Small hypercube transfer Example
Operator-norm error bounds a basis-outcome probability change Lemma
Exact and approximate success of complete-graph continuous-time search Theorem
Exact state transfer on the Boolean hypercube Theorem

Dependency tree · two levels

52 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

Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 17 §17.1 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, Chapter 9 §9.1 (standard reference, not scraped)