LemmaStatement: Literature-sourcedProof: AI-generatedPipeline-generated

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.

Acceptance-gap lower bound on final-state distance

Statement

If a common final two-outcome measurement accepts one unit state with probability at least $2 / 3$ and another with probability at most $1 / 3$ , then the distance between their state vectors is at least $1 / 6$ .

Facts & Assumptions

Given: Unit vectors $ψ, ϕ$ in the same finite-dimensional register and one common final acceptance measurement.

[F1]

A common two-outcome POVM has an acceptance effect $E$ with $0 ⪯ E ⪯ I$ and acceptance weight $tr (P_{u} E)$ on a unit state $u$ (Finite POVMs and their outcome law, Finite positive effect family).

[F4]

A non-negative operator is equivalently self-adjoint with nonnegative eigenvalues, a positive semidefinite matrix in an orthonormal basis, or an operator of the form S^*S: a finite nonnegative operator has an orthonormal eigenbasis with nonnegative real eigenvalues.

[F5]

A unit-vector outer product is a density operator: for a unit $u$ , $P_{u} : w \mapsto ⟨ w, u ⟩ u$ is its density operator. The root pairing is linear in the first argument.

[F6]

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

[F2]

Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ .

[F3]

Worst-case bounded-error correctness: bounded-error decision correctness gives acceptance probability at least $2 / 3$ on a yes input and at most $1 / 3$ on a no input.

Proof

1.1givenF1F4F5F6algebra

By [F1] and [F4], choose an orthonormal eigenbasis $(e_{j})$ for $E$ . Positivity of $E$ and $I - E$ puts each eigenvalue $λ_{j}$ in $[0, 1]$ . For $w = \sum_{j} w_{j} e_{j}$ , $∥ E w ∥^{2} = \sum_{j} λ_{j}^{2} ∣ w_{j} ∣^{2} \leq \sum_{j} ∣ w_{j} ∣^{2} = ∥ w ∥^{2}$ . For a unit $u = \sum_{j} u_{j} e_{j}$ , [F5]–[F6] give $tr (P_{u} E) = \sum_{j} λ_{j} ∣ u_{j} ∣^{2} = ⟨ E u, u ⟩$ . Thus the same acceptance expectation and contraction estimate hold for every two-outcome POVM.

2.1givenF2F3step 1.1algebra∎

Put $δ = ψ - ϕ$ . The acceptance-probability difference equals $⟨ E δ, ψ ⟩ + ⟨ E ϕ, δ ⟩$ . By [F2], unit normalization, and step 1.1, its absolute value is at most $∥ E δ ∥ ∥ ψ ∥ + ∥ E ϕ ∥ ∥ δ ∥ \leq 2 ∥ δ ∥$ . The probability gap in [F3] is at least $1 / 3$ , so $∥ ψ - ϕ ∥ \geq 1 / 6$ .

Depends on

Used by

BBBV hybrid lower bound for unstructured search Theorem

Dependency tree · two levels

43 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

Charles H. Bennett, Ethan Bernstein, Gilles Brassard, and Umesh Vazirani, Strengths and Weaknesses of Quantum Computing (standard reference, not scraped)
Andrew M. Childs, Lecture Notes on Quantum Algorithms (2025) (standard reference, not scraped)