LemmaStatement: Literature-sourcedProof: AI-adaptedPipeline-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.

Pure-state trace distance is bounded by vector error

Statement

Let $H$ be a finite-dimensional complex inner-product space and let $u, v \in H$ be unit vectors. Write $P_{u} = ∣ u ⟩ ⟨ u ∣$ and $P_{v} = ∣ v ⟩ ⟨ v ∣$ using the Dirac convention of Dirac Notation Conversion, and define their trace distance by $D (P_{u}, P_{v}) : = \frac{1}{2} ∥ P_{u} - P_{v} ∥_{1}$ , where $∥ \cdot ∥_{1}$ is the trace norm of Trace Norm. Then $D (P_{u}, P_{v}) \leq ∥ u - v ∥ .$

Facts & Assumptions

Given: A finite-dimensional complex-inner-product space $H$ and unit vectors $u, v \in H$ .

[F1]

Dirac Notation Conversion: $∣ u ⟩ ⟨ u ∣$ sends $z$ to $⟨ z, u ⟩_{r o o t} u$ , and its coordinate matrix in an orthonormal basis is $[u] [u]^{†}$ .

[F2]

The norm $∥ v ∥ = \sqrt{⟨ v, v ⟩}$ induced by a real or complex inner product: $∥ x ∥^{2} = ⟨ x, x ⟩$ for the root inner product, linear in its first argument.

[F3]

Cauchy–Schwarz: $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality exactly for linearly dependent vectors: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly when the vectors are linearly dependent.

[F4]

Trace Norm: $∥ A ∥_{1}$ is the sum of the singular values of $A$ , counted on its finite-dimensional domain.

[F5]

Trace Norm Spectral Formula: $∥ A ∥_{1} = tr \sqrt{A^{†} A}$ .

[F6]

Complex spectral theorem: a normal endomorphism of a finite-dimensional complex inner product space has an orthonormal eigenbasis, and conversely: a finite-dimensional normal endomorphism has an orthonormal eigenbasis.

Proof

Given: As in Facts & Assumptions.

1.1givenF1F2F3algebra

Put $c : = ⟨ u ∣ v ⟩_{D}$ and $s : = \sqrt{1 - ∣ c ∣^{2}}$ . By [F3], $0 \leq s \leq 1$ . If $s = 0$ , equality in Cauchy–Schwarz makes $v$ a unit scalar multiple of $u$ , so $P_{v} = P_{u}$ and $D (P_{u}, P_{v}) = 0$ .

1.2givenF1F4F5F6algebra

Suppose $s > 0$ and set $e : = (v - c u) / s$ . The Dirac convention makes $⟨ u ∣ e ⟩_{D} = 0$ , and direct expansion gives $∥ e ∥ = 1$ . In the orthonormal basis $(u, e)$ of their span, $P_{u} - P_{v}$ has matrix $(\begin{array}{cc} s^{2} & - c s \\ - \overline{c} s & - s^{2} \end{array})$ ; it is zero on the orthogonal complement. This Hermitian matrix has trace $0$ and determinant $- s^{2}$ , hence eigenvalues $s, - s$ . By [F4]–[F6], $∥ P_{u} - P_{v} ∥_{1} = 2 s$ , so $D (P_{u}, P_{v}) = s$ .

2.1givenF2step 1.1step 1.2algebra∎

For either case, $∥ u - v ∥^{2} = 2 - 2 Re c \geq 2 (1 - ∣ c ∣)$ , while $s^{2} = (1 - ∣ c ∣) (1 + ∣ c ∣) \leq 2 (1 - ∣ c ∣)$ . Thus $D (P_{u}, P_{v}) = s \leq ∥ u - v ∥$ , including the $s = 0$ case from step 1.1.

Depends on

Used by

Finite braid compilation and adiabatic error budget Lemma
Operator error bounds measured output variation Theorem
Segmented Taylor error and resources 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, §1.1.3, Eqs. (1.181)–(1.184) (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, Appendix A.5 (standard reference, not scraped)