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

Statement

For unit vectors $u, v$ in a finite-dimensional complex Hilbert space, let $P_{u} = ∣ u ⟩ ⟨ u ∣$ and $P_{v} = ∣ v ⟩ ⟨ v ∣$ . Then $D (P_{u}, P_{v}) = \sqrt{1 - ∣ ⟨ u ∣ v ⟩_{D} ∣^{2}},$ where $⟨ \cdot ∣ \cdot ⟩_{D}$ is the Dirac pairing converted from the root library's first-argument-linear inner product.

Facts & Assumptions

Given: Unit vectors $u, v$ in a finite-dimensional complex Hilbert space.

[F1]

The root inner product is linear in its first argument and conjugate-linear in its second (Real and complex inner-product spaces and their induced length).

[F2]

Cauchy–Schwarz gives $∣ ⟨ u, v ⟩ ∣ \leq ∥ u ∥ ∥ v ∥$ , with equality for nonzero vectors exactly when they are linearly dependent (Cauchy–Schwarz: $∣ ⟨ x, y ⟩ ∣ \leq ∥ x ∥ ∥ y ∥$ , with equality exactly for dependent pairs).

[L1]

Dirac notation is converted by $⟨ u ∣ v ⟩_{D} = ⟨ v, u ⟩_{r o o t}$ and $∣ u ⟩ ⟨ u ∣ x = ⟨ x, u ⟩_{r o o t} u$ (Dirac Notation Conversion). The current QC01 definition body is present and supplies the stated convention.

[F3]

$D (P_{u}, P_{v}) = \frac{1}{2} ∥ P_{u} - P_{v} ∥_{1}$ (Trace Distance).

[L2]

For Hermitian trace-zero $H$ , $∥ H ∥_{1} = 2 tr (H_{+})$ (Hermitian Trace Norm Positive Negative Parts). The current QC01 theorem body is present as a draft; it proves the positive/negative-parts identity used here.

Proof

1.1givenF1F2L1choosealgebra

Put $c = ∣ ⟨ u ∣ v ⟩_{D} ∣ \in [0, 1]$ by [F2] and the unit norms. If $c = 1$ , equality in [F2] makes $v$ a unit scalar multiple of $u$ , so $P_{u} = P_{v}$ and both sides are zero. Suppose $c < 1$ . Multiplying $v$ by a unit scalar does not change $P_{v}$ and lets us assume $⟨ v, u ⟩_{r o o t} = c$ : if $a = ⟨ v, u ⟩_{r o o t} \neq 0$ , multiply $v$ by $\overline{a} / ∣ a ∣$ . Define $w = (v - c u) / \sqrt{1 - c^{2}}$ . Direct expansion using [F1] gives $⟨ w, u ⟩_{r o o t} = 0$ and $∥ w ∥ = 1$ , so $(u, w)$ is orthonormal and $v = c u + \sqrt{1 - c^{2}} w$ .

2.1step 1.1L2algebra

In the orthonormal basis $(u, w)$ of their span, $P_{u} - P_{v}$ has matrix $(\begin{array}{cc} 1 - c^{2} & - c \sqrt{1 - c^{2}} \\ - c \sqrt{1 - c^{2}} & - (1 - c^{2}) \end{array})$ . Its trace is zero and determinant is $- (1 - c^{2})$ , so its two eigenvalues are $\sqrt{1 - c^{2}}$ and $- \sqrt{1 - c^{2}}$ ; it is zero on the orthogonal complement. By [L2], $∥ P_{u} - P_{v} ∥_{1} = 2 \sqrt{1 - c^{2}}$ .

3.1step 1.1step 2.1F3∎

Applying [F3] to step 2.1 gives $D (P_{u}, P_{v}) = \sqrt{1 - c^{2}}$ ; together with the $c = 1$ case in step 1.1 this proves the formula for every pair of unit vectors.

Depends on

Used by

One uniform phase query gives small trace-distance advantage Counterexample
Circuit-to-adiabatic simulation Theorem
Fuchs Van De Graaf Theorem

Dependency tree · two levels

28 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 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (updated July 2015) (standard reference, not scraped)