ExampleConstruction: AI-generatedVerification: 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 probability weights an otherwise imperfect one-bit key

Example

Take $K = {0, 1}$ , a one-element transcript alphabet $T = {t}$ , and a one-dimensional adversary system $H_{E} = C$ . A finite protocol aborts with probability $1 / 4$ and, on acceptance, both parties always output key $0$ . Then the correctness error is $0$ , the secrecy error is $3 / 8$ , and the abort-aware real-ideal trace distance is exactly $3 / 8$ .

Facts & Assumptions

Given: The key alphabet, transcript, adversary space, and output law in the Example.

[F1]

Finite authenticated key-distribution protocol: the output has a normalized classical state on keys, transcript and adversary; the abort flag is shared.

[F2]

Abort-aware real and ideal QKD outputs and security errors: the ideal accepted key is uniform and shared; the secrecy error is the acceptance probability times the normalized accepted secrecy distance.

[F3]

Trace Distance: for density operators, $D (ρ, σ) = \frac{1}{2} ∥ ρ - σ ∥_{1}$ .

[F4]

Trace Norm Spectral Formula and Trace Norm Unitary Invariance And Positive Formula: for a self-adjoint diagonal matrix, the trace norm is the sum of the absolute diagonal entries.

[F5]

QKD composable secrecy is bounded by correctness plus secrecy: the real-ideal trace-distance error is bounded by the sum of correctness and secrecy errors.

Verification

Given: $ρ^{r e a l} = \frac{1}{4} ∣ ⊥, ⊥, t ⟩ ⟨ ⊥, ⊥, t ∣ + \frac{3}{4} ∣ 0, 0, t ⟩ ⟨ 0, 0, t ∣$ , with the one-dimensional Eve factor suppressed.

1.1givenF1F2construct

The ideal state keeps the abort block and replaces the accepted block by $\frac{3}{4} τ_{A B}$ , so $ρ^{i d e a l} = \frac{1}{4} ∣ ⊥, ⊥, t ⟩ ⟨ ⊥, ⊥, t ∣ + \frac{3}{8} ∣ 0, 0, t ⟩ ⟨ 0, 0, t ∣ + \frac{3}{8} ∣ 1, 1, t ⟩ ⟨ 1, 1, t ∣$ .

1.2givenF1F3F4algebra

Their difference is $\frac{3}{8} ∣ 0, 0, t ⟩ ⟨ 0, 0, t ∣ - \frac{3}{8} ∣ 1, 1, t ⟩ ⟨ 1, 1, t ∣$ . The two basis vectors are orthogonal by [F1], so its nonzero eigenvalues are $3 / 8$ and $- 3 / 8$ ; [F3]–[F4] give $δ_{i d} = \frac{1}{2} (3 / 8 + 3 / 8) = 3 / 8$ .

2.1givenF2F3F4F5step 1.2algebra∎

The accepted keys always agree, so $ε_{c o r} = 0$ . The accepted Alice state is $∣ 0 ⟩ ⟨ 0 ∣$ , whose difference from $τ_{A} = \frac{1}{2} (∣ 0 ⟩ ⟨ 0 ∣ + ∣ 1 ⟩ ⟨ 1 ∣)$ has nonzero diagonal entries $1 / 2, - 1 / 2$ and trace distance $1 / 2$ . Thus $ε_{s e c} = (3 / 4) (1 / 2) = 3 / 8$ . The theorem's sum bound [F5] holds with equality.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

41 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

Christopher Portmann and Renato Renner, Cryptographic Security of Quantum Key Distribution (2014), §4.3 (standard reference, not scraped)
Joseph M. Renes, Quantum Information Theory, ETH Zürich lecture notes (HS2014; revised 4 February 2015), §8.3 (standard reference, not scraped)