TheoremStatement: Literature-sourcedProof: AI-adaptedPipeline-generatedjudge pass (gpt-6-luna)

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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.
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Trace Distance Channel Contractivity

Statement

For every finite-dimensional quantum channel $Φ : L (H) \to L (K)$ and every pair of density operators $ρ, σ$ on $H$ , $D (Φ (ρ), Φ (σ)) \leq D (ρ, σ) .$

Facts & Assumptions

Given: Nonzero finite-dimensional complex Hilbert spaces $H, K$ , a completely positive trace-preserving linear map $Φ$ , and states $ρ, σ \in D (H)$ .

[L1]

For a Hermitian trace-zero difference of states, $D$ is the maximum $\frac{1}{2} \sum_{a} ∣ p_{a} - q_{a} ∣$ over finite POVMs; each POVM's total variation is at most $D$ (Trace Distance, Measurement Variation At Most Trace Distance, Helstrom Binary Discrimination). The QC06 measurement and Helstrom results are draft suppliers in this same batch.

[L2]

A channel has a finite Kraus representation $Φ (X) = \sum_{j} K_{j} X K_{j}^{*}$ and trace-preserving Kraus operators satisfy $\sum_{j} K_{j}^{*} K_{j} = I$ (Finite-dimensional quantum channel, Kraus and Choi characterizations of complete positivity, Trace preservation is Kraus normalization). The QC05 bodies are present as drafts; their exact outputs remain subject to cross-batch reconciliation.

Proof

1.1givenL1L2algebra

Let ${E_{a}}_{a \in A}$ be any finite POVM on $K$ and choose Kraus operators from [L2]. Define $F_{a} = \sum_{j} K_{j}^{*} E_{a} K_{j}$ . For every $x \in H$ , $⟨ F_{a} x, x ⟩ = \sum_{j} ⟨ E_{a} K_{j} x, K_{j} x ⟩ \geq 0$ , so $F_{a}$ is nonnegative. Completeness follows from $\sum_{a} F_{a} = \sum_{j} K_{j}^{*} (\sum_{a} E_{a}) K_{j} = \sum_{j} K_{j}^{*} K_{j} = I_{H}$ by [L2]. Thus ${F_{a}}$ is a finite POVM on $H$ .

2.1givenL1L2step 1.1algebra∎

For each outcome, trace cyclicity and the Kraus formula give $tr (Φ (ρ) E_{a}) = \sum_{j} tr (K_{j} ρ K_{j}^{*} E_{a}) = tr (ρ F_{a})$ , and likewise for $σ$ . Their output total variation is therefore the total variation of the pulled-back POVM and is at most $D (ρ, σ)$ by [L1]. Maximizing over output POVMs gives $D (Φ (ρ), Φ (σ))$ by the Helstrom formula in [L1], proving contractivity, including $ρ = σ$ .

Depends on

Used by

Fuchs Van De Graaf Theorem

Dependency tree · two levels

35 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)