DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Degradable and antidegradable channels

Definition

Let $N : A \to B$ be a finite-dimensional channel and fix a complementary channel $N^{c} : A \to E$ coming from one finite Stinespring isometry. Relative to this chosen dilation, $N$ is degradable if there exists a quantum channel $D : B \to E$ such that

$N^{c} = D \circ N .$

It is antidegradable relative to this dilation if there exists a quantum channel $A : E \to B$ such that

$N = A \circ N^{c} .$

All hypotheses and conclusions below use the same fixed complementary dilation and the corresponding degrading or antidegrading channel.

Depends on

Used by

Dependency tree · two levels

19 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, Exercise 8.3 and §8.2.2 (standard reference, not scraped)
Mark M. Wilde, From Classical to Quantum Shannon Theory, second-edition prepublication v8, §§13.5 and 24.8 (standard reference, not scraped)