DefinitionDefinition: Literature-sourcedProof: Not applicablePipeline-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.

Finite-input memoryless cq channel

Definition

Let $X$ be a finite nonempty input alphabet and let $H_{B}$ be a nonzero finite-dimensional complex inner-product space. A finite-input cq channel is a map $x \mapsto ρ_{x}$ assigning a density operator (Density operator) $ρ_{x}$ on $H_{B}$ to each $x \in X$ . Its $n$ -use memoryless extension sends a word $x^{n} = (x_{1}, \dots, x_{n})$ to the product state $ρ_{x^{n}} : = ρ_{x_{1}} \otimes \dots \otimes ρ_{x_{n}} on H_{B}^{\otimes n} .$ For an input probability vector $p$ , write ${\overset{ˉ}{ρ}}_{p} = \sum_{x} p_{x} ρ_{x}$ and $χ_{p} = S ({\overset{ˉ}{ρ}}_{p}) - \sum_{x : p_{x} > 0} p_{x} S (ρ_{x})$ ; this is the Holevo information of the ensemble in Finite classical-quantum states and Holevo information, in nats.

Depends on

Used by

Dependency tree · two levels

23 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

Mark M. Wilde, From Classical to Quantum Shannon Theory, second-edition prepublication v8, Chapters 16 and 20 (standard reference, not scraped)
John Watrous, The Theory of Quantum Information, §8.1.2 (standard reference, not scraped)