Alphabeta MathHow 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.
Private information of a finite channel ensemble
Definition
Let be a finite-dimensional quantum channel, fix its complementary channel , and let be a finite ensemble of density operators on . Set . Its private information is
where is the Holevo information from Finite classical-quantum states and Holevo information; zero-probability terms are omitted. Equivalently, it is the mutual information between the classical label and the receiver output minus that between the label and the complementary output.
Depends on
Used by
- Coherent-information concavity and subadditivity for degradable channels Lemma
- Private information is an average coherent-information difference Lemma
- Product additivity of coherent and private information Lemma
- Regularized private-information formula for quantum channel capacity Theorem
- Single-letter quantum and private capacities of degradable channels Theorem
Dependency tree · two levels
20 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
- Igor Devetak, The private classical capacity and quantum capacity of a quantum channel, §3 (standard reference, not scraped)
- Mark M. Wilde, From Classical to Quantum Shannon Theory, second-edition prepublication v8, Chapter 23 (standard reference, not scraped)