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.

Finite row-stochastic kernel

Definition

A finite row-stochastic kernel is a finite set $X$ and a matrix $P = (P_{x y})_{x, y \in X}$ with $P_{x y} \in [0, 1]$ and

$\sum_{y \in X} P_{x y} = 1 (x \in X) .$

The row convention is $P_{x y} = \Pr (Y = y ∣ X = x)$ : distributions are row vectors and one transition sends $μ$ to $μ P$ . If $X = \emptyset$ , the row conditions are vacuous and there is no probability distribution on $X$ . No stationary distribution, reversibility, irreducibility, aperiodicity, or efficient circuit access is part of this definition.

Depends on

Transition matrices and n-step probabilities

Used by

Szegedy-walk marked-set detection Algorithm
A stationary directed cycle is not similar to its discriminant Counterexample
A zero stationary weight makes the full-state similarity undefined Counterexample
Finite reversible Markov chain with positive stationary law Definition
Finite row-preparation access model for a Szegedy walk Definition
Szegedy reflection walk and its two-reflection square Definition
Szegedy row-preparation map and discriminant Definition
Discriminant of the marked-state absorbing walk Lemma
Spectral bound for a symmetric lazy walk killed on a marked set Lemma
Szegedy reflections and walk operators are unitary Lemma
Szegedy walk invariant subspaces from discriminant eigenvectors Lemma
Szegedy spectral mapping for one and two reflections Theorem
The Szegedy discriminant is self-adjoint and contractive Theorem

Dependency tree · two levels

3 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

Andrew M. Childs, Lecture Notes on Quantum Algorithms, Chapter 18 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, Chapter 8 (standard reference, not scraped)