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.

Fidelity

Definition

Let $H$ be a nonzero finite-dimensional complex inner-product space and let $ρ, σ$ be density operators on $H$ (nonnegative operators of trace one; Density operator). Their root fidelity is $F (ρ, σ) : = ∥ \sqrt{ρ} \sqrt{σ} ∥_{1},$ where $\sqrt{ρ}$ and $\sqrt{σ}$ are the unique non-negative square roots and $∥ \cdot ∥_{1}$ is the finite-dimensional trace norm (Trace Norm). This library uses root fidelity; the squared quantity $F (ρ, σ)^{2}$ is called squared fidelity.

Remarks

The alternative formula $F (ρ, σ) = tr \sqrt{\sqrt{ρ} σ \sqrt{ρ}}$ is proved in The two root-fidelity trace expressions agree. Preskill uses the squared-fidelity convention; when using that source, its quantity is $F^{2}$ here.

Depends on

Used by

Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
Fidelity classical measurement minimum Lemma
Optimal Cloning Worst Case Upper Bound Lemma
The two root-fidelity trace expressions agree Lemma
Exact Mixed State Cloning Fidelity Criterion Theorem
Fidelity Data Processing Theorem
Fidelity Purification Characterization Theorem
Fidelity range equality and products Theorem
Fuchs Van De Graaf Theorem
Full Rank Pair No Broadcasting Theorem

Dependency tree · two levels

18 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)
John Preskill, Quantum Computation, Chapter 2: Foundations I: States and Ensembles (updated July 2015) (standard reference, not scraped)