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.

Trace Distance

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 trace distance is $D (ρ, σ) : = \frac{1}{2} ∥ ρ - σ ∥_{1},$ where $∥ \cdot ∥_{1}$ is the trace norm (Trace Norm). The factor $1 / 2$ is part of this state-distance convention. The diamond norm below is an unhalved norm on maps.

Depends on

Used by

Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
One uniform phase query gives small trace-distance advantage Counterexample
Abort-aware real and ideal QKD outputs and security errors Definition
End-to-end channel and output error contract Definition
Finite adiabatic path model Definition
Acceptance probability weights an otherwise imperfect one-bit key Example
Secrecy and correctness are separate requirements Example
Trace distance of orthogonal pure states Example
Dimension-dependent quantum entropy continuity and Holevo control Lemma
History state and padded output Lemma
Partial trace contracts trace distance Lemma
Pure state trace distance Lemma
Replacing Bob's key by a copy of Alice's costs the correctness error Lemma
Trace distance is unchanged by a fixed tensor factor Lemma
Circuit-to-adiabatic simulation Theorem
Concatenated Threshold Under Local-Stochastic Correlated Noise Theorem
Conditional Concatenated Threshold for Proper Codes Theorem
Entanglement-free, randomness-free quantum fingerprints require Ω(log n) qubits Theorem
Fuchs Van De Graaf Theorem
Helstrom Binary Discrimination Theorem
Measurement Variation At Most Trace Distance Theorem
Operator error bounds measured output variation Theorem
QKD composable secrecy is bounded by correctness plus secrecy Theorem
Tensor-product state errors add under trace distance Theorem
Trace Distance Channel Contractivity Theorem
Trace Distance Range And Equality Theorem

Dependency tree · two levels

11 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)