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 Norm

Definition

Let $A : H \to K$ be a linear map (Linear map between vector spaces over the same field) between finite-dimensional complex inner-product spaces. Its singular values are the eigenvalues of $∣ A ∣ : = \sqrt{A^{*} A} : H \to H$ , counted with multiplicity, as defined in The singular values of a linear map as the eigenvalues of the positive square root of T^*T. Define the trace norm, or Schatten- $1$ norm, by

$∥ A ∥_{1} : = \sum_{j < \dim H} s_{j} (A) .$

The sum includes zero singular values and is $0$ when $H$ is the zero space. For this definition, $A$ may be rectangular; the singular values are those of $∣ A ∣$ on the domain $H$ . The spectral trace formula is proved in Trace Norm Spectral Formula ↗.

Depends on

Used by

Fidelity and trace-distance inequalities do not imply QRE data processing Counterexample
Abort-aware real and ideal QKD outputs and security errors Definition
Diamond Norm Definition
Fidelity Definition
Trace Distance Definition
Fidelity classical measurement minimum Lemma
Gentle measurement bound in finite dimensions Lemma
Partial trace contracts the trace norm on Hermitian operators Lemma
Partial trace contracts trace distance Lemma
Pure-state trace distance is bounded by vector error Lemma
Replacing Bob's key by a copy of Alice's costs the correctness error Lemma
The two root-fidelity trace expressions agree Lemma
Trace distance is unchanged by a fixed tensor factor Lemma
Trace norm of a finite classical block operator Lemma
Trace norm of a tensor product Lemma
Trace Holder Inequality Proposition
Channel Diamond Norm One Theorem
Diamond Norm Ancilla Dimension Bound Theorem
Entanglement-free, randomness-free quantum fingerprints require Ω(log n) qubits Theorem
Hermitian Trace Norm Positive Negative Parts Theorem
Hybrid bound for a composed channel circuit Theorem
Tensor-product state errors add under trace distance Theorem
Trace Norm Duality And Norm Properties Theorem
Trace Norm Rank One Formula Theorem
Trace Norm Spectral Formula Theorem
Trace Norm Unitary Invariance And Positive Formula Theorem

Dependency tree · two levels

7 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 (2018 author-hosted draft), Chapter 1, §1.1.3 (standard reference, not scraped)
Ronald de Wolf, Quantum Computing: Lecture Notes, arXiv:1907.09415v6 (2026-08-27), Appendix A.5 (standard reference, not scraped)