TheoremStatement: AI-adaptedProof: AI-generatedPipeline-generatedjudge pass (gpt-6-luna)

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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.

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Every operator below the quantum code distance is detectable

Statement

For a finite-register quantum code of distance $d$ , every operator $E$ with $wt (E) < d$ is detectable: $P E P = λ_{E} P$ for some $λ_{E} \in C$ . No correction-map conclusion is asserted.

Facts & Assumptions

Given: A quantum code with projector $P$ , distance $d$ , and an operator $E$ on its physical register satisfying $wt (E) < d$ .

[L1]

Detectable operators and distance of a finite-register quantum code: For logical dimension at least two, $d$ is the minimum weight of a non-detectable operator; for logical dimension one, $d = n + 1$ .

[L2]

Detectable operators form a subspace and quantum distance is well defined: In logical dimension one every operator is detectable; otherwise the defining minimum exists.

Proof

1.1L2given

If the logical dimension is one, [L2] directly gives detectability.

2.1L1given∎

If the logical dimension is at least two and $E$ were non-detectable, [L1] would imply $wt (E) \geq d$ , contradicting the hypothesis. Therefore $E$ is detectable. This proves only the compression identity $P E P = λ_{E} P$ , not existence of a recovery map.

Remarks

Correction of a family of errors requires additional product conditions and is deferred to the QC19 Knill–Laflamme and recovery page.

Depends on

Used by

Dependency tree · two levels

10 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

Richard Cleve, Quantum Information Processing course notes, §§36.1, 36.3, and 37.3 (standard reference, not scraped)
John Preskill, Quantum Computation, Chapter 7, §7.4.1 (standard reference, not scraped)