Alphabeta MathHow 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.
Spacing interpretation and rate-normalization qualification
PDF p. 4: Code-comparison paragraph
Remark
In the article's fixed-jump benchmark, the selected absorption–emission (Æ) and Aydin–Barg codes correct the same coherent order-one tensor error family in their respective host blocks. A codeword's magnetic support is the set of labels with nonzero amplitude; the minimum gap between distinct labels in the union of the two supports is three for the selected Æ code and two for the selected Aydin–Barg code. The article reports that the Æ code has the larger checking time until approximately , after which the two curves nearly coincide and Aydin–Barg is marginally ahead. It suggests that greater support separation improves robustness when lifting makes emitted photons more informative. This comparison motivates that interpretation but does not establish a general monotonic relationship between support separation and performance.
Here the checking time is the infimum of times at which the recovered entanglement fidelity for the maximally mixed logical input falls strictly below , with infinity if that set is empty. The recovery used for each code and time is its transpose recovery. In the benchmark every state in a decaying block has total instantaneous downward rate , while the phenomenological lift shifts the energy of by . The article notes that real-species rates can depend strongly on and may reorder the codes. It also states that closed-form finite-time fidelities are unavailable for the compared codes and evaluates transpose fidelity numerically. These are attributed model predictions and source statements; no numerical accuracy or general impossibility of a closed form is asserted here.
Remark references
Facts & Assumptions
For the specified finite selected target blocks containing each host block, both codes satisfy the pair-product correction condition for the coherent rank-zero and rank-one tensor span. The selected support gaps are three and two, respectively. The correction statement does not cover arbitrary separately resolved matrix-unit errors or arbitrary finite-time repeated jumps. Aydin–Barg rank-one correction and lower host angular momentum J equals five absorption-emission code J equals four Aydin–Barg encoding
The comparison supplier specifies the finite benchmark, four encodings, target , time-dependent transpose recovery and the article's reported numerical ordering. It does not independently establish the numerical inequalities. Checking-time comparison under the Jm lift Transpose (Petz) recovery on a code
Checking time is the infimum of the strict sub-target fidelity set, with the empty-set convention. It is a first-crossing boundary and does not alone imply a repeated-cycle performance bound. Operational checking time
The equal-rate postulate specifies the jump coefficients, and the uniform-decay result proves total rate for every state with . The separate lift adds to its energy. The comparison in [F2] explicitly assumes that the jumps are held fixed after lifting, so the rate identity continues to apply. Different transition energies alone establish neither that continuation nor perfect environmental discrimination. Equal-rate rigid-rotor benchmark parameters Normalized jumps give a scalar total decay operator Phenomenological angular-momentum degeneracy lift Checking-time comparison under the Jm lift
Adopt the finite-emitter, downward-jump, physical and spectral hypotheses of the comparison supplier and its equal-rate model, including Choice, Countable Choice and Dependent Choice. Checking-time comparison under the Jm lift Equal-rate rigid-rotor benchmark parameters The Axiom of Choice The Axiom of Countable Choice (AC_ω) The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
Proof
Given: The selected codes and conditional benchmark in the Remark, together with the hypotheses in [A1].
Goal: Explain the scope of the article's comparison and the limits of its spacing interpretation.
The code definitions and [F1] give minimum support gaps three and two and correction for the stated coherent order-one spans in the respective host blocks. The latter is a common error-family description, not an assertion that all resolved transitions or finite-time channels are corrected.
Under [A1], [F2] specifies the numerical comparison and attributes its reported ordering to the article. By [F3], the plotted quantity concerns the first sub-target crossing for the stated recovery; by [F4], the rate normalization and lift are particular model choices. The two-code comparison changes host block and code structure as well as support gap, and its reported near-crossing depends on the specified calculation. It therefore supplies the article's stated motivation for a spacing interpretation but does not prove a universal monotonic spacing law or determine an ordering for species with different rates. The source's statement about unavailable closed forms is likewise attributed rather than proved.
Depends on
- Transpose (Petz) recovery on a code
- Checking-time comparison under the Jm lift
- Aydin–Barg rank-one correction and lower host angular momentum
- Operational checking time
- Equal-rate rigid-rotor benchmark parameters
- Phenomenological angular-momentum degeneracy lift
- J equals five absorption-emission code
- J equals four Aydin–Barg encoding
- The Axiom of Choice
- The Axiom of Countable Choice (AC_ω)
- The axiom of dependent choice: a relation in which every element is related to something admits an ℕ-indexed chain
- Normalized jumps give a scalar total decay operator
Dependency tree · 2 levels
Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.
Sources
- Monolithic Quantum Error Correction in the Presence of Distinguishability · PDF p. 4, Fig. 3 caption and code-comparison paragraph; p. 6, code definitions