Aydin–Barg rank-one correction and lower host angular momentum

PDF p. 6: Aydin–Barg bullet; PDF p. 17: Fig. S5.1 caption

Consider the displayed signed Aydin–Barg encoding in $J=4$ and the displayed Æ encoding in $J=5$. For either code choose a finite set of output spin blocks containing its input block and obeying $\mathrm{\mid }{J}^{\mathrm{\prime }}-J\mathrm{\mid }\le 1$. Both codes satisfy the pair-product Knill–Laflamme condition for the complete selected coherent order-one tensor error span. Therefore every finite trace-preserving Kraus channel whose errors lie in that span has an exact decoded recovery on the code. Tensor components change magnetic labels by $\mathrm{\mid }{m}^{\mathrm{\prime }}-m\mathrm{\mid }\le 1$; this is not a claim about every separately resolved matrix-unit transition with those label bounds.

The spin-four code uses smaller minimum magnetic-support separation, two rather than three, at lower hosting spin, four rather than five (block dimensions nine rather than eleven). In the explicit within-block distance convention of the distance lemma—the smallest spherical-tensor rank with a nonscalar code compression—both codes have exact distance three. They also both have exact spin-polynomial distance three, measured by the first undetected degree in the Cartesian spin generators. Thus the article's equal-distance comparison has this precise supported interpretation. It does not assert equality under an unspecified alternative metric or global minimality of $J=4$.

For the zero-temperature downward-branch specialization, select the lower block $J-1$ as well as the input block, and assume Choice, Countable Choice and Dependent Choice, finitely many integer-spin blocks, strictly increasing spin-manifold energies, a rank-one spherical coupling, and all bath preparation, ULE/weak-Markov, regularity, spectral-matching and compatible vacuum-limit hypotheses of Canonical angular-momentum downward jumps. In particular the chosen nonnegative spectral representative must vanish at every required nonpositive transition frequency, including zero. Under those hypotheses, the single-jump errors out of the code's host block have only $J\to J-1$ and lie in its selected order-one span. Zero temperature by itself is not used to assert a pointwise spectral value or to eliminate zero-frequency intramanifold transitions. Exact recovery above concerns channels with Kraus errors in the stated span; arbitrary finite-time repeated jumps or degeneracy-resolved errors require their own analysis.

The original article, PDF p. 6 Aydin–Barg bullet and Fig. 4, says the signed $J=4$ code “corrects the same rank-1 transitions with smaller separation between the supporting states, achieving the same code distance at lower J.” Its Fig. S5.1 caption on PDF p. 17 specifies the label bounds and says only the downward branch occurs at zero temperature. These claims are preserved with the coherent-span meaning and explicit distance convention above. The article does not define a distance metric in these cited passages; the local distance comparison is proved from the code amplitudes and tensor structure, not inferred from the wording.

Independent source context: Aydin and Barg, Class of codes correcting absorptions and emissions, arXiv:2410.03562v1, PDF p. 2, §II Eqs. (1)–(4) and the paragraph continuing on p. 3, defines coherent order-t and spin error families. It explicitly declines to use a proposed spin-code distance convention in its results. The code constructions and examples in §III, PDF pp. 3–5, are not simply identified with the target's signed $J=4$ code. Actual reading for this resolution uses those source definitions/statement passages and the already completed signed-code supplier; its imported Theorem III.1 remains subject to that supplier's honest reading limits, including no Appendix C reconstruction. This theorem supplies its distance calculation locally and does not import an unspecified distance equality.

The source's code-distance phrase is an interpretation issue resolved here by explicit formulation, not an independently assessed article defect. The original figure curves and checking-time fits are not inputs to this proof, and no numerical reproduction, experiment, separate review or scientific acceptance is asserted.