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Finite signed-code KL product test

Statement

For any two-column isometry $V$ into a fixed finite input spin block and a finite spanning list $T_{a}$ of the selected order-one tensor error span (including the identity restriction), the pair-product condition holds precisely when all $V^{*} T_{a}^{*} T_{b} V$ are scalar multiples of $I_{2}$ . For a finite trace-preserving Kraus channel whose Kraus span equals the tested span, this is equivalent to exact decoded correction; containment suffices for the forward correction implication. Retain signed amplitudes before summing cross terms.

In particular, the signed $J = 4$ isometry $V e_{0} = \sqrt{1 / 3} e_{4, - 4} + \sqrt{2 / 3} e_{4, 2}$ , $V e_{1} = \sqrt{2 / 3} e_{4, - 2} - \sqrt{1 / 3} e_{4, 4}$ satisfies the pair-product condition for the complete order-one span with every selected target block in ${3, 4, 5}$ , as defined in Order-one absorption–emission tensor error span. No separation of its error supports is assumed. Consequently every finite trace-preserving Kraus channel inside that span is exactly correctable on this code.

Facts & Assumptions

[F1]

The complete selected tensor span includes coherent rank-zero and rank-one families, with targets separated as orthogonal labelled blocks. Order-one absorption–emission tensor error span

[F2]

Necessity applies to finite trace-preserving Kraus channels. Exact recovery implies the Knill–Laflamme condition

[F3]

Sufficiency supplies an exact decoded channel for finite trace-preserving Kraus errors satisfying scalar pair products. An explicit syndrome recovery corrects a Knill–Laflamme error family

[F4]

Every tensor family in the spin-four contract is a common scalar times the coherent Clebsch–Gordan family. Tensor-family components equal coherent Clebsch–Gordan transition maps

[F5]

The finite spanning-list test and its exact channel qualification hold. Finite spanning-list compression and channel qualification

[F6]

The signed spin-four vectors give real orthonormal coefficient arrays. Real coefficient arrays for the signed spin-four isometry

[F7]

Every mixed weighted coefficient sum C3 is zero, including the signed cancellation. Signed mixed binomial sums cancel at order one

[F8]

Every diagonal-difference weighted coefficient sum C4 is zero. Signed diagonal binomial moments agree at order one

Proof

Given: The finite objects and conventions in the Statement.

Goal: Establish the stated identities and implications.

1.1F1F2F3F5

For the generic finite test, [F5] equates scalar two-by-two compressions with the span pair-product condition and gives precisely the channel qualifications in the Statement. Thus an arbitrary unnormalized spanning list is not itself asserted to be a channel.

1.2F6F7F8

For the displayed signed code, [F6] gives conditions C1 and C2 of the imported Theorem III.1. The arrays are real and $J = 8 / 2 = 4$ , $t = 1$ . Facts [F7] and [F8] give C3 and C4 for every one of the nine ordered pairs $(a, b) \in {0, 1, 2}^{2}$ , with the same zero-weight convention as the source. All hypotheses of the independent sufficient condition in External input are therefore met. It supplies scalar pair products for the coherent Eq. (3) error family with all $J^{'} = 3, 4, 5$ and the rank-zero identity.

2.1F1F4F5step 1.2algebra

By [F4], the tensor-family equations admit no additional components outside common scalar multiples of that coherent family. For the contract's operators on $H$ , their restrictions to the code are exactly those maps embedded into orthogonal target blocks; $I_{H}$ restricts to the rank-zero identity on the input. Distinct target-block pair products vanish, and the source's same-target pair products are scalar by step 1.2. Linear expansion as in [F5] now covers every operator in the complete selected span, every subset of selected target blocks, and every finite spanning list. In particular every compressed pair product for this signed code is scalar.

3.1F2F3F5step 1.1step 2.1∎

For a finite trace-preserving Kraus channel inside that span, step 2.1 supplies the scalar identities required by [F3], and hence an exact decoded recovery. Conversely [F2] and [F5] give necessity for the Kraus span and for any equal tested span. These are the stated algebraic and channel conclusions, with no separated-support assumption.

External input

Imported sufficient condition: for $n = 8$ , $t = 1$ , real arrays $α_{j}, β_{j}$ in the spin- $n / 2$ magnetic basis satisfy order-one absorption–emission pair-product correction if (C1) $\sum_{j = 0}^{8} α_{j} β_{j} = 0$ , (C2) $\sum α_{j}^{2} = \sum β_{j}^{2} = 1$ , and, for every $0 \leq a, b \leq 2$ , (C3) $\sum_{j = 0}^{6} (\binom{6}{j}) α_{j + a} β_{j + b} / \sqrt{(\binom{8}{j + a}) (\binom{8}{j + b})} = 0$ and (C4) the same weighted sum with numerator $α_{j + a} α_{j + b} - β_{j + a} β_{j + b}$ vanishes. The error family is the coherent Clebsch–Gordan sum in Eq. (3) with $r = 0, 1$ , $∣ δ J ∣, ∣ δ m ∣ \leq r$ , including its rank-zero identity. Zero numerator-binomial terms are omitted. This is the $n = 8, t = 1$ specialization of Aydin–Barg, Theorem III.1 (PDF pp. 3–4), using §II Eqs. (1)–(3) for the basis and errors.

Source: Aydin and Barg, Class of codes correcting absorptions and emissions, §II and Theorem III.1, PDF pp. 2–4. The hypotheses and honest reading limits are given in the source record.

Source notes

This proof applies the imported sufficient coefficient criterion after the five local intermediates; it does not reconstruct Appendix C or identify the source’s displayed code examples with this code. The result concerns coherent tensor components, not individually resolved magnetic-label matrix units.

Depends on

Used by · 3

Dependency tree · 2 levels

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External dependencies

Class of codes correcting absorptions and emissions · §II, PDF p. 2, Eqs. (1)–(3); §III A, Theorem III.1, conditions (C1)–(C4), PDF pp. 3–4

Imported sufficient condition: for $n=8$, $t=1$, real arrays $\alpha_j,\beta_j$ in the spin-$n/2$ magnetic basis satisfy order-one absorption–emission pair-product correction if (C1) $\sum_{j=0}^8\alpha_j\beta_j=0$, (C2) $\sum\alpha_j^2=\sum\beta_j^2=1$, and, for every $0\leq a,b\leq2$, (C3) $\sum_{j=0}^6\binom6j\alpha_{j+a}\beta_{j+b}/\sqrt{\binom8{j+a}\binom8{j+b}}=0$ and (C4) the same weighted sum with numerator $\alpha_{j+a}\alpha_{j+b}-\beta_{j+a}\beta_{j+b}$ vanishes. The error family is the coherent Clebsch–Gordan sum in Eq. (3) with $r=0,1$, $|\delta J|,|\delta m|\leq r$, including its rank-zero identity. Zero numerator-binomial terms are omitted. This is the $n=8,t=1$ specialization of Aydin–Barg, Theorem III.1 (PDF pp. 3–4), using §II Eqs. (1)–(3) for the basis and errors.

Hypotheses: Real coefficient arrays for a two-dimensional spin-n/2 code, n=8 and t=1; all C1–C4 conditions for a,b=0,1,2; coherent, not individually resolved, Clebsch–Gordan transition operators.

Reading limits: Read §II Eqs. (1)–(4), §III A Theorem III.1 and §III construction/Theorem III.3 in the retrieved v1 PDF/extracted text; inspected PDF page images 2,3,4. Theorem III.1 is imported as an established sufficient condition. Appendix C proof not read or reconstructed. Distinct displayed examples are not identified with this signed J=4 code. Local coefficient sums are proved in five-helper resolution; no independent review or acceptance is claimed.

Sources

Class of codes correcting absorptions and emissions · §II, PDF p. 2, Eqs. (1)–(3); §III A, Theorem III.1, conditions (C1)–(C4), PDF pp. 3–4