J equals four Aydin–Barg encoding

PDF p. 6: Aydin–Barg bullet and Fig. 4

In the spin-four block, let $\mathrm{\mid }4,m⟩$ be the orthonormal magnetic basis vector with $m=-4,-3,\dots ,4$. The Aydin–Barg encoding considered in the article maps the logical basis vectors $\mathrm{\mid }0⟩,\mathrm{\mid }1⟩$ to $\mathrm{\mid }\bar{0}⟩=\sqrt{1\mathrm{/}3}\mathrm{\mid }4,-4⟩+\sqrt{2\mathrm{/}3}\mathrm{\mid }4,2⟩$ and $\mathrm{\mid }\bar{1}⟩=\sqrt{2\mathrm{/}3}\mathrm{\mid }4,-2⟩-\sqrt{1\mathrm{/}3}\mathrm{\mid }4,4⟩$. Its code is their complex linear span; the minus sign is part of the encoding. The article attributes this example to its Ref. [10].

For the conditional correction statement, select any subset of target spin blocks from $\{3,4,5\}$ containing the input block $4$. A rank-$k$ spherical tensor family is a family of maps between specified spin blocks satisfying the angular-momentum commutation equations of the spin-tensor convention. The order-one absorption–emission error span is the complex span of the identity and the embedded components of all such families with $k=0$ or $1$ between the input and selected target blocks. Tensor rank specifies the transformation law, not the matrix rank. Components are coherent maps over magnetic labels; the span does not automatically contain every separately resolved matrix unit.

The code satisfies the pair-product correction condition for this span: if $P$ is its orthogonal projector, then $P{E}^{\ast }FP$ is a scalar multiple of $P$ for every pair $E,F$ in the span. Consequently every finite trace-preserving Kraus channel whose Kraus operators lie in that span has an exact decoded recovery: a channel after the noise returns every logical operator to itself after the stated encoding.

The comparator is the selected $J=5$ absorption–emission (Æ) code with codewords $\sqrt{5\mathrm{/}7}\mathrm{\mid }5,-2⟩+\sqrt{2\mathrm{/}7}\mathrm{\mid }5,5⟩$ and $\sqrt{2\mathrm{/}7}\mathrm{\mid }5,-5⟩+\sqrt{5\mathrm{/}7}\mathrm{\mid }5,2⟩$. Here the support of a codeword is the set of magnetic labels with nonzero amplitude. Define minimum magnetic-support separation as ${\min }_{m\mathrm{\ne }n}\mathrm{\mid }m-n\mathrm{\mid }$, with $m,n$ ranging over the union of the two codeword supports. It is two for the signed $J=4$ code and three for the selected $J=5$ comparator.

For an explicit distance convention, define the within-block tensor-rank distance $d_T(\mathcal{C})$ as the least integer $r\ge 0$ for which a component $T$ of a rank-$r$ spherical tensor family from $V_J$ to itself has $P_{\mathcal{C}}T{P}_{\mathcal{C}}$ not a scalar multiple of $P_{\mathcal{C}}$, with $d_T=+\mathrm{\infty }$ if no such component exists. Both selected codes have $d_T=3$. This is an error-filtration convention on a spin block, independent of a tensor-register operator-weight distance.

The article describes the signed $J=4$ code as correcting the same rank-one transitions as this selected $J=5$ Æ example, with smaller support separation and the same code distance at lower $J$. Its comparison passage does not specify a distance convention. The tensor-rank equality above gives a precise supported interpretation; it is not attributed to the article as its definition of distance. The conditional correction statement above specifies the error family actually covered. The article's zero-temperature model uses the downward $J\to J-1$ branch.

Angular-momentum basis and tensor labels

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Order-one absorption–emission tensor error span

Code restriction and decoded channel types

J equals five absorption-emission code

Spaced-support moment criterion for order-one AE correction

Both selected codes have exact tensor-rank distance three

Given: The stated definitions, hypotheses and cited interfaces.

Goal: Justify the stated construction and its scope.

1.1 By [F1], the four displayed magnetic basis vectors are distinct and orthonormal. Each codeword has squared norm $1\mathrm{/}3+2\mathrm{/}3=1$, and their supports are disjoint, so their inner product is zero. Thus the displayed assignment extends linearly to an isometry $V$ from the two-dimensional logical space. By [F4], its image is the stated code and its orthogonal projector is $P=V{V}^{\ast }$. [given, F1, F2, F3, F4]

1.2 For [F5], the union of magnetic supports is $\{-5,-2,2,5\}$, with consecutive gaps $3,4,3$; each codeword is normalized and their supports are disjoint. Both have first moment zero, since $(-10+10)\mathrm{/}7=0$, and second moment ten, since $(20+50)\mathrm{/}7=10$. Their zeroth moments are one. Thus [F6] establishes the complete selected order-one pair-product condition for this comparator, with its input block selected. For the signed spin-four code the union is $\{-4,-2,2,4\}$, with gaps $2,4,2$. Hence the stated minimum separations are two and three, at host spins four and five. [F1, F5, F6, algebra]

2.1 The span and pair-product condition are well defined by [F2]. The vectors and target-block hypotheses agree exactly with [F3], which supplies the scalar pair products and the conditional channel conclusion. In particular, the conclusion does not require disjoint error spaces and does not extend merely from the magnetic-label bounds to arbitrary individually resolved matrix units. The attributed comparison and downward-branch sentence are not premises for [F3]. [F1, F2, F3, F4, step 1.1]

3.1 The signed spin-four pair-product condition follows from [F3], and step 1.2 supplies the spin-five condition. The displayed vectors coincide with the two explicit codes in [F7], whose within-block distance definition is the one used here. Therefore [F7] yields $d_T=3$ for both. This local mathematical comparison uses the correction suppliers and the distance lemma; the article is used only for attribution, including its unspecified distance wording and downward-branch model description. [F3, F5, F7, step 1.2] ∎