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Pair-product correction detects quadratic spin errors

Statement

Fix $J \in {4, 5}$ , with spin block $V_{J}$ and Cartesian generators $J_{x}, J_{y}, J_{z}$ in the finite-spin convention. Let $S$ be a finite set of spin labels containing $J$ with $∣ J^{'} - J ∣ \leq 1$ for every $J^{'} \in S$ . On the orthogonal direct sum $H = ⨁_{J^{'} \in S} V_{J^{'}}$ , let $ι_{J} : V_{J} \to H$ and $π_{J} : H \to V_{J}$ be the input-block inclusion and coordinate projection. Let $E_{S} \subseteq L (H)$ be the complete selected order-one absorption–emission tensor error span. Let $P_{2} \subseteq L (V_{J})$ be the complex span of $I_{V_{J}}$ and all products of at most two Cartesian generators.

Let $V : C^{2} \to V_{J}$ be an isometry and put $W = ι_{J} V$ . Suppose $W^{*} E^{*} F W = c_{E, F} I_{2}$ for every $E, F \in E_{S}$ . Then $V^{*} A V$ is scalar for every $A \in P_{2}$ . In particular the code detects every component of every rank-zero, rank-one or rank-two spherical tensor family from $V_{J}$ to itself. Detection here means scalar compression of a single operator; it is distinct from correcting an arbitrary channel.

Facts & Assumptions

[F1]

The finite-spin convention fixes the magnetic basis, ladder matrices, Cartesian generators, tensor-family equations and the sum inner product on finite direct sums. Finite spin blocks, spherical tensors and Clebsch–Gordan phases

[F2]

For the selected blocks, $G_{a} = ι_{J} J_{a} π_{J}$ belongs to $E_{S}$ for $a = x, y, z$ . Every within-block tensor component of rank at most two belongs to $P_{2}$ . Low-rank tensor families are spin polynomials

[F3]

The selected order-one span contains $I_{H}$ ; its pair-product correction condition is scalar compression of $E^{*} F$ for every pair of errors in that span. Order-one absorption–emission tensor error span

Proof

Given: The spin blocks, selected set $S$ , maps $ι_{J}, π_{J}, V, W$ , spans $E_{S}, P_{2}$ , and pair-product hypothesis in the Statement.

Goal: Prove scalar compression of every quadratic spin polynomial and every within-block tensor component of rank at most two.

1.1F1F2algebra

In the magnetic basis, $J_{z}$ is real diagonal and the coefficient of $e_{J, m + 1}$ in $J_{+} e_{J, m}$ equals the real coefficient of $e_{J, m}$ in $J_{-} e_{J, m + 1}$ . Thus $J_{z}^{*} = J_{z}$ and $J_{+}^{*} = J_{-}$ , including the zero endpoints. The definitions of $J_{x}$ and $J_{y}$ give $J_{a}^{*} = J_{a}$ for $a = x, y, z$ . The sum inner product gives $ι_{J}^{*} = π_{J}$ and $π_{J} ι_{J} = I_{V_{J}}$ . Hence $G_{a}^{*} = G_{a}$ and $G_{a}^{*} G_{b} = ι_{J} J_{a} J_{b} π_{J}$ . Each $G_{a}$ is an allowed error by [F2].

2.1F2F3givenstep 1.1algebra

Applying the assumed pair-product condition to $(E, F) = (I_{H}, I_{H})$ , $(I_{H}, G_{a})$ and $(G_{a}, G_{b})$ is legitimate by [F2] and [F3]. With $W = ι_{J} V$ , the resulting compressions are respectively $I_{2}$ , $V^{*} J_{a} V$ and $V^{*} J_{a} J_{b} V$ , using $π_{J} ι_{J} = I_{V_{J}}$ and the product identity from step 1.1. All these matrices are scalar.

3.1F2step 2.1algebra∎

Every $A \in P_{2}$ is a finite complex linear combination of $I_{V_{J}}$ , the $J_{a}$ and the $J_{a} J_{b}$ . By step 2.1 and linearity, $V^{*} A V$ is therefore scalar. Fact [F2] places every within-block tensor component of rank at most two in this space, proving all the stated detection identities. This argument concerns single-operator compression and makes no assertion of correction for an arbitrary channel.

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