Order-one absorption–emission tensor error span

Fix a spin label $J$ and its finite spin block $V_J$, with magnetic basis and spherical tensor families as in Finite spin blocks, spherical tensors and Clebsch–Gordan phases. Choose a finite set $S$ of spin labels containing $J$, with $\mathrm{\mid }{J}^{\mathrm{\prime }}-J\mathrm{\mid }\le 1$ for every $J^{\mathrm{\prime }}\in S$. Let $H={⨁}_{J^{\mathrm{\prime }}\in S}{V}_{J^{\mathrm{\prime }}}$, using one labelled copy of each block, and let ${\iota }_{J^{\mathrm{\prime }}}:V_{J^{\mathrm{\prime }}}\to H$ be its inclusion and ${\pi }_J:H\to V_J$ the orthogonal projection onto the input block.

For each $J^{\mathrm{\prime }}\in S$ and $k\in \{0,1\}$, take every component $T_q:V_J\to V_{J^{\mathrm{\prime }}}$, $-k\le q\le k$, of every rank-$k$ spherical tensor family in the cited convention. The order-one absorption–emission tensor error span for these selected blocks is the complex linear span in $\mathcal{L}(H)$ of $I_H$ and all ${\iota }_{J^{\mathrm{\prime }}}{T}_q{\pi }_J$. The selected target blocks and the tensor-family equations are part of the contract. Tensor rank here means $k$, not matrix rank. A component is one coherent linear map over its magnetic labels; an individually resolved matrix unit $\mathrm{\mid }{J}^{\mathrm{\prime }},m^{\mathrm{\prime }}⟩⟨J,m\mathrm{\mid }$ is a distinct operator and is included only if it actually lies in the stated span.

Let $V:H_L\to V_J$ be an isometry from a nonzero finite-dimensional logical space and regard it as a map into $H$. Its code projector is $P=V{V}^{\ast }$ on $H$, as in Code restriction and decoded channel types. The code detects the span when, for every $E$ in it, $PEP=c_{E}P$ for some complex scalar $c_E$. It satisfies the pair-product Knill–Laflamme correction condition for the span when, for every $E,F$ in it, $P{E}^{\ast }FP=c_{E,F}P$ for some complex scalar $c_{E,F}$. These conditions concern the complete selected span; no particular channel or physical recovery is specified by this definition.

The tensor supplier defines each $T_q$ as a linear map between the named finite blocks, so ${\iota }_{J^{\mathrm{\prime }}}{T}_q{\pi }_J$ is an endomorphism of $H$. The complex span of these endomorphisms and $I_H$ is therefore a well-defined subspace of the finite-dimensional space $\mathcal{L}(H)$, even if some tensor families are zero. The code-restriction supplier gives the orthogonal projector $P$ from the isometry. Consequently $PEP$ and $P{E}^{\ast }FP$ are endomorphisms of $H$, and both displayed scalar-multiple conditions are well typed. Checking them on any spanning family suffices: the first condition is linear in $E$, and the second is conjugate-linear in $E$ and linear in $F$. Finite spin blocks, spherical tensors and Clebsch–Gordan phases Code restriction and decoded channel types

The cited paper gives the pair-product and single-operator conditions and coherent dipole operators for its fixed-$J$ model. The general finite choice of target blocks and the span convention above make the scope explicit for this collection; they do not assert that every such operator is a realized physical error.