Rank-one transition errors in the benchmark

PDF p. 4: Paragraph introducing the four candidate codes; PDF p. 17: Fig. S5.1 caption

A spin-$J$ block has orthonormal magnetic basis $\mathrm{\mid }J,m⟩$ with $m=-J,-J+1,\dots ,J$. A rank-one spherical tensor family consists of three linear maps $T_{\delta }$, $\delta =-1,0,1$, obeying the angular-momentum commutation equations in the stated spin-tensor convention. In this terminology, rank one refers to the tensor transformation law, not to the dimension of the image of an individual matrix.

Fix the input spin block $V_J$ and 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 $H={⨁}_{J^{\mathrm{\prime }}\in S}{V}_{J^{\mathrm{\prime }}}$, with one labelled copy of each selected block, let ${\iota }_{J^{\mathrm{\prime }}}:V_{J^{\mathrm{\prime }}}\to H$ be the block inclusion, and let ${\pi }_J:H\to V_J$ be the orthogonal projection onto the input block.

For each $J^{\mathrm{\prime }}\in S$ and $k\in \{0,1\}$, take every component $T_q:V_J\to V_{J^{\mathrm{\prime }}}$, $q=-k,-k+1,\dots ,k$, of every rank-$k$ spherical tensor family in the stated convention. The order-one absorption–emission error span is the complex linear span in $\mathcal{L}(H)$, the space of linear endomorphisms of $H$, of $I_H$ and all ${\iota }_{J^{\mathrm{\prime }}}{T}_q{\pi }_J$.

For the finite-block zero-temperature emitter model, assume the Axiom of Choice, Countable Choice and Dependent Choice and all system, bath, tensor and pointwise spectral hypotheses of the canonical downward-jumps result. Its three jumps have the form $L_{\delta }=\sqrt{\gamma }{\sum }_{J=1}^N{\sum }_{m=-J}^{J}f(J)C_{1,\delta ;J,m}^{J-1,m+\delta }\mathrm{\mid }J-1,m+\delta ⟩⟨J,m\mathrm{\mid }$, for $\delta =-1,0,1$, with out-of-range terms zero. Here $C$ is the Clebsch–Gordan coefficient in the stated ordered coupling and phase convention, $f(J)$ is the scalar specified by that result, and $\gamma >0$ is its coupling scale. The article calls the single-step errors represented by these jumps rank-one transitions. Each jump is a coherent sum over the allowed magnetic labels and spin blocks; it is not, in general, a single resolved matrix unit.

The article describes the Aydin–Barg code as correcting any rank-one transition error with $\mathrm{\mid }J-J^{\mathrm{\prime }}\mathrm{\mid }\le 1$ and $\mathrm{\mid }m-m^{\mathrm{\prime }}\mathrm{\mid }\le 1$, and says that only the $J\to J-1$ branch occurs in its zero-temperature model. This sentence records the article's claim; correction is not part of the definition of tensor rank. In the error-span convention above, a component is a coherent operator: the displayed label bounds alone do not assert correction of every separately resolved operator $\mathrm{\mid }{J}^{\mathrm{\prime }},m^{\mathrm{\prime }}⟩⟨J,m\mathrm{\mid }$.

Finite spin blocks, spherical tensors and Clebsch–Gordan phases

Order-one absorption–emission tensor error span

Canonical angular-momentum downward jumps

The spin-tensor convention supplies the finite blocks, magnetic bases, coupling coefficients and the three tensor-component equations. The error-span contract supplies the selected direct sum and its block maps. Each composition ${\iota }_{J^{\mathrm{\prime }}}{T}_q{\pi }_J$ is a linear endomorphism of $H$, so it and $I_H$ lie in the common operator space $\mathcal{L}(H)$ and have a well-defined complex span. The contract also distinguishes a coherent component from an individually resolved matrix unit. Under the explicitly stated hypotheses, the downward-jumps result supplies the displayed finite sums and their zero out-of-range convention. Each term maps the named basis vector in the input block to the named vector in the output block, so the finite sum is a well-defined linear operator. The correction statement is attributed to the article and is not used to justify the definition. Finite spin blocks, spherical tensors and Clebsch–Gordan phases Order-one absorption–emission tensor error span Canonical angular-momentum downward jumps