## Description

An encoding into a monolithic (i.e. non-tensor-product) Hilbert space that houses an irreducible representation of \(SU(2)\) or, more generally, another Lie group. In some cases, this space can be thought of as the permutation invariant subspace of a particular tensor-product space.

For the simplest case of \(SU(2)\), a single-spin code can be thought of as a permutation invariant qubit code encoding a \(K\)-dimensional Hilbert space into the maximally symmetric subspace or collective spin of \(2\ell\) spin-half systems. This \(2\ell+1\)-dimensional Hilbert space can be thought of as a standalone spin-\(\ell\) quantum system.

## Protection

Noise models can be categorized as those that cause the state to leave the maximally symmetric subspace and those that do not.

Noise models that do not preserve the permutation-invariant subspace correspond to models for the tensor-product case. Single spin-half errors do not preserve permutation symmetry and correspond to qubit Pauli noise.

Noise models that preserve the permutation-invariant subspace are typically relevant to the case of a monolithic spin. They include collective rotations or decays. A continuous-time single-spin noise channel akin to the depolarizing channel is the Landau-Streater channel [1]. A particular error basis of interest consists of the spherical tensors [2].

## Transversal Gates

## Parents

- Spin code
- Permutation-invariant code — Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the permutation-invariant subspace of a set of spins. A single-spin code correcting spherical tensors can be mapped into a permutation-invariant code with an analogous distance [3; Thm. 1].
- Monolithic quantum code
- Finite-dimensional quantum error-correcting code

## Children

## Cousins

- Qubit code — Certain single-spin codes yield permutation-invariant qubit codes with non-trivial distance [2] when the single spin is treated as a collective spin of several qubits.
- Quantum spherical code (QSC) — Single-spin codes whose codewords are expressed in terms of discrete sets of spin-coherent states may also be interpreted as QSCs.
- Æ code — Since Æ codes are defined in a subspace of fixed total angular momentum and protect against errors linear in the momentum generators, they can also be thought of a single-spin codes.

## References

- [1]
- L. J. Landau and R. F. Streater, “On Birkhoff’s theorem for doubly stochastic completely positive maps of matrix algebras”, Linear Algebra and its Applications 193, 107 (1993) DOI
- [2]
- S. Omanakuttan and J. A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems”, Physical Review A 108, (2023) arXiv:2304.08611 DOI
- [3]
- E. Kubischta and I. Teixeira, “The Not-So-Secret Fourth Parameter of Quantum Codes”, (2023) arXiv:2310.17652

## Page edit log

- Victor V. Albert (2022-11-13) — most recent

## Cite as:

“Single-spin code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/single_spin

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/spins/single_spin.yml.