# Æ code[1]

## Description

Code defined in a single angular-momentum subspace that is embedded in a larger direct-sum space of different angular momenta, which can arise from combinations of spin, electronic, or rotational, or nuclear angular momenta of an atom or molecule. A code is obtained by solving an over-constrained system of equations, and many solutions can be mapped into existing codes defined on other state spaces.

A simple example of an Æ code is the error-detecting code with codewords \begin{align} \begin{split} |\overline{0}\rangle&=\frac{1}{\sqrt{2}}\left(|J,-m\rangle+|J,m\rangle\right)\\ |\overline{1}\rangle&=|J,0\rangle~, \end{split} \tag*{(1)}\end{align} constructed out of states of total angular momentum \(J\) and its projection \(m\) for any \(J,m\geq 2\). This code detects a single change in \(m\) or \(J\).

## Protection

## Parent

## Cousins

- Diatomic molecular code — Diatomic molecular codes are supported on states with various total angular momenta, while the more practical Æ codes are supported on only one subspace of fixed total momentum. The latter codes are thus more practical and more applicable to other spin spaces.
- Binomial code — Many well-performing Æ codes can be mapped into shifted versions of binomial codes via the Holstein-Primakoff mapping.
- GNU permutation-invariant code — Many well-performing Æ codes can be mapped into GNU codes via the Dicke state mapping.
- Single-spin 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]
- S. P. Jain et al., “Æ codes”, (2023) arXiv:2311.12324

## Page edit log

- Victor V. Albert (2023-11-22) — most recent

## Cite as:

“Æ code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/ae

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