Okada spin code [1]

Description

Non-diagonal \(SU(2)\) single-spin code in the spin-\(J = 3m\) irrep for integer \(m \geq 1\), encoding a logical \((2m+1)\)-dimensional space. The construction uses a non-diagonal subspace (one for which the projected error space \(P_{\mathcal{B}}\mathcal{E}P_{\mathcal{B}}\) is block-diagonal rather than diagonal) to exceed the dimension bound achievable by the Tverberg theorem construction [2].

The code is defined in terms of a subspace \(\mathcal{B} = \mathrm{span}\{|k, n-k\rangle : k \equiv 0 or 1 \pmod{3}\}\) of the spin-\(n/2\) Hilbert space \(\mathcal{H}_n\) (with \(n = 2J = 6m\)), where \(|k, n-k\rangle\) denotes the state with \(k\) particles in the first mode and \(n-k\) in the second [3; Ex. 7.1]. The codespace has dimension \(\dim \mathcal{C} = 2m+1\), which is approximately \((n+1)/3\).

The \(m = 1\) (\(J = 3\)) instance encodes a logical qutrit and admits the unnormalized codewords \begin{align} \begin{split} |\overline{0}\rangle&=|_{0}^{3}\rangle\\|\overline{1}\rangle&\propto\sqrt{2}|_{-2}^{3}\rangle-|_{4}^{3}\rangle\\|\overline{2}\rangle&\propto|_{-4}^{3}\rangle+\sqrt{2}|_{2}^{3}\rangle~. \end{split}\tag*{$(1)$} \end{align}