2T-qutrit code [1]

Description

Two-mode qutrit code constructed out of superpositions of coherent states whose amplitudes make up the binary tetrahedral group \(2T\), a.k.a. the 24-cell.

The codespace is a particular three-dimensional subspace of the 24-dimensional two-mode coherent-state subspace, \begin{align} \mathrm{Span}( \{|\sqrt{2} e^{i (2k+1) \pi/4} \alpha\rangle |0\rangle, |0\rangle |\sqrt{2} e^{i (2k+1) \pi/4} \alpha\rangle, |e^{i k\pi/2} \alpha\rangle |e^{i \ell \pi/2} \alpha\rangle \: : \: 0\leq k, \ell \leq 3\})\tag*{$(1)$} \end{align}for any \(\alpha \geq 0\). A basis can be constructed whose elements are uniform superpositions of coherent states whose amplitudes make up cosets of the quaternion subgroup \(Q\) in \(2T\).