\(SU(4)\) Tverberg spin code[1; Sec. 6.3]

Description

Single-spin code family in the totally symmetric \(N\)-particle irrep \(\mathcal{H}=\mathrm{Sym}^N(\mathbb{C}^4)\) of \(\mathfrak{su}(4)\), whose weight diagram is the discrete simplex \(\Delta_{4,N}\), equivalently its centered tetrahedral realization. The construction uses the two-step Tverberg-theorem method [2]: first choose an intermediate subspace from a distance-two subset of \(\Delta_{4,N}\), then combine basis states whose convex hulls contain the origin.

For \(N\) divisible by eight, an optimal distance-two intermediate lattice is obtained as the kernel of \begin{align} a_1L_1+a_2L_2+a_3L_3+a_4L_4 \mapsto a_1+2a_2+3a_3 \pmod 4~. \tag*{(1)}\end{align} The second step pairs opposite points in the central octahedral region and can form additional triples and quadruples near the tetrahedron’s corners; for \(N=8\), this partition is optimal within the two-step framework [1; Sec. 6.3 and Ex. 6.10].

More explicitly, write the normalized monomial basis of \(\mathrm{Sym}^N(\mathbb{C}^4)\) as \(|a_1a_2a_3a_4\rangle\), where \((a_1,a_2,a_3,a_4)\in\Delta_{4,N}\), and let \(\Lambda_B^N\) be the kernel above. For each block \(Y\subset\Lambda_B^N\) whose convex hull contains the origin, choose barycentric weights \(\{\beta_{\mathbf{a}}\}_{\mathbf{a}\in Y}\) satisfying \(\beta_{\mathbf{a}}\geq0\), \(\sum_{\mathbf{a}\in Y}\beta_{\mathbf{a}}=1\), and \(\sum_{\mathbf{a}\in Y}\beta_{\mathbf{a}}\mathbf{a}=0\) in centered \(\Delta_{4,N}\) coordinates. The corresponding codeword is \begin{align} |\psi_Y\rangle=\sum_{\mathbf{a}\in Y}\sqrt{\beta_{\mathbf{a}}}\,|\mathbf{a}\rangle~. \tag*{(2)}\end{align} Thus opposite pairs give \((|p\rangle+|-p\rangle)/\sqrt{2}\); triples of the form \(\{p,p',-(p+p')\}\) give \((|p\rangle+|p'\rangle+|-(p+p')\rangle)/\sqrt{3}\); and zero-sum quadruples give equal four-term superpositions.