Factor Code Networks

For a quantum code defined by an isometry $T:H_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}\to H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, an operator map is a *-homomorphism $\phi :B(H_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}})\to B(H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})$ from the space of operators on the logical Hilbert space to the space of operators on the physical Hilbert space satisfying $\phi (\mathcal{O})T=T\mathcal{O}$ for every operator $\mathcal{O}$ on $H_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$. This can additionally be taken to be unital $\phi (I_{code})=I_{phys}$ if and only if $\dim H_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}}$ divides $\dim H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$. In this case, $\phi (B(H_{\mathrm{c}\mathrm{o}\mathrm{d}\mathrm{e}})$ is a von Neumann factor subalgebra of $B(H_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})$ and we call the pair $(T,\phi )$ a {\it factor code}. A code network is a family of Hilbert spaces $\{{\mathcal{H}}_i\}$ indexed by a partially ordered set with isometries $T_{ji}:{\mathcal{H}}_i\to {\mathcal{H}}_j$ for $i<j$ satisfying $T_{kj}{T}_{ji}=T_{ki}$ for $i<j<k$. In this paper, we ask when this can be promoted to a {\it factor code network} by defining unital operator maps ${\phi }_{ji}$ so that $(T_{ji},{\phi }_{ji})$ is a factor code and ${\phi }_{kj}{\phi }_{ji}={\phi }_{ki}$. We show that the divisibility conditions on Hilbert space dimensions are sufficient for any code network with a tree structure but not in general. For a diamond-shaped network, we show that promotion is possible if and only if the principle angles between $T_{42}{H}_2$ and $T_{43}{H}_3$ all have multiplicity divisible by $H_1$. For incomparable $i_1,i_2<j$, we characterize when ${\phi }_{j{i}_1}(B(H_{i_1}))$ and ${\phi }_{j{i}_2}(B(H_{i_2}))$ commute, when they generate all of $B(H_j)$, and when they intersect only on multiples of the identity. For $i<j_1,j_2<k$, we characterize when ${\phi }_{k{j}_1}(B(H_{j_1}))\cap {\phi }_{k{j}_2}(B(H_{j_2}))={\phi }_{ki}(B(H_i))$ and when these algebras together with $B(H_k)$ form a nondegenerate commuting square.