Location of quantum information in additive graph codes

The location of quantum information in various subsets of the qudit carriers of an additive graph code is discussed using a collection of operators on the coding space which form what we call the information group. It represents the input information through an encoding operation constructed as an explicit quantum circuit. Partial traces of these operators down to a particular subset of carriers provide an isomorphism of a subgroup of the information group, and this gives a precise characterization of what kinds of information they contain. All carriers are assumed to have the same dimension $D$, an arbitrary integer greater than 1.

Figure 1

(a) The graph state used in the example. (b) The encoding circuit: the input states $Z_1^{{\zeta }_1{m}_1}{Z}_2^{{\zeta }_2{m}_2}|++\rangle$ that correspond to the trivial code ${\mathcal{C}}_0$ are mapped by $W$ to $\mathcal{C}$, then $U$ entangles the qudits. Here $m_1=2$, $m_2=3,$ and ${\zeta }_j$ are integers such that $0⩽{\zeta }_j⩽d_j-1$, with $d_1=3$, $d_2=2$.

Figure 2

(a) The graph state for the $[[5,1,3]]{}_D$ code. (b) The graph state for the Steane $[[7,1,3]]{}_2$ code.

Figure 3

(a) Complete graph (on six qudits). (b) Bar graph with $n=2p$ carriers and $p$ bars.

Figure 4

The graph state of the $[[4,2,2]]{}_D$ code.