Framework for classifying logical operators in stabilizer codes

Entanglement, as studied in quantum information science, and nonlocal quantum correlations, as studied in condensed matter physics, are fundamentally akin to each other. However, their relationship is often hard to quantify due to the lack of a general approach to study both on the same footing. In particular, while entanglement and nonlocal correlations are properties of states, both arise from symmetries of global operators that commute with the system Hamiltonian. Here, we introduce a framework for completely classifying the local and nonlocal properties of all such global operators, given the Hamiltonian and a bipartitioning of the system. This framework is limited to descriptions based on stabilizer quantum codes, but may be generalized. We illustrate the use of this framework to study entanglement and nonlocal correlations by analyzing global symmetries in topological order, distribution of entanglement, and entanglement entropy.

Figure 1

The framework to study the nonlocal properties of stabilizer codes. The input to the framework is a stabilizer code with a bipartition. The output of the framework is nonlocal properties of the code. On the left-hand side of the figure, a stabilizer code with a bipartition is described. The dots represents qubits in the system while the dotted line represents a bipartition of qubits into two subsets $A$ and $B$. The framework consists of the three elements. First, the logical operators are classified based on their localities and nonlocalities. Here, the rectangles (red online) represent logical operators. Then, the overlapping operator group is generated from the overlap of stabilizer generators $S_i$. The circles (blue online) represent stabilizer generators at the boundary between $A$ and $B$. Finally, the framework analyzes the overlapping operator group through the canonical representation.

Figure 2

The classification of logical operators. Each rectangle represents logical operators. (a) $M_A$. This subset includes logical operators defined only inside $A$. (b) $M_B$. (c) $M_{\mathit{AB}}$. Two logical operators connected through a red arrow are equivalent, each defined inside $A$ or $B$. (d) $M_{\varphi }$. Nonlocal logical operators are defined jointly over $A$ and $B$.

Figure 3

An illustration of the differences between the projection (restriction) ${\mathcal{S}}_A$ and the overlap ${\mathcal{O}}^A$. Each circle represents the stabilizer generators supported by qubits inside the circle. (a) The restriction ${\mathcal{S}}_A$ generated from all the stabilizer generators defined inside $A$. (b) The overlapping operator group ${\mathcal{O}}^A$ generated from all the overlaps of stabilizer generators with $A$. ${\mathcal{O}}^A$ includes left parts (shaded regions) of the stabilizer generators at the boundary between $A$ and $B$ in addition to stabilizer generators inside $A$. (c) Two stabilizer generators at the boundary between $A$ and $B$. Though stabilizer generators commute with each other, their projections onto $A$ do not necessarily commute with each other.

Figure 4

Two equivalent logical operators in $M_{\mathit{AB}}$ found both inside ${\mathcal{O}}^A$ and ${\mathcal{O}}^B$. Each circle represents stabilizer generators which overlap at the boundary between $A$ and $B$.

Figure 5

An illustration of stabilizer generators and logical operators in a bipartition. Circles represent stabilizer generators while squares represent logical operators except that ${\ell }_i$ and ${\ell }_i^{\prime }$ are found inside stabilizer generators. The dotted arrows show the possibility of anticommutations. Logical operators ${\alpha }_i$ and ${\alpha }_i^{\prime }$ (${\beta }_i$ and ${\beta }_i^{\prime }$) in $M_A$ ($M_B$) form pairs just inside $A$ ($B$). Stabilizer generators for nonlocal stabilizer group ${\mathcal{S}}_{\mathit{AB}}$ are represented as products of canonical generators for ${\mathcal{O}}^A$ and ${\mathcal{O}}^B$. While logical operators ${\ell }_i$ and ${\ell }_i^{\prime }$ in $M_{\mathit{AB}}$ appear as the projection of some stabilizer generators $S_i^{(L)}$ for ${\mathcal{S}}_{\mathit{AB}}$, nonlocal logical operators ${\delta }_i$ are products of anticommuting pairs of ${\ell }_i$ and ${\ell }_i^{\prime }$.

Figure 6

(a) Geometric duality on stabilizer codes ($g_A+g_B=2k$). Rectangles represent logical operators. $g_A$ and $g_B$ represent the number of localized logical operators inside $A$ and $B$. A hatched rectangle (red online) at the boundary between $A$ and $B$ is a nonlocal logical operator which is not counted either in $g_A$ or in $g_B$. Rectangles connected by a two-sided arrow are equivalent and counted both in $g_A$ and in $g_B$. Other rectangles are localized logical operators defined only inside $A$ or $B$ and counted once in $g_A$ or in $g_B$. (b) Shrinking logical operators. When $g_A=g_{A^{\prime }}$ while $A^{\prime }$ is smaller than $A$, all the logical operators in $A$ have equivalent logical operators in $A^{\prime }$. Application of appropriate stabilizers shrinks the shapes of logical operators from $A$ to $A^{\prime }$.

Figure 7

The Toric code. Qutbis live on edges on the bonds. Periodic boundary conditions are set in $x$ and $y$ directions. (a) Stabilizers (interaction terms) in the Toric code. Stabilizers act on four qubits in either stars or plaquettes. (b) Logical operators in the Toric code.

Figure 8

Various bipartitions and geometric properties of logical operators in the Toric code. Rectangles represent logical operators and circles represent qubits. Periodic boundary conditions are set in the $\mathrm{x̂}$ and $\mathrm{ŷ}$ directions. Figures on the left-hand side show bipartitions and the number of logical operators defined inside each of the complementary regions. Figures on the right-hand side show the shapes of logical operators. (a) A bipartition into $Q_x$ and ${\mathrm{Q̄}}_x$. Each dot represents each qubit. $Q_x$ is a region of qubits which extends in the $\mathrm{x̂}$ direction. Logical operators connected by two-sided arrows are equivalent. Both $Q_x$ and ${\mathrm{Q̄}}_x$ support only $r_1$ and ${\ell }_2$. $r_2$ and ${\ell }_1$ cannot be defined whether inside $Q_x$ or ${\mathrm{Q̄}}_x$. (b) A bipartition into $Q_x$ and ${\mathrm{Q̄}}_x$. Both $Q_y$ and ${\mathrm{Q̄}}_y$ support only $r_2$ and ${\ell }_1$. $r_1$ and ${\ell }_2$ cannot be defined whether inside $Q_y$ or ${\mathrm{Q̄}}_y$. (c) A bipartition into $R_1=Q_x\cup Q_y$. $R_1$ supports all the independent logical operators $r_1$, ${\ell }_1$, $r_2$, and ${\ell }_2$. There is no logical operator defined inside ${\mathrm{R̄}}_1$.

Figure 9

The commutation relations between the four subsets of logical operators. The dotted arrows represent the anticommutations between corresponding logical operators. We may have anticommutations between two logical operators in the same column while other pairs always commute each other as in the symbolic form of canonical representations. (a) Anticommutations inside $M_A$. (b) Anticommutations inside $M_B$. (c) The pairing between $M_{\mathit{AB}}$ and $M_{\varphi }$. Panels (a) and (b) define local logical qubits while (c) defines nonlocal logical qubits.