Quantum error-correction codes and absolutely maximally entangled states

For every stabilizer $N$-qudit absolutely maximally entangled state, we present a method for determining the stabilizer generators and logical operators of a corresponding quantum error-correction code. These codes encode $k$ qudits into $N-k$ qudits, with $k\le \lfloor N/2\rfloor$, where the local dimension $d$ is prime. We use these methods to analyze the concatenation of such quantum codes and link this procedure to entanglement swapping. Using our techniques, we investigate the spread of quantum information on a tensor network code formerly used as a toy model for the AdS/CFT correspondence. In this network, we show how corrections arise to the Ryu-Takayanagi formula in the case of entangled input state, and that the bound on the entanglement entropy of the boundary state is saturated for absolutely maximally entangled input states.

Figure 1

Multiqubit state (left) vs an AME state, specifically a three-qubit GHZ state (right). Vectors corresponding to every uncontracted leg (a qubit) are taken from the corresponding column of the matrix formed by the list of generators of the two states. Therefore, different entries of a given vector describe the action of different stabilizers on the corresponding qubit.

Figure 2

After measuring the adjacent qubits in the Bell basis (contracting the leg connecting the polygon with the triangle), the state on the uncontracted legs (qubits) is fully described by the updated generators. Note that the generators of the polygon do not change (apart from the one associated with the contracted leg), while the generators formerly belonging to the AME state get updated according to the vectors of the qubits that are contracted.

Figure 3

Example of the emergence of stabilizers through concatenation. Generators are given in transparent frames and logical operators are given in blue frames. From the top: two codes, based on three-qubit GHZ states, each encoding one qubit into two qubits. The stabilizer generator of the resulting codes is $ZZ$. Then, two of the resulting qubits get encoded into a four-qubit code, based on the six-qubit AME state. The generators of this code appear. On the other hand, the previously existing stabilizers get concatenated through logical operators of the four-qubit code, and the final list of stabilizers is given in the bottom frame. Thus this total set of stabilizers consists of two classes: (i) those in the first two lines, which are “old” stabilizers, just encoded through the four-qubit code and (ii) those in the last two lines—the “new” ones—which are the stabilizers of the four-qubit code.

Figure 4

Tiling of a surface with negative curvature into pentagons. Red dots inside every pentagon, as well as the white dots coming out of the pentagons, represent qubits. Each of the pentagons represents the six-qubit AME state. Contracted legs represent entanglement swapping.

Figure 5

Spread of the operator $X$ applied to the red entry (logical qudit) of the largest pentagon. First, it gets encoded in the form of the logical operator $X_1=-ZXZ1$ of the $1\to 5$ code emerging from a six-qubit AME state. Further concatenation then results in the final form of the operator on the boundary state.

Figure 6

Labeling of logical qubits (red dots inside pentagons) and physical qubits (noncontracted links on the boundary) for studies of entropy of entanglement of the encoded states. The region $A$ consists of qubits labeled by $1,...,s_B-1$, while the region $B$ consists of qubits labeled by $13,...,s_B$. Here, $s_B=13$.

Figure 7

Entanglement entropy $S\left({Tr}_B\right|\mathrm{\Psi }\rangle \langle \mathrm{\Psi }\left|\right)$ of the boundary states of the code from Fig. 6 for different input states (AME, GHZ, singlet, or product; see the text for details) and for boundary points of the region $B$ being qubits $s_B$ and 20.