Quantum error correction codes and absolutely maximally entangled states

Abstract

For every stabiliser $N$-qudit absolutely maximally entangled state, we present a method for determining the stabiliser 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 analyse 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.

arXiv comment11 pages, 7 figures

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• Geometric measure of entanglement of multi-qubit graph states and its detection on a quantum computer
• Conformal Properties of Hyperinvariant Tensor Networks
• Local tensor-network codes
• Geometric properties of evolutionary graph states and their detection on a quantum computer
• Entanglement of multi-qubit states representing directed networks and its detection with quantum computing
• Generation of the Complete Bell Basis via Hong-Ou-Mandel Interference
• Engineering holography with stabilizer graph codes
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• Quantum Circuits for High-Dimensional Absolutely Maximally Entangled States
• Verification of entangled states under noisy measurements
• Absolutely maximally entangled pure states of multipartite quantum systems
• Studies of properties of bipartite graphs with quantum programming
• Fault-Tolerant Encoding of Logical Qudits in Spin Systems