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Physical Review Letters

Quantum Error Correction for Communication

ARTUR K. EKERT, Chiara Macchiavello

Published Sep 16, 1996• 271 citations
Publisher

Abstract

We show how procedures which can correct phase and amplitude errors are in themselves sufficient to correct errors due to quantum entanglement, generalizing earlier results. We specify general criteria for quantum error correction, introduce quantum versions of the Hamming and the Gilbert-Varshamov bounds, and comment on the practical implementations of quantum codes.

DOI
10.1103/physrevlett.77.2585

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We show how procedures which can correct phase and amplitude errors are in themselves sufficient to correct errors due to quantum entanglement, generalizing earlier results. We specify general criteria for quantum error correction, introduce quantum versions of the Hamming and the Gilbert-Varshamov bounds, and comment on the practical implementations of quantum codes.
Quantum error correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery-operator-independent definition of error-correcting codes. We relate this definition to four others: the existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state, and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct e errors and a formal proof that the classical bounds on the probability of error of e-error-correcting codes applies to e-error-correcting quantum codes, provided that the interaction is dominated by an identity component.
Abstract: Quantum Error Correction will be necessary for preserving coherent states against noise and other unwanted interactions in quantum computation and communication. We develop a general theory of quantum error correction based on encoding states into larger Hilbert spaces subject to known interactions. We obtain necessary and sufficient conditions for the perfect recovery of an encoded state after its degradation by an interaction. The conditions depend only on the behavior of the logical states. We use them to give a recovery operator independent definition of error-correcting codes. We relate this definition to four others: The existence of a left inverse of the interaction, an explicit representation of the error syndrome using tensor products, perfect recovery of the completely entangled state and an information theoretic identity. Two notions of fidelity and error for imperfect recovery are introduced, one for pure and the other for entangled states. The latter is more appropriate when using codes in a quantum memory or in applications of quantum teleportation to communication. We show that the error for entangled states is bounded linearly by the error for pure states. A formal definition of independent interactions for qubits is given. This leads to lower bounds on the number of qubits required to correct errors and a formal proof that the classical bounds on the probability of errorcorrecting codes applies to error correcting quantum codes, provided that the interaction is dominated by an identity component.