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arXiv:quant-ph/9702031 · Physical Review A

Information-theoretic interpretation of quantum error-correcting codes

Nicolas J. Cerf, Richard Cleve

Published Sep 1, 1997• 39 citations
arXivHTMLPDF

Abstract

Quantum error-correcting codes are analyzed from an information-theoretic perspective centered on quantum conditional and mutual entropies. This approach parallels the description of classical error correction in Shannon theory, while clarifying the differences between classical and quantum codes. More specifically, it is shown how quantum information theory accounts for the fact that ``redundant'' information can be distributed over quantum bits even though this does not violate the quantum ``no-cloning'' theorem. Such a remarkable feature, which has no counterpart for classical codes, is related to the property that the ternary mutual entropy vanishes for a tripartite system in a pure state. This information-theoretic description of quantum coding is used to derive the quantum analog of the Singleton bound on the number of logical bits that can be preserved by a code of fixed length which can recover a given number of errors.

DOI
10.1103/physreva.56.1721

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Quantum error-correcting codes are analyzed from an information-theoretic perspective centered on quantum conditional and mutual entropies. This approach parallels the description of classical error correction in Shannon theory, while clarifying the differences between classical and quantum codes. More specifically, it is shown how quantum information theory accounts for the fact that redundant'' information can be distributed over quantum bits even though this does not violate the quantum no-cloning'' theorem. Such a remarkable feature, which has no counterpart for classical codes, is related to the property that the ternary mutual entropy vanishes for a tripartite system in a pure state. This information-theoretic description of quantum coding is used to derive the quantum analog of the Singleton bound on the number of logical bits that can be preserved by a code of fixed length which can recover a given number of errors.
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.
Quantum error correction draws many of its principles and constructions from classical coding theory, adapted to the unique aspects of quantum mechanics. Here we extend this correspondence to a block of logical qubits, each prepared in the same quantum code, by coupling the block through a classical error correcting code: syndrome qubits then carry error information from several logical qubits at once, and collective inference recovers the errors. Algebraically, the construction defines subgroups of the product stabilizer group corresponding to duals of classical codes, and we prove that a lookup table decoder corrects every error pattern respecting the correction radii of the two constituent codes. For classical algebraic code families, the number of syndrome qubits serving LLL logical qubits scales as O(log⁡2(L+1)){\cal O}(\log_2(L+1))O(log2​(L+1)) at the code-capacity level, though at the price of stabilizer weight growing with the block length. An entropy bound shows that check weights need not grow with LLL. Under a phenomenological noise model, the same lookup table identifies measurement errors by nearest-neighbor post-processing, and classical algebraic decoders locate exactly the logical qubits carrying errors even with noisy syndromes. More broadly, we argue that quantum error correction, reduced to its functional core, is source compression in the sense of Shannon, whose source and channel coding theorems bound the overhead rates of quantum post-selection tasks.
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.
A simple and unifying method to show the perfect error-correcting condition is provided based on the quantum mutual information. The one-to-one parameterization of quantum operations and the properties of the quantum relative entropy are used effectively in this paper, where the equivalence between the subspace transmission and the entanglement transmission is clearly presented. We also revisit a variant of the no-cloning and no-deleting theorem based on an information-theoretical tradeoff between two parties for the reversibility of quantum operations, and demonstrate that the no-cloning and no-deleting theorem leads to the perfect error-correcting condition on Kraus operators.
We show that a relatively simple reasoning using von Neumann entropy inequalities yields a robust proof of the quantum Singleton bound for quantum error-correcting codes (QECC). For entanglement-assisted quantum error-correcting codes (EAQECC) and catalytic codes (CQECC), a type of generalized quantum Singleton bound [Brunet al., IEEE Trans. Inf. Theory 60(6):3073–3089 (2014)] was believed to hold for many years until recently one of us found a counterexample [MG, Phys. Rev. A 103, 020601 (2021)]. Here, we rectify this state of affairs by proving the correct generalized quantum Singleton bound, extending the above-mentioned proof method for QECC; we also prove information-theoretically tight bounds on the entanglement-communication tradeoff for EAQECC. All of the bounds relate block lengthnnnand code lengthkkkfor given minimum distancedddand we show that they are robust, in the sense that they hold with small perturbations for codes which only correct most of the erasure errors of less thandddletters. In contrast to the classical case, the bounds take on qualitatively different forms depending on whether the minimum distance is smaller or larger than half the block length. We also provide a propagation rule: any pure QECC yields an EAQECC with the same distance and dimension, but of shorter block length.