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arXiv:2007.07958 · IEEE Transactions on Information Theory

Generalized Perfect Codes for Symmetric Classical-Quantum Channels

Andreu Blasco Coll, Gonzalo Vazquez-Vilar, J.R. Fonollosa

Published Apr 28, 2022• 4 citations
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Abstract

We define a new family of codes for symmetric classical-quantum channels and establish their optimality. To this end, we extend the classical notion of generalized perfect and quasi-perfect codes to channels defined over some finite dimensional complex Hilbert output space. The resulting optimality conditions depend on the channel considered and on an auxiliary state defined on the output space of the channel. For certain$N$-qubit classical-quantum channels, we show that codes based on a generalization of Bell states are quasi-perfect and, therefore, they feature the smallest error probability among all codes of the same blocklength and cardinality.

DOI
10.1109/tit.2022.3170868

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We investigate converse bounds on the average decoding error probability in finite-blocklength classical-quantum channel coding. We first present a lower bound for multiple quantum hypothesis testing in terms of pairwise trace distances and derive a corresponding fidelity bound. We then obtain a spectrum-based converse that depends only on the a priori probabilities and spectra of the states. We show that the converse bound remains tight for the quantum depolarizing channel under suitable conditions. For codes with product-state outputs, this converse takes an explicit form involving products of output-state eigenvalues. We apply it to binary codes over the quantum amplitude damping channel using the input states $\vert+\rangle$ and $\vert-\rangle$. For this setting, we also discuss a normal approximation to the spectrum-based converse in the large blocklength regime. In all numerical examples considered, the spectrum-based converse is tighter than the other converse bounds at low noise levels.

quant-pharXiv:2610.06840•Oct 6, 2026•0 citations
We define a new family of codes for symmetric classical-quantum channels and establish their optimality. To this end, we extend the classical notion of generalized perfect and quasi-perfect codes to channels defined over some finite dimensional complex Hilbert output space. The resulting optimality conditions depend on the channel considered and on an auxiliary state defined on the output space of the channel. For certainNNN-qubit classical-quantum channels, we show that codes based on a generalization of Bell states are quasi-perfect and, therefore, they feature the smallest error probability among all codes of the same blocklength and cardinality.
In this article, we explore perfect and quasi-perfect codes for the Bosonic channel, where information is generated by a laser and conveyed in the form of coherent states. In particular, we consider the phase-modulation codebook for coherent states in a Bosonic channel. We show that these phase-modulation codes are quasi-perfect as long as the cardinality of the code is the same as the dimension of the coherent states. These codes feature the smallest error probability among all codes of the same cardinality and the same dimension of the channel Hilbert space. We study the performance of these codes in terms of error probability, incorporating the degradation caused by a depolarizing or an erasure quantum channel.
An important part of the information theory folklore had been about the output statistics of codes that achieve the capacity and how the empirical distributions compare to the output distributions induced by the optimal input in the channel capacity problem. Results for a variety of such empirical output distributions of good codes have been known in the literature, such as the comparison of the output distribution of the code to the optimal output distribution in vanishing and non-vanishing error probability cases. Motivated by these, we aim to achieve similar results for the quantum codes that are used for classical communication, that is the setting in which the classical messages are communicated through quantum codewords that pass through a noisy quantum channel. We first show the uniqueness of the optimal output distribution, to be able to talk more concretely about the optimal output distribution. Then, we extend the vanishing error probability results to the quantum case, by using techniques that are close in spirit to the classical case. We also extend non-vanishing error probability results to the quantum case on block codes, by using the second-order converses for such codes based on hypercontractivity results for the quantum generalized depolarizing semi-groups.
We prove a one-shot "minimax" converse bound for quantum channel coding assisted by positive partial transpose channels between sender and receiver. The bound is similar in spirit to the converse by Polyanskiy, Poor, and Verdu [IEEE Trans. Info. Theory 56, 2307-2359 (2010)] for classical channel coding, and also enjoys the saddle point property enabling the order of optimizations to be interchanged. Equivalently, the bound can be formulated as a semidefinite program satisfying strong duality. The convex nature of the bound implies channel symmetries can substantially simplify the optimization, enabling us to explicitly compute the finite blocklength behavior for several simple qubit channels. In particular, we find that finite blocklength converse statements for the classical erasure channel apply to the assisted quantum erasure channel, while bounds for the classical binary symmetric channel apply to both the assisted dephasing and depolarizing channels. This implies that these qubit channels inherit statements regarding the asymptotic limit of large blocklength, such as the strong converse or second-order converse rates, from their classical counterparts. Moreover, for the dephasing channel, the finite blocklength bounds are as tight as those for the classical binary symmetric channel, since coding for classical phase errors yields equivalently-performing unassisted quantum codes.
(English) Quantum computers use the laws of quantum mechanics to do complex operations that classical computers can't solve. Quantum computers and other physical quantum systems require quantum information to be completely isolated from the environment in order to be perfectly functional. Quantum systems that are not completely isolated suffer from decoherence noise and may suffer from errors. Implementing these systems is a huge challenge and development of suboptimal systems with a low error rate seems to be the best way forward to building these systems. In quantum communications, designing codes that are able to reduce or protect against errors is a necessity. This work focuses on perfect and quasi-perfect quantum codes, which are a family of codes that exists in the classical setting and are optimum in terms of minimizing the error probability for a given number of channel uses. In the first part of the work we generalize the definition of quasi-perfect codes to include both classical codes and quantum codes. We also provide an example of a quantum quasi-perfect code for 2-qubits classical-quantum channels that make use of quantum entanglement and that can be extrapolated to an N-dimensional classical-quantum channel. The second part of this work focuses on quasi-perfect codes in optical communications, where coherent states are used to convey information through an optical channel, known as the Bosonic channel. The Bosonic channel has infinite dimension, so instead we consider a finite-dimensional approximation of the Bosonic channel with a negligible approximation error for a sufficiently large channel dimension. We show that phase-modulated coherent states constitute a codebook that is quasi-perfect for the approximated channel, and thus are close to optimal for the Bosonic channel. The last part of the work focuses on stabilizer error correction codes, which are practical codes that use redundancy to protect against errors. Error correction is not specifically used to transmit classical information, but to protect quantum information instead. However, it is possible that we require to obtain classical information from a quantum state after performing error correction. For these particular cases we may be able to prove that an error correction code is quasi-perfect and thus optimum. (Català) Els ordinadors quàntics fan servir les lleis de la mecànica quàntica per a realitzar operacions complexes que els ordinadors clàssics no poden resoldre. Els ordinadors quàntics i els altres sistemes quàntics requereixen que la informació quàntica estigui completament aïllada de l'entorn per tal de ser perfectament funcionals. Els sistemes quàntics que no estan completament aïllats són afectats per soroll de decoherència i poden patir errors. Implementar aquests sistemes és un gran repte i l'única manera de construir sistemes quàntics és desenvolupar sistemes subòptims amb una taxa d'error baixa. En comunicacions quàntiques, dissenyar codis capaços de reduir o protegir contra errors és una necessitat. Aquest treball es centra en codis quàntics perfectes i quasi-perfectes que ja existeixen en sistemes clàssics i que són òptims en termes de minimitzar la probabilitat d'error per un determinat nombre d'usos del canal. A la primera part del treball generalitzem la definició de codis quasi-perfectes per incloure codis clàssics i codis quàntics. També proveïm un exemple de codi quàntic quasi-perfecte per a canals clàssic-quàntics de 2-qubits que fa servir entrellaçat quàntic i que pot extrapolar-se a un canal de dimensió N. La segona part d'aquest treball es centra en codis quasi-perfectes en comunicacions òptiques, on estats coherents es fan servir per a transmetre informació a través d'un canal òptic, conegut com a canal Bosònic. El canal Bosònic té dimensió infinita, per això considerem en canvi una aproximació del canal Bosònic de dimensió finita, amb un error d'aproximació negligible amb una dimensió de canal suficientment gran. Demostrem que una modulació de fase dels estats coherents constitueix un codi quasi-perfecte pel canal aproximat, i per tant és pràcticament òptim per al canal Bosònic. L'última part del treball es centra en codis estabilitzadors de correcció d'errors, que són codis pràctics que fan servir redundància per a protegir contra errors. La correcció d'errors no es fa servir específicament per a transmetre informació clàssica, sinó per a protegir la informació quàntica. Tot i això, és possible que requerim obtenir informació clàssica d'un estat quàntic després de fer servir correcció d'errors. Per a aquests casos, pot ser possible demostrar que un codi de correcció d'errors és quasi-perfecte i per tant òptim. (Español) Los ordenadores cuánticos hacen uso de las leyes de la mecánica cuántica para realizar operaciones complejas que los ordenadores clásicos no pueden resolver. Los ordenadores cuánticos y otros sistemas cuánticos requieren que la información cuántica esté completamente aislada del entorno para poder ser perfectamente funcionales. Los sistemas cuánticos que no están completamente aislados son afectados por ruido de decoherencia y pueden sufrir errores. Implementar estos sistemas es un gran reto y la única manera de construir sistemas cuánticos es desarrollar sistemas subóptimos con una tasa de error baja. En comunicaciones cuánticas, diseñar códigos capaces de reducir o proteger contra errores es una necesidad. Este trabajo se centra en códigos cuánticos perfectos y quasi-perfectos que ya existen en sistemas clásicos y que son óptimos en términos de minimizar la probabilidad de error para un determinado número de usos del canal. En la primera parte del trabajo generalizamos la definición de códigos quasi-perfectos para incluir códigos clásicos y códigos cuánticos. También proveemos un ejemplo de código cuántico quasi-perfecto para canales clásico-cuánticos de 2-qubits que usa entrelazado cuántico y que puede extrapolarse a un canal de dimensión N. La segunda parte de este trabajo se centra en códigos quasi-perfectos en comunicaciones ópticas, donde se usan estados coherentes para transmitir información a través de un canal óptico, conocido como canal Bosónico. El canal Bosónico tiene dimensión infinita, por eso consideramos en cambio una aproximación del canal Bosónico de dimensión finita, con un error de aproximación negligible con una dimensión del canal suficientemente grande. Demostramos que una modulación de fase de los estados coherentes constituye un código quasi-perfecto para el canal aproximado y, por lo tanto es prácticamente óptimo para el canal Bosónico. La última parte del trabajo se centra en códigos estabilizadores de corrección de errores, que son códigos prácticos que usan redundancia para proteger contra errores. La corrección de errores no se usa específicamente para transmitir información clásica, sino para proteger la información cuántica. Aún así, es posible que requiramos obtener información clásica de un estado cuántico después de usar corrección de errores. Para estos casos, puede ser posible demostrar que un código de corrección de errores es quasi-perfecto y, por lo tanto, óptimo.
We investigate converse bounds on the average decoding error probability in finite-blocklength classical-quantum channel coding. We first present a lower bound for multiple quantum hypothesis testing in terms of pairwise trace distances and derive a corresponding fidelity bound. We then obtain a spectrum-based converse that depends only on the a priori probabilities and spectra of the states. We show that the converse bound remains tight for the quantum depolarizing channel under suitable conditions. For codes with product-state outputs, this converse takes an explicit form involving products of output-state eigenvalues. We apply it to binary codes over the quantum amplitude damping channel using the input states ∣+⟩\vert+\rangle∣+⟩ and ∣−⟩\vert-\rangle∣−⟩. For this setting, we also discuss a normal approximation to the spectrum-based converse in the large blocklength regime. In all numerical examples considered, the spectrum-based converse is tighter than the other converse bounds at low noise levels.