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.
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.
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 certainN-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.
(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.
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.
A pure-loss bosonic channel is a simple model for communication over free-space or fiber-optic links. More generally, phase-insensitive bosonic channels model other kinds of noise, such as thermalizing or amplifying processes. Recent work has established the classical capacity of all of these channels, and furthermore, it is now known that a strong converse theorem holds for the classical capacity of these channels under a particular photon-number constraint. The goal of the present paper is to initiate the study of second-order coding rates for these channels, by beginning with the simplest one, the pure-loss bosonic channel. In a second-order analysis of communication, one fixes the tolerable error probability and seeks to understand the back-off from capacity for a sufficiently large yet finite number of channel uses. We find a lower bound on the maximum achievable code size for the pure-loss bosonic channel, in terms of the known expression for its capacity and a quantity called channel dispersion. We accomplish this by proving a general “one-shot” coding theorem for channels with classical inputs and pure-state quantum outputs which reside in a separable Hilbert space. The theorem leads to an optimal second-order characterization when the channel output is finite-dimensional, and it remains an open question to determine whether the characterization is optimal for the pure-loss bosonic channel.
The extrinsic information transfer (EXIT) and generalized EXIT (GEXIT) area theorems provide fundamental conservation laws that connect the rate of classical codes to the performance of maximum-a-posteriori decoding on binary memoryless channels. In this work, we generalize these area theorems to quantum error correction. We first develop a coset-entropy formalism for nested linear codes, where the message specifies a coset and encoder randomization selects a codeword within that coset. The area under the resulting coset (G)EXIT function equals the normalized dimension of the quotient. For stabilizer codes, the measured syndrome restricts the Pauli error but does not always determine its logical effect, and the remaining uncertainty is an entropy over logical correction classes. We show that this entropy equals a classical coset entropy associated with the stabilizer group and the Pauli operators that commute with it. The resulting quantum GEXIT function has area equal to the number of encoded qubits per channel use for any stabilizer code, and this applies to the depolarizing channel through its joint four-valued Pauli-error distribution. For CSS codes on the quantum erasure channel and under independent bit- and phase-flip noise, the logical correction problem separates into two binary coset-decoding problems. On the quantum erasure channel, the duality between the stabilizer group and the Pauli operators that commute with it gives a folding identity at complementary erasure probabilities. Using this and the area theorem, we derive an upper bound on the rate of code families with vanishing logical error probability that matches the capacity of the quantum erasure channel. For the depolarizing channel, we also derive a rate-dependent converse, using a connection between correction-class entropy and coherent information to establish an inequality involving the complementary channel.