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arXiv:1310.2856 · IEEE Transactions on Information Theory

Quantum Subdivision Capacities and Continuous-Time Quantum Coding

Alexander Müller‐Hermes, David Reeb, Michael M. Wolf

Published Nov 5, 2014• 13 citations
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Abstract

Quantum memories can be regarded as quantum channels that transmit information through time without moving it through space. Aiming at a reliable storage of information, we may thus not only encode at the beginning and decode at the end, but also intervene during the transmission-a possibility not captured by the ordinary capacities in quantum Shannon theory. In this paper, we introduce capacities that take this possibility into account and study them, in particular, for the transmission of quantum information via dynamical semigroups of Lindblad form. When the evolution is subdivided and supplemented by additional continuous semigroups acting on arbitrary block sizes, we show that the capacity of the ideal channel can be obtained in all cases. If the supplementary evolution is reversible, however, this is no longer the case. Upper and lower bounds for this scenario are proven. Finally, we provide a continuous coding scheme and simple examples showing that adding a purely dissipative term to a Liouvillian can sometimes increase the quantum capacity.

DOI
10.1109/tit.2014.2366456

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Quantum memories can be regarded as quantum channels that transmit information through time without moving it through space. Aiming at a reliable storage of information, we may thus not only encode at the beginning and decode at the end, but also intervene during the transmission-a possibility not captured by the ordinary capacities in quantum Shannon theory. In this paper, we introduce capacities that take this possibility into account and study them, in particular, for the transmission of quantum information via dynamical semigroups of Lindblad form. When the evolution is subdivided and supplemented by additional continuous semigroups acting on arbitrary block sizes, we show that the capacity of the ideal channel can be obtained in all cases. If the supplementary evolution is reversible, however, this is no longer the case. Upper and lower bounds for this scenario are proven. Finally, we provide a continuous coding scheme and simple examples showing that adding a purely dissipative term to a Liouvillian can sometimes increase the quantum capacity.
Quantum memories can be regarded as quantum channels that transmit information through time without moving it through space. Aiming at a reliable storage of information we may thus not only encode at the beginning and decode at the end, but also intervene during the transmission – a possi-bility not captured by the ordinary capacities in Quantum Shannon Theory. In this work we introduce capacities that take this possibility into account and study them in particular for the transmission of quantum information via dynamical semigroups of Lindblad form. When the evolution is subdivided and supplemented by additional continuous semigroups acting on arbitrary block sizes, we show that the capacity of the ideal channel can be obtained in all cases. If the supplementary evolution is reversible, however, this is no longer the case. Upper and lower bounds for this scenario are proven. Fi-nally, we provide a continuous coding scheme and simple examples showing that adding a purely dissipative term to a Liouvillian can sometimes increase
We study the information transmission capacities of quantum Markov semigroups acting on finite-dimensional quantum systems. We show that, in the limit of infinite time, the capacities can be efficiently computed in terms of the structure of the peripheral space of the semigroup, are strongly additive, and satisfy the strong converse property. We also establish convergence bounds to show that the infinite-time capacities are reached after time scaling quadratically with the system dimension. From a data storage perspective, our analysis provides tight bounds on the number of bits or qubits that can be reliably stored for long times in a quantum memory experiencing Markovian noise. From a practical standpoint, we show that typically, a quantum memory with Markovian noise acting independently and identically on all qubits and a fixed time-independent global error correction mechanism becomes useless for storage after time scaling exponentially with the number of qubits. In contrast, if the error correction is local, the memory becomes useless much more quickly, i.e., after time scaling logarithmically with the number of qubits. In the setting of point-to-point communication between two spatially separated parties, our analysis provides efficiently computable bounds on the optimal rate at which bits or qubits can be reliably transmitted via long Markovian communication channels, both in the finite block-length and asymptotic regimes.
A basic question in quantum information theory is to quantify the optimal rate of transmitting quantum information via a given quantum channel using certain coding schemes. We study this question for discrete- and continuous-time quantum Markov processes. For continuous-time processes modeling a quantum memory we introduce new capacities where coding channels may be applied during the storage time. We prove upper bounds on these quantities by studying the entropy production of quantum Markov processes.
Shannon's channel coding theorem describes the maximum possible rate of reliable information transfer through a classical noisy communication channel. It, together with the source coding theorem, characterizes lossless channel communication in the classical regime. Lossy compression scenarios require the additional description provided by rate-distortion theory, which characterizes the trade-off between compression rate and the distortion of the compressed signal. Even in this context, the capacity characterizes the usefulness of a channel -- a channel with more capacity will always outperform a channel with less capacity. We show that this is no longer true when sending classical information over a quantum channel. In particular, we find a pair of quantum channels where the channel with the lower capacity causes less distortion than the higher capacity channel when both are used at a fixed rate.
Quantum memory systems are vital in quantum information processing for dependable storage and retrieval of quantum states. Inspired by classical reliability theories that synthesize reliable computing systems from unreliable components, we formalize the problem of reliable storage of quantum information using noisy components. We introduce the notion of stable quantum memories and define the storage rate as the ratio of the number of logical qubits to the total number of physical qubits as well as the circuit complexity of the decoder, which includes both quantum gates and measurements. We demonstrate that a strictly positive storage rate can be achieved by constructing a quantum memory system with quantum expander codes. Moreover, by reducing the reliable storage problem to reliable quantum communication, we provide upper bounds on the achievable storage capacity. In the case of physical qubits corrupted by noise satisfying hypercontractivity conditions, we provide a tighter upper bound on storage capacity using an entropy dissipation argument. Furthermore, observing that the time complexity of the decoder scales nontrivially with the number of physical qubits, achieving asymptotic rates may not be possible due to the induced dependence of the noise on the number of physical qubits. In this constrained nonasymptotic setting, we derive upper bounds on storage capacity using finite blocklength communication bounds. Finally, we numerically analyze the gap between upper and lower bounds in both asymptotic and nonasymptotic cases, and provide suggestions to tighten the gap.
Quantum capacity, as the key figure of merit for a given quantum channel, upper bounds the channel's ability in transmitting quantum information. Identifying different types of channels, evaluating the corresponding quantum capacity, and finding the capacity-approaching coding scheme are the major tasks in quantum communication theory. Quantum channel in discrete variables has been discussed enormously based on various error models, while error model in the continuous variable channel has been less studied due to the infinite dimensional problem. In this paper, we investigate a general continuous variable quantum erasure channel. By defining an effective subspace of the continuous variable system, we find a continuous variable random coding model. We then derive the quantum capacity of the continuous variable erasure channel in the framework of decoupling theory. The discussion in this paper fills the gap of a quantum erasure channel in continuous variable setting and sheds light on the understanding of other types of continuous variable quantum channels.