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Journal of Physics Communications

The n-shot classical capacity of the quantum erasure channel

Matteo Rosati

Published Aug 1, 2024• 1 citation
Publisher

Abstract

Abstract We compute the n-shot classical capacity of the quantum erasure channel, providing upper bounds and almost-matching lower bounds for it, the latter achievable via large-minimum-distance classical linear codes for any n. The protocols are in full product form, i.e. no entanglement is needed either at the encoder or decoder to attain the capacity, and they explicitly adapt to the transition between different error regimes as the erasure probability increases. Finally, we show that our upper and lower bounds on the capacity are tighter than those obtainable from the general theory of finite-size capacity via generalized divergences.

DOI
10.1088/2399-6528/ad6f6b

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Abstract We compute the n-shot classical capacity of the quantum erasure channel, providing upper bounds and almost-matching lower bounds for it, the latter achievable via large-minimum-distance classical linear codes for any n. The protocols are in full product form, i.e. no entanglement is needed either at the encoder or decoder to attain the capacity, and they explicitly adapt to the transition between different error regimes as the erasure probability increases. Finally, we show that our upper and lower bounds on the capacity are tighter than those obtainable from the general theory of finite-size capacity via generalized divergences.
We consider the probability of decoding error, equivalently the fidelity, of finite length quantum codes assuming quantum information transmission via qubit erasure channel. First, we obtain an analytical expression on this probability via combinatorial invariants of quantum codes. Next, we derive a generalization of the MacWilliams identities for those invariants and use them to formulate a linear programming optimization problem, which allows us to obtain lower bounds on the number of unrecoverable sets of i erasures and the probability of decoding error. Our example shows that the obtain bounds can be very tight. Next, we derive upper (achievability) bounds on the probability of decoding error for finite length stabilizer and CSS codes. CSS codes have many advantages for practical applications. Our results show that CSS codes of small lengths, like 100 qubits, visibly lose in performance to generic stabilizer codes.
We determine the optimal success probability for transmitting a fixed number of classical messages through a finite number of uses of the quantum erasure channel, assisted by noiseless classical feedback and without initial shared entanglement. For an input dimension ddd, an erasure probability ppp, a blocklength nnn, and MMM equiprobable messages, the optimal success probability is equal to E[min⁡{1,dK/M}]E[\min\{1,d^K/M\}]E[min{1,dK/M}], where KKK is binomial with parameters nnn and 1−p1-p1−p. The converse allows arbitrary adaptive quantum encoders, quantum memories, and receiver instruments. Its main ingredient is an elementary dimension bound for noiseless quantum communication with classical feedback, proved by fixing the classical controls without conditioning the sender's state on the observed transcript. A classical protocol that transmits the base-ddd digits of an integer representing the message, repeating each digit until the receiver acknowledges its reception or the prescribed blocklength is reached, attains the bound for every integer MMM. We also establish a relative-majorization property of erasure-channel outputs and exactly evaluate a hypothesis-testing converse, recovering the same numerical bound without feedback. That converse need not be achievable without feedback: four uses of the qubit erasure channel and four messages give a strict gap. A product-state code employing tetrahedral qubit states nevertheless outperforms every classical binary erasure code with these parameters. We give an explicit message-size formula and show that the bounded-remainder normal approximation and the average-success strong-converse exponent are unchanged without feedback. We also determine the feedback-assisted error exponent below capacity, prove that it agrees with the no-feedback exponent above a critical rate, and give no-feedback bounds at lower rates.
In this paper, we propose an elegant and simple one-shot achievability proof for classical-quantum channel coding. The established one-shot bound admits a closed-form expression as in the celebrated Holevo–Helstrom Theorem. Namely, the average error probability of sending M messages through a classical-quantum channel is upper bounded by the minimum error of distinguishing the joint state between channel input and output against (M−1)-many products of its marginals. Our result provides the tightest achievable one-shot capacity for classical-quantum channel coding, and it improves the third-order coding rate in the asymptotic scenario. Moreover, the derived bound directly leads to asymptotic achievability results in the large deviation, small deviation, and moderate deviation regimes in a unified manner.The full version of the manuscript can be found at [arXiv:2208.02132] [1].
The aim of this work is to study the zero-error capacity of pure-state classical-quantum channels in the setting of list decoding. We provide an achievability bound for list-size two and a converse bound holding for every fixed list size. The two bounds coincide for channels whose pairwise absolute state overlaps form a positive semi-definite matrix. Finally, we discuss a remarkable peculiarity of the classical-quantum case: differently from the fully classical setting, the rate at which the sphere-packing bound diverges might not be achievable by zero-error list codes, even when we take the limit of fixed but arbitrarily large list size.
Quantum queue-channels arise naturally in the context of buffering in quantum networks, wherein the noise suffered by the quantum states depends on the time spent waiting in the buffer. Mandayam (2020) derived a simple upper bound on the classical capacity of an additive queue-channel and showed it to be achievable for the erasure and depolarizing channels. In this article, we characterize the classical capacity for the class of unital qubit queue-channels and show that a simple product (nonentangled) decoding strategy is capacity achieving. As an intermediate step, we present a simpler derivation of an explicit capacity-achieving product decoding strategy for any independent identically distributed unital qubit channel, which may be of interest. As an important special case, we also derive the capacity and optimal decoding strategies for a symmetric generalized amplitude damping queue-channel. Our results provide useful insights toward designing practical quantum communication networks and highlight the need to explicitly model the impact of buffering.
We consider a setting where a stream of qubits is processed sequentially. We derive fundamental limits on the rate at which classical information can be transmitted using qubits that decohere as they wait to be processed. Specifically, we model the sequential processing of qubits using a single server queue, and derive expressions for the classical capacity of such a quantum `queue-channel.' Focusing on two important noise models, namely the erasure channel and the depolarizing channel, we obtain explicit single-letter capacity formulas in terms of the stationary waiting time of qubits in the queue. Our capacity proof also implies that a `classical' coding/decoding strategy is optimal, i.e., an encoder which uses only orthogonal product states, and a decoder which measures in a fixed product basis, are sufficient to achieve the classical capacity of both queue-channels. Our proof technique for the converse theorem generalizes readily - in particular, whenever the underlying quantum noise channel is additive, we can obtain a single-letter upper bound on the classical capacity of the corresponding quantum queue-channel. More broadly, our work begins to quantitatively address the impact of decoherence on the performance limits of quantum information processing systems.