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The Classical Capacity of Additive Quantum Queue-Channels Prabha Mandayam, Member, IEEE, Krishna Jagannathan, Member, IEEE, and Avhishek Chatterjee

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The Classical Capacity of Additive Quantum Queue-Channels Prabha Mandayam, Member, IEEE, Krishna Jagannathan, Member, IEEE, and Avhishek Chatterjee 1 The Classical Capacity of Additive Quantum Queue-Channels Prabha Mandayam, Member, IEEE, Krishna Jagannathan, Member, IEEE, and Avhishek Chatterjee Abstract—We consider a setting where a stream of qubits is in the context of quantum computation, subsets of qubits are processed sequentially. We derive fundamental limits on the rate often ‘idling’ in a quantum circuit, waiting their turn while at which classical information can be transmitted using qubits other qubits are undergoing gate operations. Such idle qubits that decohere as they wait to be processed. Specifically, we model the sequential processing of qubits using a single server undergo decoherence as they wait to get processed, and the queue, and derive expressions for the classical capacity of such resulting errors are modelled as ‘storage errors’ in the context a quantum ‘queue-channel.’ Focusing on two important noise of quantum fault tolerance [5]. models, namely the erasure channel and the depolarizing channel, In the context of quantum communication, such a sequential we obtain explicit single-letter capacity formulas in terms of the processing of qubits arises naturally when we study consider stationary waiting time of qubits in the queue. Our capacity proof also implies that a ‘classical’ coding/decoding strategy is optimal, the problem of entanglement distribution over quantum net- i.e., an encoder which uses only orthogonal product states, and works [6]. In this case, classical information is encoded in a decoder which measures in a fixed product basis, are sufficient pairs of entangled photons which are transmitted over lossy to achieve the classical capacity of both queue-channels. Our fibre. To sustain the entanglement over long distances, the proof technique for the converse theorem generalizes readily — photons undergo an entanglement swap operation at quantum in particular, whenever the underlying quantum noise channel is additive, we can obtain a single-letter upper bound on the repeaters that are placed at intermediate distances. Every pho- classical capacity of the corresponding quantum queue-channel. ton that arrives at a repeater has to wait for its partner to arrive More broadly, our work begins to quantitatively address the in order to be processed, in which time they may get erased impact of decoherence on the performance limits of quantum or depolarized, depending on the nature of communication information processing systems. channel as the quantum memory used to realise the quantum repeater [7]. We derive fundamental limits on the rate at which classical information can be transmitted using qubits I. INTRODUCTION that decohere as they wait to be processed. Unlike classical bits, quantum bits (or qubits) undergo To be more precise, we model the sequential processing rapid decoherence in time, due to certain unavoidable physical of a stream of qubits using a single server queue [8]. The interactions with their environment [3]. Information stored in qubits arrive to be processed at a ‘server’ according to a qubit may be completely or partially lost as the qubit deco- some stationary point process. For example, in the context heres. The nature and the rate of the noise process depends of quantum communication, qubits are generated by optical on the particular physical implementation of the quantum sources which have an inherent randomness due to the under- state, as well as other factors such as the environment and lying physical processes. The commonly used heralded single- temperature. For example, superconducting Josephson junction photon sources, for example, rely on a nonlinear physical based qubits have average coherence times that are typically interaction which is probabilistic in nature [9]. The server of the order of a few tens of microseconds [4, Table 2]. processes the qubits at a fixed average rate. The qubits undergo Decoherence or noise poses a major challenge to the scalability decoherence (leading to errors) as they wait to be processed, of quantum information processing systems — therefore, it and the probability of error/erasure of each qubit is modeled is imperative to obtain a quantitative understanding of the as a function of the time spent in the queue by that qubit. impact of decoherence on the performance limits of quantum After the processing completes, the qubits are measured and information processing systems. interpreted as classical bits. We call this system a ‘quantum In this paper, we consider a setting where a stream of queue-channel’ and characterise the classical capacity of such qubits is processed sequentially. There are several possible a quantum queue-channel. scenarios where qubits may wait to be processed. For example, P. Mandayam is with the Department of Physics, IIT Madras, Chennai, A. Related Work India (e-mail: [email protected]). K. Jagannathan and A. Chatterjee are with the Department An information theoretic notion of reliability of a queu- of Electrical Engineering, IIT Madras, Chennai , India (e-mail: ing system with state-dependent errors was introduced and fkrishnaj,[email protected]). AC acknowledges the Department of Science and Technology, Govt. of India for its support through studied in [10], where the authors considered queue-length SERB/SRG/2019/001809 and INSPIRE/04/2016/001171. dependent errors motivated mainly by human computation and The material in this paper was presented in part at the 2019 IEEE 20th crowd-sourcing. The classic paper of Anantharam and Verdu´ International Workshop on Signal Processing Advances in Wireless Com- munications (SPAWC) [1], and the National Conference on Communications considered timing channels where information is encoded in 2019 [2]. the times between consecutive information packets, and these 2 packets are subsequently processed according to some queue- channels, for any given sequence of the waiting times. Next, ing discipline [11]. Due to randomness in the sojourn times we use a general capacity upper bound proved in [12, Lemma of packets through servers, the encoded timing information is 5] for non-stationary channels, which we then simplify using distorted, which the receiver must decode. In contrast to [11], the conditional independence result from the first step. we are not concerned with information encoded in the timing 2) The Quantum Erasure Queue-Channel: The general between packets — in our work, all the information is in the upper bound proved for any additive quantum queue-channel, qubits. along with the celebrated additivity result of Holevo [13] for In a recent paper [2], we considered the queue-channel the quantum erasure channel immediately gives us a single- problem described above, and derived the capacity of an letter capacity upper bound. For the erasure queue-channel, erasure queue-channel under certain technically restrictive we show that the upper bound is indeed the same as the conditions. Specifically, [2] restricts the encoder to using only capacity expression derived in [2, Theorem 1] for the induced orthogonal product states, and the decoder measures in a fixed classical channel. That is, the capacity of the quantum erasure product basis. In this restricted setting, qubits are essentially queue-channel does not increase by allowing for entangled made to behave like ‘classical bits that decohere,’ and the channel uses by the encoder, and arbitrary measurements at the underlying quantum channel effectively simulates a classical decoder. In other words, the classical coding/decoding strategy channel known as the induced classical channel. In general, in [2] (proposed for the induced classical channel), is indeed the classical capacity of the underlying quantum channel could sufficient to realise the classical capacity of the underlying be larger than the capacity of the induced classical channel, quantum erasure queue-channel. Furthermore, we show that because the former allows for entangled channel uses and more the capacity remains the same, regardless of whether the arrival general (joint) measurements at the decoder. and departure times of the qubits (and hence their waiting As an aside, we remark that the erasure queue-channel times) is available at the decoder. treated in [2] can be used to model a multimedia-streaming In hindsight, this result may not be altogether surprising, scenario, where information packets become useless (erased) considering that the classical capacity of the memoryless after a certain time. quantum erasure channel is the same as the capacity of the classical erasure channel. In other words, a classical coding strategy is sufficient to realise the classical capacity of the B. Our Contributions (memoryless) quantum erasure channel — see [14]. In this paper, we completely characterise the classical capac- 3) The Depolarising Queue-Channel: When the underlying ity of quantum queue-channel for two important noise models, noise model is a depolarising channel, we obtain the capacity namely the erasure channel and the depolarizing channel. expression for the queue channel, assuming that the decoder Specifically, we allow for possibly entangled channel uses by has access to the sequence of arrival and departure times the encoder, and arbitrary measurements at the decoder. (and hence the waiting times) of the qubits. The upper bound Obtaining the queue-channel capacity for an arbitrary noise proof is technically similar to the erasure case, except that model involves overcoming some technical challenges. First, we appeal to the additivity result for the depolarising channel, the quantum queue-channel
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