One-Shot Rate Regions Bring Sgarro Coding Into the Quantum Era

Every time two devices try to compress and reconstruct correlated data, information theory quietly decides how many bits must change hands. For nearly half a century, the benchmark problem for this scenario with multiple receivers has been the Sgarro coding problem, first formulated by Andrea Sgarro in 1977 in a short paper in the IEEE Transactions on Information Theory. In that setting, a source produces data that must be compressed once and sent to several decoders, each of which holds its own side information about the source. The question is how small the compressed description can be while still allowing every decoder to reconstruct the source reliably with the help of what it already knows. A new study by Xiaomin Liu of Fujian Normal University and Zhengjun Xi of Shaanxi Normal University, published in Quantum Information Processing, revisits this classical problem and pushes it into the quantum domain, delivering precise one-shot bounds that hold even when only a single copy of the source is available.

The traditional analysis of source coding problems relies on the asymptotic i.i.d. regime, where encoders and decoders are allowed to operate on long blocks of independently and identically distributed data. In that limit, the law of large numbers smooths out statistical fluctuations, and achievable rates converge to clean expressions involving Shannon entropies and mutual informations. But the asymptotic idealization is increasingly out of step with modern practice. Quantum cryptographic systems, for example, must often work with finite blocks of quantum states, and the security of quantum key distribution depends on tight finite-key analyses of exactly the kind pioneered by Renner and collaborators. When block lengths are short, the asymptotic formulas no longer tell the whole story, and one-shot information theory takes over.

The heart of the one-shot approach is a family of refined information quantities known as smooth entropies. Unlike the Shannon entropy, which averages over the whole probability distribution, smooth min- and max-entropies capture the extreme statistical behavior of a single sample: the min-entropy measures how much randomness can be extracted with high probability, while the max-entropy quantifies how much compression is possible without exceeding a given error tolerance. The smoothing parameter allows a small, controlled amount of probability mass to be neglected, which makes these quantities tractable and well behaved. These tools, developed extensively by Renner, Tomamichel, Datta, and others, have become the standard currency for finite-blocklength analysis in both classical and quantum information theory, appearing everywhere from quantum state redistribution to hypothesis testing and channel coding.

Liu and Xi’s first contribution is a set of one-shot achievable rate bounds for the classical Sgarro problem, in which a single compressed description of the source must serve two decoders, each equipped with its own classical side information. The technical engine behind their result is what they call the one-shot maximal simultaneous code lemma. The idea behind maximal coding constructions is well established in one-shot analysis: rather than fixing a code and averaging over random choices, one builds the largest possible codebook with a small error probability and then shows that any maximal code can be converted into one of the desired size. The word simultaneous is crucial here, because the same encoded message must be decodable by both receivers at once, each using a different body of side information. Combining this lemma with smooth information quantities yields explicit bounds on how many bits the encoder must transmit so that both decoders succeed simultaneously with high probability.

The rate region that emerges has a natural interpretation. For each decoder, the required rate is governed by a smooth conditional entropy of the source given that decoder’s side information, with the smoothing parameter tied to the tolerated error probability. Because the encoder produces a single description for both decoders, the achievable region is shaped by the interplay between the two requirements: the transmitted rate must satisfy both decoders’ reconstruction demands simultaneously, and the bounds make this trade-off precise at the level of a single use of the system. This is exactly the kind of statement that asymptotic theory cannot deliver, since asymptotic rate regions describe averages over infinitely long blocks rather than the behavior of any particular finite instance.

The second, and arguably most significant, contribution extends the framework to the genuinely quantum setting: classical source coding with full quantum side information available at two decoders. Here the side information is no longer a classical random variable but a quantum system correlated with the source, so each decoder holds a quantum state whose joint state with the source carries the correlation. Analyzing this scenario requires quantum versions of the one-shot tools, and the authors prove a quantum maximal simultaneous code lemma to serve that purpose. This lemma generalizes techniques from the one-shot classical-quantum capacity literature, notably the work of Wang and Renner on one-shot classical-quantum capacity and hypothesis testing, and from the Hayashi-Nagaoka analysis of classical-quantum channel capacities. With the quantum lemma in hand, the authors derive achievable rate bounds for the two-decoder quantum side information problem, giving the quantum analogue of the classical Sgarro rate region.

The third variant addresses a practical complication: what if the decoders cannot access their quantum side information directly? In the quantum helper model considered by Liu and Xi, the quantum systems held by the helpers are not handed to the decoders. Instead, the helpers perform measurements on their quantum systems and transmit coded classical outcomes to the decoders. This turns the problem into one of measurement compression, a task with its own rich one-shot theory. By applying one-shot measurement compression techniques, the authors derive an achievable rate region for this helper variant, specifying how many bits each helper must communicate for the decoders to reconstruct the source. The model is conceptually important because it separates the quantum resources from the classical communication, reflecting situations in distributed quantum networks where quantum systems may be too fragile or too costly to distribute directly to every party that needs the information they contain.

A key consistency check for any one-shot result is its behavior in the asymptotic limit, and the new paper passes it. The authors show that when the one-shot bounds are evaluated on sequences of i.i.d. sources and the block length is allowed to grow, the smooth information quantities converge, via the fully quantum asymptotic equipartition property proved by Tomamichel, Colbeck, and Renner, to their Shannon and von Neumann entropy counterparts. In other words, the one-shot rate regions recover the corresponding asymptotic rate bounds, including the quantum Sgarro rate regions previously established by Han, Liu, and Xi in 2023. This bridging role is one of the main virtues of the one-shot framework: it provides finite-blocklength bounds that are provably compatible with the classical asymptotic theory while remaining meaningful for any single instance of the problem.

The significance of this work lies in its systematic unification. The Sgarro problem sits at the intersection of several strands of information theory: source coding with side information, multi-terminal networks as systematized by El Gamal and Kim, and the one-shot quantum information program driven by smooth Rényi entropies and hypothesis testing relative entropies. By proving both a classical and a quantum maximal simultaneous code lemma, and by handling the helper variant through measurement compression, Liu and Xi assemble a coherent toolkit that can be applied to any scenario in which one description must serve multiple quantum-assisted receivers. Related tools have already proven their worth in quantum state redistribution, where Berta, Christandl, and Touchette derived smooth entropy bounds on one-shot state merging and splitting, and in the distillation of common randomness and secret keys from quantum correlations studied by Renes and Renner.

Looking forward, one-shot rate regions of this kind are the raw material for finite-blocklength analyses of realistic quantum communication protocols. As quantum networks mature, the questions they raise increasingly concern single instances: a single distribution round, a single sensor measurement, a single entanglement distillation attempt. Bounds expressed in smooth entropies translate directly into statements about error probabilities and resource costs for those single instances, which is precisely what protocol designers need. The work of Liu and Xi, supported by the National Natural Science Foundation of China, extends the reach of that program to the multi-decoder Sgarro setting in both its classical and quantum forms, and its helper variant hints at architectures for quantum networks in which measurement, rather than state distribution, is the scarce resource. The 1977 problem posed by Sgarro has now been given a fully quantum, fully finite-blocklength treatment, and the rate regions that result are likely to inform the design of the next generation of quantum communication schemes.

Subject of Research: One-shot achievable rate regions for classical and quantum source coding with side information at multiple decoders

Article Title: One-shot achievable rate regions for classical and quantum Sgarro coding

Article References: Liu, X., & Xi, Z. (2026). One-shot achievable rate regions for classical and quantum Sgarro coding. Quantum Information Processing, 25(10), Article 334. https://doi.org/10.1007/s11128-026-05361-4

Image Credits: AI Generated

DOI: 10.1007/s11128-026-05361-4

Keywords: one-shot information theory, Sgarro coding, source coding with side information, smooth Rényi entropy, quantum side information, maximal simultaneous codes, measurement compression, finite blocklength, quantum information processing, rate regions, quantum helper, asymptotic equipartition property