The asymptotic rates of information-theoretic protocols - including error exponents, data-compression rates, and channel capacities - are traditionally derived under the idealised assumption that the underlying resources are independent and identically distributed (i.i.d.). Somewhat surprisingly, even slight departures from the exact i.i.d. structure can drastically alter the asymptotic behaviour predicted by the i.i.d. theory. If the precise nature of the perturbation is known, for instance in the case of a pointwise defect, one can design a bespoke protocol that compensates for it, e.g. by discarding the corrupted subsystem. In realistic physical settings, however, exact i.i.d. behaviour cannot be guaranteed, and deviations from the ideal regime cannot generally be identified precisely. This raises a fundamental question: which notions of almost i.i.d. structure are sufficiently robust to preserve the asymptotic predictions of quantum Shannon theory? We investigate this question for three central information-theoretic tasks: asymmetric hypothesis testing, classical and quantum data compression, and classical communication through quantum channels. Rather than designing protocols tailored to specific defects, we seek robust protocols that remain asymptotically optimal and that are universal within a broad class of almost i.i.d. resources whose precise deviations from the ideal regime are unknown. To this end, we study three inequivalent notions of almost i.i.d. structure, and determine which of them preserve the asymptotic rates and error exponents predicted by the i.i.d. theory. Along the way, we introduce the notion of an almost i.i.d. process and a new distance measure between quantum channels - the club distance - designed to capture stability under local perturbations. These notions may be of independent interest.
The asymptotic rates of information-theoretic protocols - including error exponents, data-compression rates, and channel capacities - are traditionally derived under the idealised assumption that the underlying resources are independent and identically distributed (i.i.d.). Somewhat surprisingly, even slight departures from the exact i.i.d. structure can drastically alter the asymptotic behaviour predicted by the i.i.d. theory. If the precise nature of the perturbation is known, for instance in the case of a pointwise defect, one can design a bespoke protocol that compensates for it, e.g. by discarding the corrupted subsystem. In realistic physical settings, however, exact i.i.d. behaviour cannot be guaranteed, and deviations from the ideal regime cannot generally be identified precisely. This raises a fundamental question: which notions of almost i.i.d. structure are sufficiently robust to preserve the asymptotic predictions of quantum Shannon theory? We investigate this question for three central information-theoretic tasks: asymmetric hypothesis testing, classical and quantum data compression, and classical communication through quantum channels. Rather than designing protocols tailored to specific defects, we seek robust protocols that remain asymptotically optimal and that are universal within a broad class of almost i.i.d. resources whose precise deviations from the ideal regime are unknown. To this end, we study three inequivalent notions of almost i.i.d. structure, and determine which of them preserve the asymptotic rates and error exponents predicted by the i.i.d. theory. Along the way, we introduce the notion of an almost i.i.d. process and a new distance measure between quantum channels - the club distance - designed to capture stability under local perturbations. These notions may be of independent interest.
The generalised quantum Stein's lemma is a key result in quantum hypothesis testing, and connects this fundamental primitive of quantum information processing with quantum resource manipulation, a task that is central for technological applications. Prior works have proved this statement in the idealised setting of independent and identically distributed (i.i.d.) sequences of quantum states, and recent extensions consider also sources that are 'close' to i.i.d., according to the strict notion put forth by Mazzola, Sutter, and Renner. For several applications, however, one would need to consider yet more general sources. We establish a version of the generalised quantum Stein's lemma that is conceptually much simpler and general, as it applies to any source that is asymptotically close to an i.i.d. state with respect to the normalised quantum Wasserstein distance of order 1. As an immediate consequence, we solve the Stein exponent of a scenario where the null hypothesis is arbitrarily varying, expressing it in terms of i.i.d. Stein exponents corresponding to arbitrary states in the convex hull of the null hypothesis base set.
The Quantum Reverse Shannon Theorem has been a milestone in quantum information theory. It states that asymptotically reliable simulation of a quantum channel, assisted by unlimited shared entanglement, requires a rate of classical communication equal to the channel’s entanglement-assisted classical capacity. In this paper, we study the error exponent of quantum channel simulation, which characterizes the optimal speed of exponential convergence of the performance towards the perfect, as the blocklength increases. Based on channel purified distance, we derive lower and upper bounds for the error exponent. Then we show that the two bounds coincide when the classical communication rate is below a critical value, and hence, we have determined the exact formula of the error exponent in the low-rate case. This enables us to obtain an operational interpretation to the channel’s sandwiched Rényi information of order from 1 to 2, since our formula is expressed as a transform of this quantity. In the derivation, we have also obtained an achievability bound for quantum channel simulation in the finite-blocklength setting, which is of realistic significance.
Information-theoretic techniques are based on the assumption that resources are well characterized by independent and identically distributed (iid) states. This assumption cannot be justified operationally, since, for example, correlations between subsequent systems emitted by a source cannot be detected by any practical tomographic protocol. Operationally motivated symmetry assumptions still imply, via de Finetti theorems, that the resources are described by almost-iid states. This raises the question: Are almost-iid resources as effective as perfect iid resources for information-processing tasks? Here we address this question and prove that the conditional entropy of almost-iid states asymptotically coincides with that of iid states. As an application, this implies that squashed entanglement is robust for almost-iid states, asymptotically matching its value on iid states.
A fundamental quantity of interest in Shannon theory, classical or quantum, is the error exponent of a given channel W and rate R: the constantE(W,R)which governs the exponential decay of decoding error when using ever larger optimal codes of fixed rate R to communicate over ever more (memoryless) instances of a given channel W. Nearly matching lower and upper bounds are well-known for classical channels. Here I show a lower bound on the error exponent of communication over arbitrary classical-quantum (CQ) channels which matches Dalai’s sphere-packing upper bound for rates above a critical value, exactly analogous to the case of classical channels. This proves a conjecture made by Holevo in his investigation of the problem. Unlike the classical case, however, the argument does not proceed via a refined analysis of a suitable decoder, but instead by leveraging a bound by Hayashi on the error exponent of the cryptographic task of privacy amplification. This bound is then related to the coding problem via tight entropic uncertainty relations and Gallager’s method of constructing capacity-achieving parity-check codes for arbitrary channels. Along the way, I find a lower bound on the error exponent of the task of compression of classical information relative to quantum side information that matches the sphere-packing upper bound of Cheng et al. In turn, the polynomial prefactors to the sphere-packing bound found by Cheng et al. may be translated to the privacy amplification problem, sharpening a recent result by Li, Yao, and Hayashi, at least for linear randomness extractors.
We study the robustness of asymptotic entanglement manipulation for Mazzola--Sutter--Renner (MSR) almost-iid sources, which allow a sublinear number of deviations from tensor-power structure. For pure sources along ∣φ⟩AB, every concentration rate below S(φA) is achievable by a single Schur--Weyl protocol depending only on the rate and local Hilbert spaces. For general sources along ρAB, every distillation rate below ED(ρAB) is achievable by a single LOCC protocol sequence independent of the particular source and defect size. Its error vanishes exponentially and uniformly for each prescribed defect bound rn=o(n), yielding an optimal universal distillation rate equal to ED(ρAB). For dilution, the entanglement cost of every MSR almost-iid target sequence along ρAB is at most EF∞(ρAB), the regularized entanglement of formation of the reference state. Consequently, every MSR almost-iid sequence along a pure reference state ∣φ⟩AB is asymptotically reversible, with distillable entanglement and entanglement cost equal to S(φA), even when its individual states are mixed. The proofs establish structural and entropic properties of MSR almost-iid sources that may be useful in other information-theoretic settings.
A recurring structural pattern appears across three apparently distinct areas of quantum information theory: the asymptotic rates at which quantum resources can be converted, concentrated, or distilled depend sensitively on the *operational class* of allowed processes — and this sensitivity is not captured by the leading-order rate alone. This paper synthesizes findings from quant-ph and cond-mat.stat-mech preprints to argue, as a heuristic reading rather than a derivation, that a shared geometry underlies asymptotic resource conversion: the relevant figure of merit is not the optimal rate but the *reliability* or *precision* of that rate, and different operational classes produce characteristically different reliability profiles even when their leading rates coincide. Specifically, we draw on five corpus findings: (1) the demonstration that Gibbs-preserving and thermal operations achieve identical leading-order work-extraction rates but are distinguished by Petz versus sandwiched Rényi divergences at the level of error-suppression speed [corpus:arxiv:2606.06318]; (2) the proof that almost-i.i.d. entanglement manipulation is robust at the rate level but may require source-dependent protocols at the distillation level [corpus:arxiv:2606.06392]; (3) the discovery that magic state distillation protocols can achieve near-linear asymptotic rates while carrying overhead exponents larger than one, decoupling rate from overhead in a previously unappreciated way [corpus:arxiv:2605.30108]; (4) the universal thermokinetic decomposition showing that energy dissipation suppresses *fluctuations* in information predictability rather than raising its mean [corpus:arxiv:2605.29957]; and (5) the finding that excessive entanglement can *reduce* channel discriminability, establishing that resource abundance does not monotonically improve task performance [corpus:arxiv:2605.31519]. The falsification path is concrete: if the heuristic is correct, then for any resource theory admitting two operational classes with equal leading-order rates, the error-exponent functions must be distinguishable by a divergence-class argument analogous to Petz vs. sandwiched. A computation showing that two such classes share both their leading rate *and* their full error-exponent function would falsify the central claim. --- Authorship: Saluca Agentic AI Research Team (Saluca LLC). AI-drafted from arXiv preprint corpus on the date in the filename. Cited arXiv preprints: 2605.29957, 2605.30108, 2605.30199, 2605.30558, 2605.30806, 2605.31379, 2605.31519, 2606.06107, 2606.06318, 2606.06392, 2606.06455, 2606.06456