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arXiv:2605.15114

New approaches to almost i.i.d. information theory

Filippo Girardi, Giacomo De Palma, Ludovico Lami

quant-phmath-phSubmitted May 14, 2026• Announced May 15, 2026• 0 citations• Revised Aug 4, 2026 (v2)
v1 · May 14, 2026v2 · Aug 4, 2026
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Abstract

Independent and identically distributed (i.i.d.) states are ubiquitous in quantum information theory. However, in a practical setting, the i.i.d. assumption is too stringent, and possibly not realistic. A physically more compelling class of 'almost i.i.d.' sources was recently proposed by [Mazzola/Sutter/Renner, arXiv:2603.15792]. In this paper, we introduce two alternative definitions of almost i.i.d. states, based on the normalised quantum Wasserstein distance and on the idea of looking at the average $k$-body marginal. We explore some basic properties of these notions and prove a strict hierarchical relation among them, with Mazzola et al.'s notion being the strictest, the one based on $k$-body marginals the loosest, and the one based on the quantum Wasserstein distance in between. Strict separation is established by means of explicit examples.

Comments
25 pages, 1 figure. Sections 3.4 and 3.5 are new

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Independent and identically distributed (i.i.d.) states are ubiquitous in quantum information theory. However, in a practical setting, the i.i.d. assumption is too stringent, and possibly not realistic. A physically more compelling class of 'almost i.i.d.' sources was recently proposed by [Mazzola/Sutter/Renner, arXiv:2603.15792]. In this paper, we introduce two alternative definitions of almost i.i.d. states, based on the normalised quantum Wasserstein distance and on the idea of looking at the average kkk-body marginal. We explore some basic properties of these notions and prove a strict hierarchical relation among them, with Mazzola et al.'s notion being the strictest, the one based on kkk-body marginals the loosest, and the one based on the quantum Wasserstein distance in between. Strict separation is established by means of explicit examples.
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 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 independent and identically distributed (iid) paradigm for asymptotic communication through quantum channels describes nnn uses of a channel ΦΦΦ by Φ⊗nΦ^{\otimes n}Φ⊗n. Since exact tensor-power behaviour is an idealization, we ask whether the asymptotic capacities of ΦΦΦ remain achievable for channel sequences that are almost-iid along ΦΦΦ. We consider sequences of Choi-almost-iid channels whose normalized Choi states are Mazzola-Sutter-Renner (MSR) almost-iid along the normalized Choi state of ΦΦΦ, with defect parameters rn=o(n)r_n=o(n)rn​=o(n). In Renner's exponential de Finetti theorem for quantum states, almost-iid states appear in the approximating mixture and motivate the MSR almost-iid class. Choi-almost-iid channels play the corresponding role in an exponential de Finetti theorem for quantum channels established in a companion paper. In this paper, we prove that the universal unassisted classical and quantum capacities of the class of Choi-almost-iid channels along ΦΦΦ equal those of the channel ΦΦΦ. Here universality is within this class: for each rate below the relevant capacity, a code sequence depending on ΦΦΦ and the rate, but not on the particular channel sequence, achieves vanishing error for every sequence in the class.
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.
Classical shadow tomography has emerged as a powerful framework for predicting properties of quantum many-body systems with favorable sample complexity. Standard theoretical guarantees, however, rely on the assumption that experimental rounds are independent and identically distributed (i.i.d.). This idealization is often violated in practice, where parameter drift, environmental noise, and active feedback generate history-dependent sequences of states or channels. To address this, we introduce a robust classical shadow protocol based on a truncated mean estimator. We prove that its sample complexity for predicting properties of the time-averaged state or channel matches the standard i.i.d. scaling governed by the shadow norm, even when experimental rounds depend arbitrarily on the past. Our results establish the robustness of the shadow formalism beyond the i.i.d. regime.
We establish two concentration principles for almost i.i.d. quantum sources under the weakly almost i.i.d. condition, the broadest condition in the established hierarchy of almost i.i.d. sources. Our results therefore apply also to Wasserstein- and Mazzola-Sutter-Renner almost i.i.d. sources. Weakly almost i.i.d. sources are sequences of multipartite states whose fixed-size marginals converge, on average, to tensor powers of a reference state, while allowing arbitrary global correlations and entanglement. We prove a noncommutative weak law of large numbers for empirical observables, and a universal entropy-concentration principle showing asymptotic concentration on subspaces of exponential dimension governed by the von Neumann entropy of the reference state. These concentration principles provide a unified and conceptually transparent approach to several information-theoretic applications beyond the i.i.d. setting, including direct proofs of universal compression within classes of weakly almost i.i.d. sources sharing a common reference state and asymmetric quantum hypothesis-testing bounds. They also yield concentration of empirical local-measurement frequencies and of empirical averages of commuting single-site observables, with applications to single-site generalized Gibbs ensembles, together with bounds on smooth and spectral entropies.