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FG\(^\diamond\), Jacopo Rizzo\(^\diamond\), Leah Turner\(^\diamond\), Gerardo Adesso, Ludovico Lami
arXiv:2609.40307
A central yet mysterious problem in quantum information is the asymptotic distinguishability of quantum states under restricted measurements. Here we look at asymptotic asymmetric hypothesis testing for general multi-mode bosonic Gaussian states restricted to arbitrary collective Gaussian measurements. We establish a single-letter formula for the Stein exponent for many pairs of states, featuring two distinct contributions, corresponding to first and second moments, respectively. Remarkably, the Gaussian-measured relative entropy exhibits genuine non-additivity, but the only effect of additivity violations is to asymptotically decouple first and second moments. We prove this by establishing sufficient conditions for additivity of the second-moment term via novel log-determinant matrix inequalities. Our results not only identify a single-letter expression for the Gaussian Stein exponent in a wide range of practical situations, but also rule out any possible advantage from collective Gaussian measurements when considering centred Gaussian states.
FG, Kuan-Yi Lee, Masahito Hayashi, Ludovico Lami
arXiv:2609.17309
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.
FG, Nilanjana Datta, Giacomo De Palma, Ludovico Lami
arXiv:2605.18726
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.
Lukas Schmitt, FG, Laura Burri
arXiv:2603.17005
We study a doubly minimized variant of the lautum information - a reversed analogue of the mutual information - defined as the minimum relative entropy between any product state and a fixed bipartite quantum state; we refer to this measure as the tumula information. In addition, we introduce the corresponding Petz Rényi version, which we call the doubly minimized Petz Rényi lautum information (PRLI). We derive several general properties of these correlation measures and provide an operational interpretation in the context of hypothesis testing. Specifically, we show that the reverse direct exponent of certain binary quantum state discrimination problems is quantified by the doubly minimized PRLI of order \(\alpha \in (0,1/2)\), and that the Sanov exponent is determined by the tumula information. Furthermore, we investigate the extension of the tumula information to channels and compare its properties with previous results on the channel umlaut information [Girardi et al., arXiv:2503.21479].
Marco Dalai, FG, Ludovico Lami
2026 IEEE International Symposium on Information Theory (ISIT)
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.
FG\(^\ast\), Francesco Anna Mele\(^\ast\), Haimeng Zhao\(^\ast\), Marco Fanizza\(^\dagger\), Ludovico Lami\(^\dagger\)
arXiv:2512.20599
The recently introduced random purification channel, which converts $n$ copies of an arbitrary mixed quantum state into $n$ copies of the same uniformly random purification, has emerged as a powerful tool in quantum information theory. Motivated by this development, we introduce a channel-level analogue, which we call the random Stinespring superchannel. This consists in a procedure to transform \(n\) parallel queries of an arbitrary quantum channel into \(n\) parallel queries of the same uniformly random Stinespring isometry, via universal encoding and decoding operations that are efficiently implementable. When the channel is promised to have Choi rank at most \(r\), the procedure can be tailored to yield a Stinespring environment of dimension \(r\). We present two applications of the random Stinespring superchannel, one in quantum Shannon theory and one in quantum learning theory. In quantum Shannon theory, we prove a channel-level analogue of Uhlmann's theorem for quantum divergences. In quantum learning theory, our construction shows that tomography of quantum channels reduces to tomography of isometries. This yields a simple channel learning algorithm, based on existing isometry learning protocols, that matches the performance of the two recently proposed channel tomography algorithms. Complementarily, whereas the optimality of these algorithms had previously been established only up to a logarithmic factor in the dimension, we close this gap by removing this logarithmic factor from the lower bound. Taken together, our results fully establish the optimality of these recently introduced channel learning algorithms, showing that the optimal query complexity of learning a quantum channel with input dimension \(d_A\), output dimension \(d_B\), and Choi rank \(r\) is \(\Theta(d_A d_B r)\).
FG, Francesco Anna Mele, Ludovico Lami
arXiv:2511.23451
The recently introduced random purification channel, which converts \(n\) i.i.d. copies of any mixed quantum state into a uniform convex combination of \(n\) i.i.d. copies of its purifications, has proved to be an extremely useful tool in quantum learning theory. Here we give a remarkably simple construction of this channel, making its known properties -- and several new ones -- immediately transparent. In particular, we show that the channel also purifies non-i.i.d. states: it transforms any permutationally symmetric state into a uniform convex combination of permutationally symmetric purifications, each differing only by a tensor-product unitary acting on the purifying system. We then apply the channel to give a one-line proof of (a stronger version of) the recently established Uhlmann's theorem for quantum divergences.
FG, Aadil Oufkir, Bartosz Regula, Marco Tomamichel, Mario Berta, Ludovico Lami
arXiv:2503.21479
We study the quantum umlaut information, a correlation measure defined for bipartite quantum states \(\rho_{AB}\) as a reversed variant of the quantum mutual information: \(U(A;B)_\rho=\min_{\sigma_{B}}D(\rho_A\otimes\sigma_B\|\rho_{AB})\) in terms of the quantum relative entropy \(D\). As in the classical case [Girardi et al., arXiv:2503.18910], this definition allows for a closed-form expression and has an operational interpretation as the asymptotic error exponent in the hypothesis testing task of deciding whether a given bipartite state is product or not. We generalise the umlaut information to quantum channels, where it also extends the notion of 'oveloh information' [Nuradha et al., arXiv:2404.16101]. We prove that channel umlaut information is additive for classical-quantum channels, while we observe additivity violations for fully quantum channels. Inspired by recent results in entanglement theory, we then show as our main result that the regularised umlaut information constitutes a fundamental measure of the quality of classical information transmission over a quantum channel - as opposed to the capacity, which quantifies the quantity of information that can be sent. This interpretation applies to coding assisted by activated non-signalling correlations, and the channel umlaut information is in general larger than the corresponding expression for unassisted communication as obtained by Dalai for the classical-quantum case [IEEE Trans. Inf. Theory 59, 8027 (2013)]. Combined with prior works on non-signalling-assisted zero-error channel capacities, our findings imply a dichotomy between the settings of zero-rate error exponents and zero-error communication. While our results are single-letter only for classical-quantum channels, we also give a single-letter bound for fully quantum channels in terms of the 'geometric' version of umlaut information.
FG, Aadil Oufkir, Bartosz Regula, Marco Tomamichel, Mario Berta, Ludovico Lami
2026 IEEE International Symposium on Information Theory (ISIT)
The sphere-packing bound quantifies the error exponent for noisy channel coding for rates above a critical value. Here, we study the zero-rate limit of the sphere-packing bound and show that it has an intriguing single-letter form, which we call the umlaut information of the channel, inspired by the lautum information introduced by Palomar and Verdú. Unlike the latter quantity, we show that the umlaut information is additive for parallel uses of channels. We show that it has a twofold operational interpretation: as the zero-rate error exponent of non-signalling-assisted coding on the one hand, and as the zero-rate error exponent of list decoding in the large list limit on the other.