arXiv subjects

Giacomo De Palma

Publications and source records attributed to Giacomo De Palma.

At least 19 recordsLinked to original sources

arXiv2026

Convex combinations of bosonic pure-loss channels

The pure-loss channel is a fundamental noise model for bosonic quantum platforms, characterised by a single parameter, the transmissivity. In realistic scenarios such as free-space quantum communication, the transmissivity fluctuates from one channel use to another, and the channel is a convex combination of pure-loss channels, known as a fading channel. Despite its practical relevance, its quantum Shannon theory has remained largely unexplored. Here we investigate degradability, anti-degradability, entanglement breakingness, and capacities of the fading channel. We prove that entanglement distribution and quantum key distribution can be achieved at a strictly positive rate over any fading channel that is not completely noisy. When the transmissivity takes a finite set of values, we determine the energy-unconstrained two-way quantum and secret-key capacities exactly, as the averages of those of the pure-loss components. We prove that thermal states, optimal for a broad class of static bosonic Gaussian channels, do not in general achieve the entanglement-assisted classical capacity of fading channels: for a binary fading model we derive the capacity-achieving state in closed form, and we exhibit channels for which non-Gaussian Fock-diagonal states strictly outperform every Gaussian encoding. For the quantum capacity, we give a simple sufficient condition on the transmissivity distribution under which weak thermal inputs yield a strictly positive rate. Outside this condition, we numerically identify parameter regions where no thermal input compatible with the energy constraint has a positive coherent information, while optimized non-Gaussian inputs do. For general fading distributions, we design an iterative variational algorithm to optimize the coherent and mutual information. Our work advances the study of quantum communication in the non-Gaussian regime.

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arXiv2026

Lazy training of quantum physics informed neural networks

We study the gradient-flow training dynamics of quantum physics-informed neural networks (QPINNs) for the solution of second-order elliptic partial differential equations with Dirichlet boundary conditions. We consider parameterized quantum circuits as function approximators and analyze their overparameterized regime through the lens of the neural tangent kernel (NTK). Our contribution is a nonasymptotic lazy-training theory for QPINNs and their variational formulation: we prove that, for sufficiently large circuit width, the nonlinear gradient flow is quantitatively approximated by a linearized NTK model, with explicit bounds depending on the number of qubits, circuit depth, circuit light-cone geometry, and the dimension of the domain of the solution to the PDE.

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arXiv2026

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

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.

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arXiv2026

Quantum Shannon theory made robust: a tale of three protocols for almost i.i.d. sources

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.

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arXiv2026

Trained quantum neural networks are Gaussian processes

We study quantum neural networks made by parametric one-qubit gates and fixed two-qubit gates in the limit of infinite width, where the generated function is the expectation value of the sum of single-qubit observables over all the qubits. First, we prove that the probability distribution of the function generated by the untrained network with randomly initialized parameters converges in distribution to a Gaussian process whenever each measured qubit is correlated only with few other measured qubits. Then, we analytically characterize the training of the network via gradient descent with square loss on supervised learning problems. We prove that, as long as the network is not affected by barren plateaus, the trained network can perfectly fit the training set and that the probability distribution of the function generated after training still converges in distribution to a Gaussian process. Finally, we consider the statistical noise of the measurement at the output of the network and prove that a polynomial number of measurements is sufficient for all the previous results to hold and that the network can always be trained in polynomial time.

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arXiv2026

Efficient classical computation of the neural tangent kernel of quantum neural networks

We propose an efficient classical algorithm to estimate the Neural Tangent Kernel (NTK) associated with a broad class of quantum neural networks. These networks consist of arbitrary unitary operators belonging to the Clifford group interleaved with parametric gates given by the time evolution generated by an arbitrary Hamiltonian belonging to the Pauli group. The proposed algorithm leverages a key insight: the average over the distribution of initialization parameters in the NTK definition can be exactly replaced by an average over just four discrete values, chosen such that the corresponding parametric gates are Clifford operations. This reduction enables an efficient classical simulation of the circuit. Combined with recent results establishing the equivalence between wide quantum neural networks and Gaussian processes [Girardi \emph{et al.}, Comm. Math. Phys. 406, 92 (2025); Melchor Hernandez \emph{et al.}, Ann. Henri Poincar{é} (2025)], our method enables efficient computation of the expected output of wide, trained quantum neural networks, and therefore shows that such networks cannot achieve quantum advantage.

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arXiv2026

Mean-field limit from general mixtures of experts to quantum neural networks

In this work, we study the asymptotic behavior of Mixture of Experts (MoE) trained via gradient flow on supervised learning problems. Our main result establishes the propagation of chaos for a MoE as the number of experts diverges. We demonstrate that the corresponding empirical measure of their parameters is close to a probability measure that solves a nonlinear continuity equation, and we provide an explicit convergence rate that depends solely on the number of experts. We apply our results to a MoE generated by a quantum neural network.

math-ph↗

arXiv2026

Failure of the mean-field Hartree approximation for a bosonic many-body system with non-Hermitian Hamiltonian

Mean-field Hartree theory is a central tool for reducing interacting many-body dynamics to an effective nonlinear one-particle evolution. This approximation has been employed also when the Hamiltonian that governs the many-body dynamics is not Hermitian. Indeed, non-Hermitian Hamiltonians model particle gain/loss or the evolution of open quantum systems between consecutive quantum jumps. Furthermore, the validity of the Hartree approximation for generic non-Hermitian Hamiltonians lies at the basis of a quantum algorithm for nonlinear differential equations. In this work, we show that this approximation can fail. We analytically solve a model of $N$ bosonic qubits with two-body interactions generated by a purely anti-Hermitian Hamiltonian, determine an analytic expression for the limit for $N\to\infty$ of the one-particle marginal state and show that such a limit does not agree with the solution of the non-Hermitian Hartree evolution equation. We further show that there exists an initial condition such that the exact one-particle marginal state undergoes a finite-time transition to a mixed state, a phenomenon that is completely absent in the case of Hermitian Hamiltonians. Our findings challenge the validity of the mean-field Hartree approximation for non-Hermitian Hamiltonians, and call for additional conditions for the validity of the mean-field regime to model the dynamics of particle gain and loss and the open-system dynamics in bosonic many-body systems.

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arXiv2026

Exponential concentration of fluctuations in mean-field boson dynamics

We study the mean-field dynamics of a system of $N$ interacting bosons starting from an initially condensated state. For a broad class of mean-field Hamiltonians, including models with arbitrary bounded interactions and models with unbounded interaction potentials, we prove that the probability of having $n$ particles outside the condensate decays exponentially in $n$ for any finite evolution time. Our results strengthen previously known bounds that provide only polynomial control on the probability of having $n$ excitations.

math-ph↗

arXiv2025

Non-iid hypothesis testing: from classical to quantum

We study hypothesis testing (aka state certification) in the non-identically distributed setting. A recent work (Garg et al. 2023) considered the classical case, in which one is given (independent) samples from $T$ unknown probability distributions $p_1, \dots, p_T$ on $[d] = \{1, 2, \dots, d\}$, and one wishes to accept/reject the hypothesis that their average $p_{\mathrm{avg}}$ equals a known hypothesis distribution $q$. Garg et al. showed that if one has just $c = 2$ samples from each $p_i$, and provided $T \gg \frac{\sqrt{d}}{ε^2} + \frac{1}{ε^4}$, one can (whp) distinguish $p_{\mathrm{avg}} = q$ from $d_{\mathrm{TV}}(p_{\mathrm{avg}},q) > ε$. This nearly matches the optimal result for the classical iid setting (namely, $T \gg \frac{\sqrt{d}}{ε^2}$). Besides optimally improving this result (and generalizing to tolerant testing with more stringent distance measures), we study the analogous problem of hypothesis testing for non-identical quantum states. Here we uncover an unexpected phenomenon: for any $d$-dimensional hypothesis state $σ$, and given just a single copy ($c = 1$) of each state $ρ_1, \dots, ρ_T$, one can distinguish $ρ_{\mathrm{avg}} = σ$ from $D_{\mathrm{tr}}(ρ_{\mathrm{avg}},σ) > ε$ provided $T \gg d/ε^2$. (Again, we generalize to tolerant testing with more stringent distance measures.) This matches the optimal result for the iid case, which is surprising because doing this with $c = 1$ is provably impossible in the classical case. We also show that the analogous phenomenon happens for the non-iid extension of identity testing between unknown states. A technical tool we introduce may be of independent interest: an Efron-Stein inequality, and more generally an Efron-Stein decomposition, in the quantum setting.

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arXiv2025

The multimode conditional quantum Entropy Power Inequality and the squashed entanglement of the multimode extreme bosonic Gaussian channels

We prove the multimode conditional quantum Entropy Power Inequality for bosonic quantum systems. This inequality determines the minimum conditional von Neumann entropy of the output of the most general linear mixing of bosonic quantum modes among all the input states of the modes with given conditional entropies. Bosonic quantum systems constitute the mathematical model for the electromagnetic radiation in the quantum regime, which provides the most promising platform for quantum communication and quantum key distribution. We apply our multimode conditional quantum Entropy Power Inequality to determine new lower bounds to the squashed entanglement of a large family of bosonic Gaussian states. The squashed entanglement is one of the main entanglement measures in quantum communication theory, providing the best known upper bound to the distillable key. Exploiting this result, we determine a new lower bound to the squashed entanglement of the multimode bosonic Gaussian channels that are extreme, i.e., that cannot be decomposed as a non-trivial convex combination of quantum channels. The squashed entanglement of a quantum channel provides an upper bound to its secret-key capacity, i.e., the capacity to generate a secret key shared between the sender and the receiver. Lower bounds to the squashed entanglement are notoriously hard to prove. Our results contribute to break this barrier and will stimulate further research in the field of quantum communication with bosonic quantum systems.

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arXiv2025

Quantum concentration inequalities and equivalence of the thermodynamical ensembles: an optimal mass transport approach

We prove new concentration inequalities for quantum spin systems which apply to any local observable measured on any product state or on any state with exponentially decaying correlations. Our results do not require the spins to be arranged in a regular lattice, and cover the case of observables that contain terms acting on spins at arbitrary distance. Moreover, we introduce a local W1 distance, which quantifies the distinguishability of two states with respect to local observables. We prove a transportation-cost inequality stating that the local W1 distance between a generic state and a state with exponentially decaying correlations is upper bounded by a function of their relative entropy. Finally, we apply such inequality to prove the equivalence between the canonical and microcanonical ensembles of quantum statistical mechanics and the weak eigenstate thermalization hypothesis for the Hamiltonians whose Gibbs states have exponentially decaying correlations.

math-ph↗

arXiv2025

Solving MNIST with a globally trained Mixture of Quantum Experts

We propose a new quantum neural network for image classification, which is able to classify the parity of the MNIST dataset with full resolution with a test accuracy of up to 97.5% without any classical pre-processing or post-processing. Our architecture is based on a mixture of experts whose model function is the sum of the model functions of each expert. We encode the input with amplitude encoding, which allows us to encode full-resolution MNIST images with 10 qubits and to implement a convolution on the whole image with just a single one-qubit gate. Our training algorithm is based on training all the experts together, which significantly improves trainability with respect to training each expert independently. In fact, in the limit of infinitely many experts, our training algorithm can perfectly fit the training data. Our results demonstrate the potential of our quantum neural network to achieve high-accuracy image classification with minimal quantum resources, paving the way for more scalable and efficient quantum machine learning models.

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arXiv2025

Statistical Characterization of Entanglement Degradation Under Markovian Noise in Composite Quantum Systems

Understanding how noise degrades entanglement is crucial for the development of reliable quantum technologies. While the Markovian approximation simplifies the analysis of noise, it remains computationally demanding, particularly for high-dimensional systems like quantum memories. In this paper, we present a statistical approach to study the impact of different noise models on entanglement in composite quantum systems. By comparing global and local noise scenarios, we quantify entanglement degradation using the Positive Partial Transpose Time (PPTT) metric, which measures how long entanglement persists under noise. When the sampling of different noise scenarios is performed under controlled and homogeneous conditions, our analysis reveals that systems subjected to global noise tend to exhibit longer PPTTs, whereas those influenced by independent local noise models display the shortest entanglement persistence. To carry out this analysis, we employ a computational method proposed by Cao and Lu, which accelerates the simulation of PPTT distributions and enables efficient analysis of systems with dimensions up to $D=8$. Our results demonstrate the effectiveness of this approach for investigating the resilience of quantum systems under Markovian noise.

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arXiv2025

Optical fibres with memory effects and their quantum communication capacities

The development of quantum repeaters poses significant challenges in terms of cost and maintenance, prompting the exploration of alternative approaches for achieving long-distance quantum communication. In the absence of quantum repeaters and under the memoryless (iid) approximation, it has been established that some fundamental quantum communication tasks are impossible if the transmissivity of an optical fibre falls below a known critical value, resulting in a severe constraint on the achievable distance for quantum communication. However, if the memoryless assumption does not hold -- e.g. when input signals are separated by a sufficiently short time interval -- the validity of this limitation is put into question. In this paper we introduce a model of optical fibre that can describe memory effects for long transmission lines. We then solve its quantum capacity, two-way quantum capacity, and secret-key capacity exactly. By doing so, we show that -- due to the memory cross-talk between the transmitted signals -- reliable quantum communication is attainable even for highly noisy regimes where it was previously considered impossible. As part of our solution, we find the critical time interval between subsequent signals below which quantum communication, two-way entanglement distribution, and quantum key distribution become achievable.

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arXiv2024

Quantitative convergence of trained quantum neural networks to a Gaussian process

We study quantum neural networks where the generated function is the expectation value of the sum of single-qubit observables across all qubits. In [Girardi \emph{et al.}, arXiv:2402.08726], it is proven that the probability distributions of such generated functions converge in distribution to a Gaussian process in the limit of infinite width for both untrained networks with randomly initialized parameters and trained networks. In this paper, we provide a quantitative proof of this convergence in terms of the Wasserstein distance of order $1$. First, we establish an upper bound on the distance between the probability distribution of the function generated by any untrained network with finite width and the Gaussian process with the same covariance. This proof utilizes Stein's method to estimate the Wasserstein distance of order $1$. Next, we analyze the training dynamics of the network via gradient flow, proving an upper bound on the distance between the probability distribution of the function generated by the trained network and the corresponding Gaussian process. This proof is based on a quantitative upper bound on the maximum variation of a parameter during training. This bound implies that for sufficiently large widths, training occurs in the lazy regime, \emph{i.e.}, each parameter changes only by a small amount. While the convergence result of [Girardi \emph{et al.}, arXiv:2402.08726] holds at a fixed training time, our upper bounds are uniform in time and hold even as $t \to \infty$.

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arXiv2024

The generalized strong subadditivity of the von Neumann entropy for bosonic quantum systems

We prove a generalization of the strong subadditivity of the von Neumann entropy for bosonic quantum Gaussian systems. Such generalization determines the minimum values of linear combinations of the entropies of subsystems associated to arbitrary linear functions of the quadratures, and holds for arbitrary quantum states including the scenario where the entropies are conditioned on a memory quantum system. We apply our result to prove new entropic uncertainty relations with quantum memory, a generalization of the quantum Entropy Power Inequality, and the linear time scaling of the entanglement entropy produced by quadratic Hamiltonians.

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arXiv2024

Entanglement Degradation in the Presence of Markovian Noise: a Statistical Analysis

Adopting a statistical approach we study the degradation of entanglement of a quantum system under the action of an ensemble of randomly distributed Markovian noise. This enables us to address scenarios where only limited information is available on the mechanisms that rule the noisy evolution of the model. As an application, we characterize the statistic of entanglement deterioration for a quantum memory formed by $n$ qudits that undergo randomly distributed local, uniform, Markovian noise evolution.

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