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Genuine multipartite entanglement of a given multipartite pure quantum state can be quantified through its geometric measure of entanglement, which, up to logarithms, is simply the maximum overlap of the corresponding unit tensor with product unit tensors, a quantity which is also known as the injective norm of the tensor. Our general goal in this work is to estimate this injective norm for randomly sampled tensors. To this end, we study and compare various algorithms, based either on the widely used alternating least squares method or on a novel normalized gradient descent approach, and suited to either symmetrized or non-symmetrized random tensors. We first benchmark their respective performances on the case of symmetrized real Gaussian tensors, whose asymptotic average injective norm is known analytically. Having established that our proposed normalized gradient descent algorithm generally performs best, we then use it to provide approximate numerical values for the average injective norm of complex Gaussian tensors (i.e.~up to normalization uniformly distributed multipartite pure quantum states), with or without permutation-invariance. Finally, we are also able to estimate the average injective norm of random matrix product states constructed from Gaussian local tensors, with or without translation-invariance. All these results constitute the first numerical estimates on the amount of genuinely multipartite entanglement typically present in various models of random multipartite pure states.

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Building on the matrix cube problem, inclusions of free spectrahedra have been used successfully to obtain relaxations of hard spectrahedral inclusion problems. The quality of such a relaxation is quantified by the inclusion constant associated with each free spectrahedron. While optimal values of inclusion constants were known in certain highly symmetric cases, no general method for computing them was available. In this work, we show that inclusion constants for Cartesian products of free simplices can be computed using methods from non-commutative polynomial optimization, together with a detailed analysis of the extreme points of the associated free spectrahedra. This analysis also yields new closed-form analytic expressions for these constants. As an application to quantum information theory, we prove new bounds on the amount of white noise that incompatible measurements can tolerate before they become compatible. In particular, we study the case of one dichotomic and one k-outcome measurement, as well as the case of four dichotomic qubit measurements.

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Quantum theory exhibits various nonclassical features, such as measurement incompatibility, contextuality, steering, and Bell nonlocality, which distinguish it from classical physics. These phenomena are often studied separately, but they possess deep interconnections. This work introduces a unified mathematical framework based on commuting diagrams that unifies them. By representing collections of measurements (multimeters) as maps to the set of column-stochastic matrices, we show that measurement compatibility and simulability correspond to specific factorizations of these maps through intermediate systems. We apply this framework to put forward connections between different nonclassical notions and provide factorization-based characterizations for steering assemblages and Bell correlations, including a new perspective on the CHSH inequality witnessing measurement incompatibility. We also investigate the symmetric n-extensions of multimeters and no-signaling behaviors and connect these extensions to a notion of n-wise compatibility and to the existence of n-wise LHV models, respectively. Furthermore, we investigate robustness to noise of nonlocal features by examining factorization conditions for maps involving noisy state spaces, providing geometric criteria for when noisy multimeters can be simulated by simpler measurement settings.

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In this work, we investigate the incompatibility of random quantum measurements. Most previous work has focused on characterizing the maximal amount of white noise that any fixed number of incompatible measurements with a fixed number of outcomes in a fixed dimension can tolerate before becoming compatible. This can be used to quantify the maximal amount of incompatibility available in such systems. The present article investigates the incompatibility of several classes of random measurements, i.e., the generic amount of incompatibility available. In particular, we show that for an appropriate choice of parameters, both random dichotomic projective measurements and random basis measurements are close to being maximally incompatible. We use the technique of incompatibility witnesses to certify incompatibility and combine it with tools from random matrices and free probability.

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We study properties of opinion formation on Wikipedia Ising Networks. Each Wikipedia article is represented as a node and links are formed by citations of one article to another generating a directed network of a given language edition with millions of nodes. Ising spins are placed at each node and their orientation up or down is determined by a majority vote of connected neighbors. At the initial stage there are only a few nodes from two groups with fixed competing opinions up and down while other nodes are assumed to have no initial opinion with no effect on the vote. The competition of two opinions is modeled by an asynchronous Monte Carlo process converging to a spin polarized steady-state phase. This phase remains stable with respect to small fluctuations induced by an effective temperature of the Monte Carlo process. The opinion polarization at the steady-state provides opinion (spin) preferences for each node. In the framework of this Ising Network Opinion Formation model we analyze the influence and competition between political leaders, world countries and social concepts. This approach is also generalized to the competition between three groups of different opinions described by three colors, for example Donald Trump, Vladimir Putin, Xi Jinping or USA, Russia, China within English, Russian and Chinese editions of Wikipedia of March 2025. We argue that this approach provides a generic description of opinion formation in various complex networks.

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328