The geometric separability probability of the composite quantum systems has been extensively studied in the recent decades. One of the simplest but strikingly difficult problem is to compute the separability probability of qubit-qubit and rebit-rebit quantum states with respect to the Hilbert-Schmidt measure. A lot of numerical simulations confirm the P rebit-rebit = 29 64 and P qubit-qubit = 8 33 conjectured probabilities. Milz and Strunz studied the separability probability with respect to given subsystems. They conjectured that the separability probability of qubitqubit (and qubit-qutrit) states of the form of D1 C C * D2 depends on D = D1 + D2 (on single qubit subsystems), moreover it depends only on the Bloch radii (r) of D and it is constant in r. Using the Peres-Horodecki criterion for separability we give mathematical proof for the 29 64 probability and we present an integral formula for the complex case which hopefully will help to prove the 8 33 probability, too. We prove Milz and Strunz's conjecture for rebit-rebit and qubitqubit states. The case, when the state space is endowed with the volume form generated by the operator monotone function f (x) = √ x is also studied in detail. We show that even in this setting the Milz and Strunz's conjecture holds true and we give an integral formula for separability probability according to this measure.

Artificial neural networks (ANNs) are typically highly nonlinear systems which are finely tuned via the optimization of their associated, non-convex loss functions. Typically, the gradient of any such loss function fails to be dissipative making the use of widely-accepted (stochastic) gradient descent methods problematic. We offer a new learning algorithm based on an appropriately constructed variant of the popular stochastic gradient Langevin dynamics (SGLD), which is called tamed unadjusted stochastic Langevin algorithm (TUSLA). We also provide a nonasymptotic analysis of the new algorithm's convergence properties in the context of non-convex learning problems with the use of ANNs. Thus, we provide finite-time guarantees for TUSLA to find approximate minimizers of both empirical and population risks. The roots of the TUSLA algorithm are based on the taming technology for diffusion processes with superlinear coefficients as developed in Sabanis (2013developed in Sabanis ( , 2016 and for MCMC algorithms in Brosse et al. (2019). Numerical experiments are presented which confirm the theoretical findings and illustrate the need for the use of the new algorithm in comparison to vanilla SGLD within the framework of ANNs.

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We prove the existence of limiting distributions for a large class of Markov chains on a general state space in a random environment. We assume suitable versions of the standard drift and minorization conditions. In particular, the system dynamics should be contractive on the average with respect to the Lyapunov function and large enough small sets should exist with large enough minorization constants. We also establish that a law of large numbers holds for bounded functionals of the process. Applications to queuing systems and to machine learning algorithms are presented.

The simplest building blocks for quantum computations are the qubitqubit quantum channels. In this paper, we analyze the structure of these channels via their Choi representation. The restriction of a quantum channel to the space of classical states (i.e. probability distributions) is called the underlying classical channel. The structure of quantum channels over a fixed classical channel is studied, the volume of general and unital qubit channels with respect to the Lebesgue measure is computed and explicit formulas are presented for the distribution of the volume of quantum channels over given classical channels. We study the state transformation under uniformly random quantum channels. If one applies a uniformly random quantum channel (general or unital) to a given qubit state, the distribution of the resulted quantum states is presented. * quantum channel, volume; MSC2010: 81P16, 81P45, 94A17 † lovas@math.bme.hu

The objective of the present study was to identify risk factors among epidemiological factors and meteorological conditions in connection with fatal pulmonary embolism. Information was collected from forensic autopsy records in sudden unexpected death cases where pulmonary embolism was the exact cause of death between 2001 and 2010 in Budapest. Meteorological parameters were detected during the investigated period. Gender, age, manner of death, cause of death, place of death, post-mortem pathomorphological changes and daily meteorological conditions (i.e. daily mean temperature and atmospheric pressure) were examined. We detected that the number of registered pulmonary embolism (No 467, 211 male) follows power law in time regardless of the manner of death. We first described that the number of registered fatal pulmonary embolism up to the nth day can be expressed as Y(n) = α ⋅ n (β) where Y denotes the number of fatal pulmonary embolisms up to the nth day and α > 0 and β > 1 are model parameters. We found that there is a definite link between the cold temperature and the increasing incidence of fatal pulmonary embolism. Cold temperature and the change of air pressure appear to be predisposing factors for fatal pulmonary embolism. Meteorological parameters should have provided additional information about the predisposing factors of thromboembolism.

Multiplying a likelihood function with a positive number makes no difference in Bayesian statistical inference, therefore after normalization the likelihood function in many cases can be considered as probability distribution. This idea led Aitchison to define a vector space structure on the probability simplex in 1986. Pawlowsky-Glahn and Egozcue gave a statistically relevant scalar product on this space in 2001, endowing the probability simplex with a Hilbert space structure. In this paper we present the noncommutative counterpart of this geometry. We introduce a real Hilbert space structure on the quantum mechanical finite dimensional state space. We show that the scalar product in quantum setting respects the tensor product structure and can be expressed in terms of modular operators and Hamilton operators. Using the quantum analogue of the logratio transformation it turns out that all the newly introduced operations emerge naturally in the language of Gibbs states. We show an orthonormal basis in the state space and study the introduced geometry on the space of qubits in details.

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