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Based on the theory of quantum mechanics, intrinsic randomness in measurement distinguishes quantum effects from classical ones. From the perspective of states, this quantum feature can be summarized as coherence or superposition in a…
In this work, we study the H\"older regularity of the K\"ahler- Ricci flow on compact K\"ahler manifolds with semi-ample canonical line bundle. By adapting the method in the work of Hein-Tosatti on collapsing Calabi-Yau metrics, we…
The first part I talk about the motivation for Lu Qi-Keng conjecture and the results about the presence or absence of zeroes of the Bergman kernel function of a bounded domain in ${\bf{C^n}}$. The second part I summarize the main results on…
It is shown that the classical book by von Neumann proposing dynamics of measured systems with "reduction (or collapse) of system's wave packet" contains also hints how to avoid this discontinuity in time evolution of the measured system…
A new formulation of potential scattering in quantum mechanics is developed using a close structural analogy between partial waves and the classical dynamics of many non-interacting fields. Using a canonical formalism we find non-linear…
Let $X$ be a compact hyperbolic Riemann surface equipped with the Poincar\'e metric. For any integer $k\geq 2$, we investigate the Bergman kernel associated to the holomorphic Hermitian line bundle $\Omega^{\otimes k}_X$, where $\O$ is the…
We develop a framework for function classes generated by parametric ridge kernels: one-dimensional kernels composed with affine projections and averaged over a parameter measure. The induced kernels are positive definite, and the resulting…
We introduce a measure of average dimensionality (or coherence) for high-dimensional quantum devices. This includes sets of quantum measurements, steering assemblages, and quantum channels. For measurements and channels, our measure…
We derive a semi-analytical form for the Wigner transform for the canonical density operator of a discrete system coupled to a harmonic bath based on the path integral expansion of the Boltzmann factor. The introduction of this simple and…
In ordinary Quantum Mechanics only ideally instantaneous observations of a quantity or a set of compatible quantities are usually considered. In an old paper of our group in Milano a formalism was introduced for the continuous monitoring of…
In their seminal paper (Phys. Rev.121, 920 (1961)) Sudarshan, Mathews and Rau investigated properties of the dynamical $A$ and $B$ maps acting on $n$ dimensional quantum systems. Nature of the dynamical maps in open quantum system…
We develop a novel framework for Monge--Kantorovi\v{c} metrics using Schatten ideals and commutators of fractional Laplacians on Ahlfors regular spaces. Notably, for those metrics we derive closed formulas in terms of spectra of…
The continuous quantum measurement within the probability representation of quantum mechanics is discussed. The partial classical propagator of the symplectic tomogram associated to a particular measurement outcome is introduced, for which…
We study the space of Sasaki metrics on a compact manifold $M$ by introducing an odd-dimensional analogue of the $J$-flow. That leads to the notion of critical metric in the Sasakian context. In analogy to the K\"ahler case, on a polarised…
We propose the following way of constructing quantum measure in Regge calculus: the full discrete Regge manifold is made continuous in some direction by tending corresponding dimensions of simplices to zero, then functional integral measure…
Measurement in quantum mechanics is generally described as an irreversible process that perturbs the wavefunction describing a quantum system. In this work we establish a formal connection between the measurement description within the…
Inspired by work of Besson-Courtois-Gallot, we construct a flow called the natural flow on a non-positively curved Riemannian manifold $M$. As with the natural map, the $k$-Jacobian of the natural flow is directly related to the critical…
We present a formalism for which a dissipative system is given by a variational principle. The formalism applies to dynamical systems where its trajectory is monotonic. Subsequently, we derive its Lagrangian and Hamiltonian. From the…
During a continuous measurement, quantum systems can be described by a stochastic Schr\"odinger equation which, in the appropriate limit, reproduces the von Neumann wave-function collapse. The average behavior on the ensemble of all…
We develop a statistical model of microscopic stochastic deviation from classical mechanics based on a stochastic processes with a transition probability that is assumed to be given by an exponential distribution of infinitesimal stationary…