Joint measurability through Naimark's theorem

We use Naimark's dilation theorem in order to characterize the joint measurability of two POVMs. Then, we analyze the joint measurability of two commutative POVMs $F_1$ and $F_2$ which are the smearing of two self-adjoint operators $A_1$ and $A_2$ respectively. We prove that the compatibility of $F_1$ and $F_2$ is connected to the existence of two compatible self-adjoint dilations $A_1^+$ and $A_2^+$ of $A_1$ and $A_2$ respectively. As a corollary we prove that each couple of self-adjoint operators can be dilated to a couple of compatible self-adjoint operators. Next, we analyze the joint measurability of the unsharp position and momentum observables and show that it provides a master example of the scheme we propose. Finally, we give a sufficient condition for the compatibility of two effects

Stochastic matrices and a property of the infinite sequences of linear functionals

Abstract Our starting point is the proof of the following property of a particular class of matrices. Let T = { T i , j } be a n × m non-negative matrix such that ∑ j T i , j = 1 for each i . Suppose that for every pair of indices ( i , j ) , there exists an index l such that T i , l ≠ T j , l . Then, there exists a real vector k = ( k 1 , k 2 , … , k m ) T , k i ≠ k j , i ≠ j ; 0 k i ⩽ 1 , such that, ( T k ) i ≠ ( T k ) j if i ≠ j . Then, we apply that property of matrices to probability theory. Let us consider an infinite sequence of linear functionals { T i } i ∈ N , T i f = ∫ f ( t ) d μ t ( i ) , corresponding to an infinite sequence of probability measures { μ ( · ) ( i ) } i ∈ N , on the Borel σ -algebra B ( [ 0 , 1 ] ) such that, μ ( · ) ( i ) ≠ μ ( · ) ( j ) , i , j ∈ N , i ≠ j . The property of matrices described above allows us to construct a real bounded one-to-one piecewise continuous and continuous from the left function f such that T i f = ∫ f ( t ) d μ t ( i ) ≠ ∫ f ( t ) d μ t ( j ) = T j f , i , j ∈ N , i ≠ j . The relevance to quantum mechanics is showed

Forest fire spreading: a nonlinear stochastic model continuous in space and time

Forest fire spreading is a complex phenomenon characterized by a stochastic behavior. Nowadays, the enormous quantity of georeferenced data and the availability of powerful techniques for their analysis can provide a very careful picture of forest fires opening the way to more realistic models. We propose a stochastic spreading model continuous in space and time that is able to use such data in their full power. The state of the forest fire is described by the subprobability densities of the green trees and of the trees on fire that can be estimated thanks to data coming from satellites and earth detectors. The fire dynamics is encoded into a density probability kernel which can take into account wind conditions, land slope, spotting phenomena and so on, bringing to a system of integro-differential equations for the probability densities. Existence and uniqueness of the solutions is proved by using Banach's fixed point theorem. The asymptotic behavior of the model is analyzed as well. Stochastic models based on cellular automata can be considered as particular cases of the present model from which they can be derived by space and/or time discretization. Suggesting a particular structure for the kernel, we obtain numerical simulations of the fire spreading under different conditions. For example, in the case of a forest fire evolving towards a river, the simulations show that the probability density of the trees on fire is different from zero beyond the river due to the spotting phenomenon. Firefighters interventions and weather changes can be easily introduced into the model.Comment: 25 pages, 27 figure

Classical Mechanics in Hilbert Space, Part 1

We consider the Hamilton formulation as well as the Hamiltonian flows on a symplectic (phase) space. These symplectic spaces are derivable from the Lie group of symmetries of the physical system considered. In Part 2 of this work, we then obtain the Hamiltonian formalism in the Hilbert spaces of square integrable functions on the symplectic spaces so obtained

An operational link between MUBs and SICs

We exhibit an operational connection between mutually unbiased bases and symmetric infomationally complete positive operator-valued measures. Assuming that the latter exists, we show that there is a strong link between these two structures in all prime power dimensions. We also demonstrate that a similar link cannot exists in dimension 6.Comment: 17 pages, 2 figure