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Register a new userNon-local Markovian symmetric forms on infinite dimensional spaces; Part 2. Examples: non local stochastic quantization of space cut-off quantum fields and infinite particle systems
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The general framework on the non-local Markovian symmetric forms on weighted $l^p$ $(p in [1, infty])$ spaces constructed by [A,Kagawa,Yahagi,Y 2020], by restricting the situation where $p =2$, is applied to such measure spaces as the space cut-off $P(phi)_2$ Euclidean quantum field, the $2$-dimensional Euclidean quantum fields with exponential and trigonometric potentials, and the field describing a system of an infinite number of classical particles. For each measure space, the Markov process corresponding to the {it{non-local}} type stochastic quantization is constructed.
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General theorems on the closability and quasi-regularity of non-local Markovian symmetric forms on probability spaces $(S, {cal B}(S), mu)$, with $S$ Fr{e}chet spaces such that $S subset {mathbb R}^{mathbb N}$, ${cal B}(S)$ is the Borel $sigma$-field of $S$, and $mu$ is a Borel probability measure on $S$, are introduced. Firstly, a family of non-local Markovian symmetric forms ${cal E}_{(alpha)}$, $0 < alpha < 2$, acting in each given $L^2(S; mu)$ is defined, the index $alpha$ characterizing the order of the non-locality. Then, it is shown that all the forms ${cal E}_{(alpha)}$ defined on $bigcup_{n in {mathbb N}} C^{infty}_0({mathbb R}^n)$ are closable in $L^2(S;mu)$. Moreover, sufficient conditions under which the closure of the closable forms, that are Dirichlet forms, become strictly quasi-regular, are given. Finally, an existence theorem for Hunt processes properly associated to the Dirichlet forms is given. The application of the above theorems to the problem of stochastic quantizations of Euclidean $Phi^4_d$ fields, for $d =2, 3$, by means of these Hunt processes is indicated.
In this paper a general definition of quantum conditional entropy for infinite-dimensional systems is given based on recent work of Holevo and Shirokov arXiv:1004.2495 devoted to quantum mutual and coherent informations in the infinite-dimensional case. The properties of the conditional entropy such as monotonicity, concavity and subadditivity are also generalized to the infinite-dimensional case.
The thermal equilibrium distribution over quantum-mechanical wave functions is a so-called Gaussian adjusted projected (GAP) measure, $GAP(rho_beta)$, for a thermal density operator $rho_beta$ at inverse temperature $beta$. More generally, $GAP(rho)$ is a probability measure on the unit sphere in Hilbert space for any density operator $rho$ (i.e., a positive operator with trace 1). In this note, we collect the mathematical details concerning the rigorous definition of $GAP(rho)$ in infinite-dimensional separable Hilbert spaces. Its existence and uniqueness follows from Prohorovs theorem on the existence and uniqueness of Gaussian measures in Hilbert spaces with given mean and covariance. We also give an alternative existence proof. Finally, we give a proof that $GAP(rho)$ depends continuously on $rho$ in the sense that convergence of $rho$ in the trace norm implies weak convergence of $GAP(rho)$.
The coding theorem for the entanglement-assisted communication via infinite-dimensional quantum channel with linear constraint is extended to a natural degree of generality. Relations between the entanglement-assisted classical capacity and the $chi$-capacity of constrained channels are obtained and conditions for their coincidence are given. Sufficient conditions for continuity of the entanglement-assisted classical capacity as a function of a channel are obtained. Some applications of the obtained results to analysis of Gaussian channels are considered. A general (continuous) version of the fundamental relation between the coherent information and the measure of privacy of classical information transmission by infinite-dimensional quantum channel is proved.
We give a stochastic proof of the finite approximability of a class of Schru007fodinger operators over a local field, thereby completing a program of establishing in a non-Archimedean setting corresponding results and methods from the Archimedean (real) setting. A key ingredient of our proof is to show that Brownian motion over a local field can be obtained as a limit of random walks over finite grids. Also, we prove a Feynman-Kac formula for the finite systems, and show that the propagator at the finite level converges to the propagator at the infinite level.
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