The hypothesis of stochastic space and the general connection of stochastic theory

M. Consoli, Alessandro Pluchino

Daniel Alonso, Juan Gonzalo Muga, R. Sala Mayato

Recently, Olavo has proposed several derivations of the Schr\"odinger equation from different sets of hypothesis [Phys. Rev. A 61, 052109 (2000)]. One of them is based on the infinitesimal inverse Weyl transform of a classically evolved phase-space density. We show, however, that the Schr\"odinger equation can only be obtained in that manner for linear or quadratic potential functions.

Jacob A. Barandes

This paper introduces several new classes of mathematical structures that have close connections with physics and with the theory of dynamical systems. The most general of these structures, called indivisible stochastic processes, collectively encompass many important kinds of stochastic processes, including Markov chains and random dynamical systems. This paper then states and proves a new theorem that establishes a precise correspondence between any indivisible stochastic process and a unitarily evolving quantum system. This theorem therefore leads to a new formulation of quantum theory, alongside the Hilbert-space, path-integral, and quasi-probability formulations. The theorem also provides a first-principles explanation for why quantum systems are based on the complex numbers, Hilbert spaces, linear-unitary time evolution, and the Born rule. In addition, the theorem suggests that by selecting a suitable Hilbert space, together with an appropriate choice of unitary evolution, one can simulate any indivisible stochastic process on a quantum computer, thereby potentially opening up an extensive set of novel applications for quantum computing.

L. de la Peña-Auerbach

Mankei Tsang

I propose a general quantum hypothesis testing theory that enables one to test hypotheses about any aspect of a physical system, including its dynamics, based on a series of observations. For example, the hypotheses can be about the presence of a weak classical signal continuously coupled to a quantum sensor, or about competing quantum or classical models of the dynamics of a system. This generalization makes the theory useful for quantum detection and experimental tests of quantum mechanics in general. In the case of continuous measurements, the theory is significantly simplified to produce compact formulas for the likelihood ratio, the central quantity in statistical hypothesis testing. The likelihood ratio can then be computed efficiently in many cases of interest. Two potential applications of the theory, namely, quantum detection of a classical stochastic waveform and test of harmonic-oscillator energy quantization, are discussed.