Quantum noise, bits and jumps: uncertainties, decoherence, measurements and filtering

Abstract

A brief review of the history and drama of the development of quantum theory is given starting from Plank's quantum hypothesis exactly a 100 years ago all the way up to the modern developments in the theory of quantum measurement. It is shown that before the rise of quantum mechanics 75 years ago, the quantum theory had appeared first in the form of the statistics of quantum thermal noise and quantum spontaneous jumps which have never been explained by quantum mechanics. This led to numerous quantum paradoxes, some of them due to the great inventors of quantum theory such as Einstein and Schrödinger. They are reconsidered in this paper. The development of quantum measurement theory, initiated by von Neumann, indicated a possibility for resolution of this interpretational crisis by divorcing, the algebra of the dynamical generators from the subalgebra of the actual observables. It is shown that within this approach quantum causality can be rehabilitated in the form of a superselection rule for compatibility of past observables with the potential future. This rule, together with the self-compatibility of measurements insuring the consistency of histories, is called the nondemolition principle. The application of this rule in the form of the dynamical commutation relations leads to the derivation of the von Neumann projection postulate, and also to the more general reductions, instantaneous, spontaneous, and even continuous in time. This gives a quantum probabilistic solution, in the form of the dynamical filtering equations, of the notorious measurement problem which was tackled unsuccessfully by many famous physicists starting with Schrödinger and Bohr. The simplest Markovian quantum stochastic model for the continuous-time measurements involves a boundary-value problem in second quantization for input “offer” waves in one extra dimension, and a reduction of the algebra of “actual” observables to an Abelian subalgebra for the output waves.