A probabilistic framework for the continuous observation of a quantum process

✍ Scribed by Marek Kanter

• Publisher: Springer Netherlands
• Year: 1991
• Tongue: English
• Weight: 981 KB
• Volume: 23
• Category: Article
• ISSN: 0167-8019
• DOI: 10.1007/bf00047136

✦ Synopsis

This paper treats quantum measurement within von Neumann's abstract framework. Specifically, observation is defined as a fixed self-adjoint operator with countable spectrum and nondegenerate eigenstates. Suppose scenarios for the observation of a quantum process over time are expanded by adding extra observations at time points interspersed among those of a previous scenario. If each observation leads to a mixture of eigenstates rather than a pure state, then the naturally defined joint probability measures on observed results are not consistent as scenarios vary. Nevertheless, we characterize the limiting 'subprobability' measure when the times of observation become infinitely dense in any finite interval. This limiting measure corresponds to a continuous-time 'sub-stochastic' process which decays with exponential rate out of any initial state and never reappears in any other state. Thus the process loses probability exponentially over time, and this loss occurs equally fast in the case of 'nonselective' observation as for 'selective' observation.

Previous treatments of this problem have concentrated on the special case when 'Zeno's Paradox' is in force, i.e. the rate of decay out of any state is zero and the process is immobilized by continuous observation. This situation exists, for instance, when the initial state is in the domain of the generator for the unitary group underlying the quantum process.