Abstract
Suppose we make a series of measurements on a chosen quantum system. The outcomes of the measurements form a sequence of random events, which occur in a particular order. The system, together with a meter or meters, can be seen as following the paths of a stochastic network connecting all possible outcomes. The paths are shaped from the virtual paths of the system, and the corresponding probabilities are determined by the measuring devices employed. If the measurements are highly accurate, the virtual paths become “real”, and the mean values of a quantity (a functional) are directly related to the frequencies with which the paths are traveled. If the measurements are highly inaccurate, the mean (weak) values are expressed in terms of the relative probabilities’ amplitudes. For pre- and post-selected systems they are bound to take arbitrary values, depending on the chosen transition. This is a direct consequence of the uncertainty principle, which forbids one from distinguishing between interfering alternatives, while leaving the interference between them intact.
Highlights
A sequence of measurements made on a quantum system leads to a number of random outcomes, observed by an experimentalist
In the simplest case of von Neumann measurements [1], the transition probabilities, as well as the observed events themselves are determined by the virtual paths available to the studied system, the nature of the measured quantity and the accuracy of the meter(s)
The weakness of the measurement inevitably causes the meter’s readings to be spread, covering the whole real axis in the limit when the perturbation is sent to zero [2]. This gives an operational meaning to the uncertainty principle, which states that the value of  in a superposition of its eigenstates must be indeterminate
Summary
A sequence of measurements made on a quantum system leads to a number of random outcomes, observed by an experimentalist. The weakness of the measurement inevitably causes the meter’s readings to be spread, covering the whole real axis in the limit when the perturbation is sent to zero [2] This gives an operational meaning to the uncertainty principle, which states that the value of  in a superposition of its eigenstates must be indeterminate. To which we will refer as “Feynman’s uncertainty principle”, one may conclude that interfering alternatives cannot be told apart and must, form a single indivisible pathway [20] Another corollary to the principle is that interference must be destroyed by a physical agent, e.g., a meter coupled to the observed system. We will not follow this matter any further, and we discuss possible realizations of quantum measurements
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