Can a quantum process be oblivious to observation?

Abstract

In this paper time is indexed by a “time group” G that may be either the real line or the integers. A quantum process is a homomorphic representation of G via unitary operators on a Hilbert space L, with the usual continuity restrictions if time is indexed by the reals. Consider a quantum process subject to nonselective observation at all points in time via an observation operator with purely discrete eigenvalues and corresponding simple eigenvectors. Suppose that the process evolves in such a way that the results of observation at a given time are the same, whether or not there have been previous observations. Suppose further that L is spanned by the set of vectors invariant modulo phase under the quantum evolution. If the time group is the integers, then the process is observed to evolve in a classical manner. If the time group is the real line, then the process is never observed to undergo a change of state. The results of this paper are essentially a corollary of von Neumann’s mean ergodic theorem.