On the repeated measurement of continuous observables in quantum mechanics

Masanao Ozawa

Toru Fuda

Masanao Ozawa

The purpose of this paper is to provide a basis of theory of measurements of continuous observables. We generalize von Neumann’s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables, and show how every such measuring process determines the state change caused by the measurement. We establish a one-to-one correspondence between completely positive instruments in the sense of Davies and Lewis and the state changes determined by the measuring processes. We also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics, so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis. A proof of the Wigner–Araki–Yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework. We also discuss the implication of these results for the recent results due to Srinivas and due to Mercer on measurements of continuous observables.

Masanao Ozawa

In order to develop a statistical theory of quantum measurements including continuous observables, a concept of a posteriori states is introduced, which generalizes the notion of regular conditional probability distributions in classical probability theory. Its statistical interpretation in measuring processes is discussed and its existence is proved. As an application, we also give a complete proof of the Davies-Lewis conjecture that there are no (weakly) repeatable instruments for non-discrete observables in the standard formulation of quantum mechanics, using the notion of a posteriori states.

Alberto Barchielli

Francesco Buscemi, Giacomo Mauro D’Ariano, Paolo Perinotti

We show that, contrary to the widespread belief, in quantum mechanics repeatable measurements are not necessarily described by orthogonal projectors--the customary paradigm of observable. Nonorthogonal repeatability, however, occurs only for infinite dimensions. We also show that, when a nonorthogonal repeatable measurement is performed, the measured system retains some "memory" of the number of times that the measurement has been performed.