Conditional expectation and repeated measurements of continuous quantum observables

E. B. Davies

Alexander Bach

We analyze a quantum central limit theorem without centering and a quantum Poisson limit theorem on an array of indistinguishable quanta and investigate the emergence of the continuous and discrete structure of the quantized electromagnetic field in classical states. Moreover, we identify the observables that converge to the random variables which induce the mixing measure of a classical state. Finally, we show that the transformation to normal order is a conditional expectation replacing expectations of functions of the annihilation and creation operator by expectations with respect to global classical observables of the array.

Xing Min Wang

After a brief introduction to Probability Bracket Notation (PBN), indicator operator and conditional density operator (CDO), we investigate probability spaces associated with various quantum systems: system with one observable (discrete or continuous), system with two commutative observables (independent or dependent) and a system of indistinguishable non-interacting many-particles. In each case, we derive unified expressions of conditional expectation (CE), conditional probability (CP), and absolute probability (AP): they have the same format for discrete or continuous spectrum; they are defined in both Hilbert space (using Dirac notation) and probability space (using PBN); and they may be useful to deal with CE of non-commutative observables.

Alberto Barchielli

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.

Maicol A. Ochoa, Wolfgang Belzig, Abraham Nitzan