Consistency of quantum phase measurements

Joan A. Vaccaro, David Thomas Pegg

In the Ψ-space limiting procedure limits of expectation values, rather than operator or states, are found as the state space dimensionality tends to infinity. This approach has been applied successfully to the calculation of the phase properties of various states of light, but its status as a valid quantum mechanical thory equivalent to the usual infinite Hilbert space (H-space) approach has not yet been fully accepted. Here we address this issue by investigating the formal relationship between the two approaches. We establish the consistency between the Ψ-space and H-space approaches for observables which are amenable to an H-space treatment. Such observables are represented in H by operators which are strong limits of Ψ-space operators and which obey the same algebra as the corresponding Ψ-space operators. The phase operator, however, exists in H only as a weak limit of a Ψ-space operator. For such limits the Ψ-space operator algebra is not preserved, which is the fundamental reason for the difficulties in constructing a consistent quantum description of phase in H. We show that for the phase observable the Ψ-space approach is consistent with the probability-operator measure (POM) method with the important distinction that, whereas the relation between non-orthogonal POMs and probability has to be accepted in the latter method as a postulate, the corresponding relation is derived in the Ψ-space approach. We conclude that the Ψ-space approach is not only equivalent to, but is also more fundamental than both the H-space and POM approaches.

Jaewoo Noh

J. R. Torgerson, L. Mandeļ

We show that the measured probability distribution of the phase difference between two electromagnetic fields can be sharpened progressively as data involving more and more detected photons are discarded, and that the phase difference can be well-defined even when the mean photon number n → 0. In our operational phase theory the variance of the directly measured phase difference has no lower bound other than zero, and this is consistent with the experimental results.

Robert B. Griffiths

Andrew M. Steane

A discussion of quantum measurement theory is presented. It is argued that the quantum measurement problem consists in the failure of any formalism based on unitary evolution to describe fully the observed behaviour in measurement-like processes. It is shown that the problem is essentially concerned with the phase of the coherence between orthogonal eigenstates of an object/measuring device system. It is then proposed that the observed behaviour is fully described if the value of this phase is formally undecidable.

Z. Hradil