On the moment limit of quantum observables, with an application to the balanced homodyne detection

• Published in: Journal of modern optics Vol. 55; no. 7; pp. 1175 - 1198
• Main Authors: Kiukas, J., Lahti†, P.
• Format: Journal Article
• Language: English
• Published: London Taylor & Francis Group 10.04.2008; Taylor & Francis
• Subjects: balanced homodyne detection; Classical and quantum physics: mechanics and fields; Exact sciences and technology; Foundations, theory of measurement, miscellaneous theories (including aharonov-bohm effect, bell inequalities, berry's phase); high amplitude limit; moment problem; operator integral; Physics; positive operator measure; Quantum mechanics; Measuring methods; Homodyne detection; Observables; Quantum tomography; Quantum measurement theory
• ISSN: 0950-0340; 1362-3044
• DOI: 10.1080/09500340701624658

We consider the moment operators of the observable (i.e. a semispectral measure or POM) associated with the balanced homodyne detection statistics, whilst paying attention to the correct domains of these unbounded operators. We show that the high amplitude limit, when performed on the moment operators, actually determines uniquely the entire statistics of a rotated quadrature amplitude of the signal field, thereby verifying the usual assumption that the homodyne detection achieves a measurement of that observable. We also consider, in a general setting, the possibility of constructing a measurement of a single quantum observable from a sequence of observables by taking the limit on the level of moment operators of these observables. In this context, we show that under some natural conditions (each of which is satisfied by the homodyne detector example), the existence of the moment limits ensures that the underlying probability measures converge weakly to the probability measures of the limiting observable. The moment approach naturally requires that the observables be determined by their moment operator sequences (which does not automatically happen), and it turns out, in particular, that this is the case for the balanced homodyne detector.