TY - JOUR

T1 - On the moment limit of quantum observables, with an application to the balanced homodyne detection

AU - Kiukas, J.

AU - Lahti, P.

N1 - Funding Information:
The authors thank Dr Paul Busch and Dr Kari Ylinen for useful discussions and comments. One of us (JK) was supported by Finnish Cultural Foundation.
Copyright:
Copyright 2008 Elsevier B.V., All rights reserved.

PY - 2008/5/12

Y1 - 2008/5/12

N2 - 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.

AB - 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.

KW - Balanced homodyne detection

KW - High amplitude limit

KW - Moment problem

KW - Operator integral

KW - Positive operator measure

UR - http://www.scopus.com/inward/record.url?scp=45849137863&partnerID=8YFLogxK

U2 - 10.1080/09500340701624658

DO - 10.1080/09500340701624658

M3 - Article

AN - SCOPUS:45849137863

SN - 0950-0340

VL - 55

SP - 1175

EP - 1198

JO - Journal of Modern Optics

JF - Journal of Modern Optics

IS - 7

ER -