On Covariant Instruments in Quantum Measurement Theory

Franklin E. Schroeck

B. Roy Frieden

Introduction In preceding chapters, EPI has been used as a computational procedure for establishing the physical laws governing various measurement scenarios. We showed, by means of the optical measurement model of Sec. 3.8, that EPI is, as well, a physical process that is initiated by a measurement. Specifically, it arises out of the interaction of the measuring instrument's probe particle with the object under measurement. This perturbs the system probability amplitudes, which perturbs the informations I and J , etc., as indicated in Figs. 3.3 and 3.4. The result is that EPI derives the phenomenon's physics as it exists at the input space to the measuring device. We also found, in Sec. 3.8, the form of the phenomenon's physics at the output to the measuring instrument. This was given by Eq. (3.51) for the output probability amplitude function. The analysis in Sec. 3.8 was, however, severely limited in dimensionality. A One-dimensional analysis of the measurement phenomenon was given. A full, covariant treatment would be preferable, i.e., where the space-time behavior of all probability amplitudes were determined. Such an analysis will be given next. It constitutes a covariant quantum theory of measurement . This covariant theory will be developed from an entirely different viewpoint than that in Sec. 3.8. The latter was an analysis that focussed attention upon the probe-particle interaction and the resulting wave propagation through the instrument. The covariant approach will concentrate, instead, on the meaning of the acquired data to the observer as it reflects upon the quantum state of the measured particle.

Bernard d’Espagnat

G. C. Ghirardi, A. Rimini

B. Roy Frieden

Introduction In preceding chapters, EPI has been used as a computational procedure for establishing the physical laws governing various measurement scenarios. We showed, by means of the optical measurement model of Sec. 3.8, that EPI is, as well, a physical process that is initiated by a measurement. Specifically, it arises out of the interaction of the measuring instrument's probe particle with the object under measurement. This perturbs the system probability amplitudes, which perturbs the informations I and J , etc., as indicated in Figs. 3.3 and 3.4. The result is that EPI derives the phenomenon's physics as it exists at the input space to the measuring device. We also found, in Sec. 3.8, the form of the phenomenon's physics at the output to the measuring instrument. This was given by Eq. (3.51) for the output probability amplitude function. The analysis in Sec. 3.8 was, however, severely limited in dimensionality. A one-dimensional analysis of the measurement phenomenon was given. A full, covariant treatment would be preferable, i.e., where the space-time behaviors of all probability amplitudes were determined. Such an analysis will be given next. It constitutes a covariant quantum theory of measurement . This covariant theory will be developed from an entirely different viewpoint than that in Sec. 3.8. The latter was an analysis that focussed attention upon the probe–particle interaction and the resulting wave propagation through the instrument. The covariant approach will concentrate, instead, on the meaning of the acquired data to the observer as it reflects upon the quantum state of the measured particle. Hence, whereas the previous analysis was purely physical in nature, this one will be “knowledge-based” as well as physical.

Alexander Semenovich Holevo

A convenient representation for Radon–Nikodym derivatives of completely positive (c.p.) instruments on B(H) with respect to a scalar measure is suggested, similar to the Stinespring–Kraus representation for c.p. maps, but involving possibly nonclosable unbounded operators. The structure of covariant c.p. instruments is studied in detail. In particular, an exhaustive description is given to instruments covariant with respect to shifts or rotations, corresponding to “position” or “angle” measurements.