Abstract

A class of Lüders instruments representing quantum measurement is defined, and some their properties are investigated. A generalisation of Lüders theorem is shown to hold for these instruments. It is also shown that the fixed-point algebra of the generalised Lüders operation is sufficient for the family of states determined by the observable associated with the instrument.

Highlights

  • IntroductionLuders [11] proved that for a complex separable Hilbert space H and selfadjoint operator A with discrete spectrum and the spectral decomposition A = aiEi, i the relation B = Ei BEi i holds for a self-adjoint operator B if and only if B commutes with all Ei. Luders [11] proved that for a complex separable Hilbert space H and selfadjoint operator A with discrete spectrum and the spectral decomposition A = aiEi, i the relation B = Ei BEi i holds for a self-adjoint operator B if and only if B commutes with all Ei

  • In the present paper, guided by the Luders theorem, we define a Luders instrument, compare it with another known classes of instruments such as ideal and strongly repeatable, and obtain as a corollary the Luders theorem for these instruments

  • We show a result about sufficiency of some important subalgebra for a family of states determined by the observable associated with the instrument

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Summary

Introduction

Luders [11] proved that for a complex separable Hilbert space H and selfadjoint operator A with discrete spectrum and the spectral decomposition A = aiEi, i the relation B = Ei BEi i holds for a self-adjoint operator B if and only if B commutes with all Ei. Luders [11] proved that for a complex separable Hilbert space H and selfadjoint operator A with discrete spectrum and the spectral decomposition A = aiEi, i the relation B = Ei BEi i holds for a self-adjoint operator B if and only if B commutes with all Ei This result can be interpreted as follows: if A and B commute, the outcomes of measurement of an observable represented by B do not depend on whether A has been measured first. In 1998 this result was generalised in [2] for unsharp observable A represented by a semispectral measure and for observable B of a special form. We show a result about sufficiency of some important subalgebra for a family of states determined by the observable associated with the instrument

Instruments in Quantum Measurement Theory
Repeatable and Ideal Measurements
Sufficiency
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