On the mathematical Structure of Quantum Measurement Theory

TL;DR

This paper demonstrates that core issues in quantum measurement, such as wave packet reduction and state specification, can be rigorously addressed within finite conservative quantum systems, using a model of a microsystem coupled to a macroscopic measuring instrument.

Contribution

It provides a rigorous framework within finite quantum mechanics to resolve wave packet reduction and measurement state determination, supported by a finite Coleman-Hepp model.

Findings

Wave packet reduction can be achieved within finite systems.

Pointer positions correspond uniquely to system states after measurement.

Conditions for measurement consistency are satisfied by the finite Coleman-Hepp model.

Abstract

We show that the key problems of quantum measurement theory, namely the reduction of the wave packet of a microsystem and the specification of its quantum state by a macroscopic measuring instrument, may be rigorously resolved within the traditional framework of the quantum mechanics of finite conservative systems. The argument is centred on the generic model of a microsystem, S, coupled to a finite macroscopic measuring instrument, I, which itself is an N-particle quantum system. The pointer positions of I correspond to the macrostates of this instrument, as represented by orthogonal subspaces of the Hilbert space of its pure states. These subspaces, or 'phase cells', are the simultaneous eigenspaces of a set of coarse grained intercommuting macroscopic observables, M, and, crucially, are of astronomically large dimensionalities, which incease exponentially with N. We formulate conditions on the conservative dynamics of the composite (S+I) under which it yields both a reduction of the wave packet describing the state of S and a one-to-one correspondence, following a measurement, between the pointer position of I and the resultant state of S; and we show that these conditions are fulfilled by the finite version of the Coleman-Hepp model.