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arXiv:1808.02421 · Philosophical Transactions of the Royal Society A Mathematical Physical and Engineering Sciences

Dynamics of quantum measurements employing two Curie–Weiss apparatuses

Martí Perarnau-Llobet, Theo M. Nieuwenhuizen

Published Oct 2, 2017• 0 citations
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Abstract

Two types of quantum measurements, measuring the spins of an entangled pair and attempting to measure a spin at either of two positions, are analysed dynamically by apparatuses of the Curie-Weiss type. The outcomes comply with the standard postulates.This article is part of the themed issue 'Second quantum revolution: foundational questions'.

DOI
10.1098/rsta.2016.0386

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Two types of quantum measurements, measuring the spins of an entangled pair and attempting to measure a spin at either of two positions, are analysed dynamically by apparatuses of the Curie-Weiss type. The outcomes comply with the standard postulates.This article is part of the themed issue 'Second quantum revolution: foundational questions'.
Quantum measurement is a dynamical process of an apparatus coupled to a test system. Ideal measurement of the zzz-component of a spin-12\frac{1}{2}21​ (sz=±12s_z=\pm\frac{1}{2}sz​=±21​) has been modeled by the Curie-Weiss model for quantum measurement. Recently, the model was generalized to higher spin and the thermodynamics was solved. Here the dynamics is considered. To this end, the dynamics for spin-12\frac{1}{2}21​ case are cast in general notation. The dynamics of the measurement of the zzz-component of a spin-1 (sz=0,±1s_z=0,\pm 1sz​=0,±1) are solved in detail and evaluated numerically. Energy costs of the measurement, which are macroscopic, are evaluated. Generalization to higher spin is straightforward.
The Curie–Weiss model for quantum measurement describes the dynamical measurement of a spin-12 by an apparatus consisting of an Ising magnet of many spins 12 coupled to a thermal phonon bath. To measure the z-component s=−l,−l+1,⋯,l of a spin l, a class of models is designed along the same lines, which involve 2l order parameters. As required for unbiased measurement, the Hamiltonian of the magnet, its entropy and the interaction Hamiltonian possess an invariance under the permutation s→s+1 mod 2l+1. The theory is worked out for the spin-1 case, where the thermodynamics is analyzed in detail, and, for spins 32,2,52, the thermodynamics and the invariance are presented.
A measuring apparatus is described by quantum mechanics while it interacts with the quantum system under observation and then it must be given a classical description so that the result of the measurement appears as objective reality. Alternatively, the apparatus may always be treated by quantum mechanics and be measured by a second apparatus, which has such a dual description. This article examines whether these two different descriptions are mutually consistent. It is shown that if the dynamical variable used in the first apparatus is represented by an operator of the Weyl-Wigner type (for example, if it is a linear coordinate), then the conversion from quantum to classical terminology does not affect the final result. However, if the first apparatus encodes the measurement in a different type of operator (e.g., the phase operator), the two methods of calculation may give different results.
By considering the measurement of a quantum spin by two different models of the measuring apparatus, we try to throw some light on the nature of the apparatus that can perform quantum measurement. Our scheme is to consider a quantum spin in interaction with another quantum system playing the role of an apparatus, which in turn is dissipatively coupled to an environment. The coupling with the environment drives the density matrix of the apparatus to a diagonal form. However, measurement is performed only when the reduced density matrix of the apparatus contains correlations with the states of the spin. The two examples considered here show that this happens when the quantum apparatus has a well-defined classical limit. If the apparatus does not have a well-defined classical limit, the reduced density matrix of the apparatus is not correlated with the states of the measured system.