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arXiv:1901.01035 · Entropy

Statistics of the Bifurcation in Quantum Measurement

Karl-Erik Eriksson, Kristian Lindgren

Published Aug 26, 2019• 2 citations
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Abstract

We model quantum measurement of a two-level system μ . Previous obstacles for understanding the measurement process are removed by basing the analysis of the interaction between μ and the measurement device on quantum field theory. This formulation shows how inverse processes take part in the interaction and introduce a non-linearity, necessary for the bifurcation of quantum measurement. A statistical analysis of the ensemble of initial states of the measurement device shows how microscopic details can influence the transition to a final state. We find that initial states that are efficient in leading to a transition to a final state result in either of the expected eigenstates for μ , with ensemble averages that are identical to the probabilities of the Born rule. Thus, the proposed scheme serves as a candidate mechanism for the quantum measurement process.

DOI
10.3390/e21090834

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We model quantum measurement of a two-level system μ . Previous obstacles for understanding the measurement process are removed by basing the analysis of the interaction between μ and the measurement device on quantum field theory. This formulation shows how inverse processes take part in the interaction and introduce a non-linearity, necessary for the bifurcation of quantum measurement. A statistical analysis of the ensemble of initial states of the measurement device shows how microscopic details can influence the transition to a final state. We find that initial states that are efficient in leading to a transition to a final state result in either of the expected eigenstates for μ , with ensemble averages that are identical to the probabilities of the Born rule. Thus, the proposed scheme serves as a candidate mechanism for the quantum measurement process.
We present a generic model of (non-destructive) quantum measurement. Being formulated within reversible quantum mechanics, the model illustrates a mechanism of a measurement process --- a transition of the measured system to an eigenstate of the measured observable. The model consists of a two-level system μμμ interacting with a larger system AAA, consisting of smaller subsystems. The interaction is modelled as a scattering process. Restricting the states of AAA to product states leads to a bifurcation process: In the limit of a large system AAA, the initial states of AAA that are efficient in leading to a final state are divided into two separated subsets. For each of these subsets, μμμ ends up in one of the eigenstates of the measured observable. The probabilities obtained in this branching confirm the Born rule.
An investigation is made of the interaction of a microsystem μ, subject to measurement and a macroscopic apparatus M responsible for preparation of μ and measurement on μ of an observable T with a finite number of non-degenerate eigenvalues. It is shown that the statistics of stochastic properties in M leads to a bifurcation process which forces μ into an eigenstate | j ⟩ μ of T combined with registration in M of the measurement result T j with branching ratios in agreement with the usual postulate. The process may be fast, so that only a relatively small number of degrees of freedom in M are active in the bifurcation.
A Hamiltonian model is solved, which satisfies all requirements for a realistic ideal quantum measurement. The system S is a spin-½, whose z -component is measured through coupling with an apparatus A = M + B, consisting of a magnet M formed by a set of N >> 1 spins with quartic infinite-range Ising interactions, and a phonon bath B at temperature T . Initially A is in a metastable paramagnetic phase. The process involves several time-scales. Without being much affected, A first acts on S, whose state collapses in a very brief time. The mechanism differs from the usual decoherence. Soon after its irreversibility is achieved. Finally, the field induced by S on M, which may take two opposite values with probabilities given by Born's rule, drives A into its up or down ferromagnetic phase. The overall final state involves the expected correlations between the result registered in M and the state of S. The measurement is thus accounted for by standard quantum-statistical mechanics and its specific features arise from the macroscopic size of the apparatus.
The quantum mechanics postulate called the Born Rule attributes a probabilistic meaning to a wave function. This paper derives the Born Rule from other quantum principles along with a model of the measurement process. The nondeterministic nature of quantum measurements is hypothesized to arise from an ignorance of the quantum states of a measuring device's microscopic components. Their interactions with a system to be measured are modeled heuristically with any member of a particular class of stochastic processes, each of which generate the Born Rule. One member of the class appears particularly compatible with properties expected of quantum interactions.
The thesis of this paper is that quantum measurement can be analyzed and understood within quantum mechanics itself. The statistics of the measurement process comes from the unknown details of the macroscopic measurement device. Quantum measurement is analyzed within the framework of scattering theory of quantum field theory with the aim of finding a physical rather than a metaphysical understanding. The measurement interaction is treated together with the quantum process to be measured. The evaluation of a Feynman diagram for the total process leads to one factor from the measurement interaction for each channel multiplying the basic scattering amplitudes. With increasing entanglement taken into account, these factors compete for dominance. They depend on unknown details of the measurement apparatus. The statistics of this competition is studied under the assumption that the measurement interaction does not introduce any bias. A binary quantum system is analyzed and after that the n-channel case. As a background, a couple of classical examples are shown, leading to the same selective mechanisms. The result of this analysis is that the quantum measurement process can be understood, whether a single measurement or an ensemble of measurements, as a result of an ordinary unitary time development, based on the interaction beteen the system subjet to measurement and the measurement apparatus. This result makes it possible to view the quantum-mechanical state as describing reality rather than merely a potentiality. This opens again for an ontology of physics, and hence for science in general.
We describe a quantum mechanical measurement as a variational principle including interaction between the system under measurement and the measurement apparatus. Augmenting the action with a nonlocal term (a double integration over the duration of the interaction) results in a theory capable of describing both the measurement process (agreement between system state and pointer state) and the collapse of both systems into a single eigenstate (or superposition of degenerate eigenstates) of the relevant operator. In the absence of the interaction, a superposition of states is stable, and the theory agrees with the predictions of standard quantum theory. Because the theory is nonlocal, the resulting wave equation is an integrodifferential equation (IDE). We demonstrate these ideas using a simple Lagrangian for both systems, as proof of principle. The variational principle is time-symmetric and retrocausal, so the solution for the measurement process is determined by boundary conditions at both initial and final times; the initial condition is determined by the experimental preparation and the final condition is the natural boundary condition of variational calculus. We hypothesize that one or more hidden variables (not ruled out by Bell's Theorem, due both to the retrocausality and the nonlocality of the theory) influence the outcome of the measurement, and that distributions of the hidden variables that arise plausibly in a typical ensemble of experimental realizations give rise to outcome frequencies consistent with Born's rule. We outline steps in a theoretical validation of the hypothesis. We discuss the role of both initial and final conditions to determine a solution at intermediate times, the mechanism by which a system responds to measurement, time symmetry of the new theory, causality concerns, and issues surrounding solution of the IDE.