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arXiv:quant-ph/9707053

Approximate Consistency and Prediction Algorithms in Quantum Mechanics

Jim McElwaine

Published Jul 29, 1997• 1 citation
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Abstract

This dissertation investigates questions arising in the consistent histories formulation of the quantum mechanics of closed systems. Various criteria for approximate consistency are analysed. The connection between the Dowker-Halliwell criterion and sphere packing problems is shown and used to prove several new bounds on the violation of probability sum rules. The quantum Zeno effect is also analysed within the consistent histories formalism and used to demonstrate some of the difficulties involved in discussing approximate consistency. The complications associated with null histories and infinite sets are briefly discussed. The possibility of using the properties of the Schmidt decomposition to define an algorithm which selects a single, physically natural, consistent set for pure initial density matrices is investigated. The problems that arise are explained, and different possible algorithms discussed. Their properties are analysed with the aid of simple models. A set of computer programs is described which apply the algorithms to more complicated examples. Another algorithm is proposed that selects the consistent set (formed using Schmidt projections) with the highest Shannon information. This is applied to a simple model and shown to produce physically sensible histories. The theory is capable of unconditional probabilistic prediction for closed quantum systems, and is strong enough to be falsifiable. Ideas on applying the theory to more complicated examples are discussed.

DOI
10.48550/arxiv.quant-ph/9707053

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This dissertation investigates questions arising in the consistent histories formulation of the quantum mechanics of closed systems. Various criteria for approximate consistency are analysed. The connection between the Dowker-Halliwell criterion and sphere packing problems is shown and used to prove several new bounds on the violation of probability sum rules. The quantum Zeno effect is also analysed within the consistent histories formalism and used to demonstrate some of the difficulties involved in discussing approximate consistency. The complications associated with null histories and infinite sets are briefly discussed. The possibility of using the properties of the Schmidt decomposition to define an algorithm which selects a single, physically natural, consistent set for pure initial density matrices is investigated. The problems that arise are explained, and different possible algorithms discussed. Their properties are analysed with the aid of simple models. A set of computer programs is described which apply the algorithms to more complicated examples. Another algorithm is proposed that selects the consistent set (formed using Schmidt projections) with the highest Shannon information. This is applied to a simple model and shown to produce physically sensible histories. The theory is capable of unconditional probabilistic prediction for closed quantum systems, and is strong enough to be falsifiable. Ideas on applying the theory to more complicated examples are discussed.
The consistent histories formalism is discussed using path-projected states. These are used to analyze various criteria for approximate consistency. The connection between the Dowker-Halliwell criterion and sphere packing problems is shown and used to prove several bounds on the violation of probability sum rules. The quantum Zeno effect is also analyzed within the consistent histories formalism and used to demonstrate some of the difficulties involved in discussing approximate consistency. The complications associated with null histories and infinite sets are briefly discussed. \textcopyright{} 1996 The American Physical Society.
The consistent histories formulation of the quantum theory of a closed system with a pure initial state defines an infinite number of incompatible consistent sets, each of which gives a possible description of the physics. We investigate the possibility of using the properties of the Schmidt decomposition to define an algorithm which selects a single, physically natural, consistent set. We explain the problems which arise, set out some possible algorithms, and explain their properties with the aid of simple models. Though the discussion is framed in the language of the consistent histories approach, it is intended to highlight the difficulty in making any interpretation of quantum theory based on decoherence into a mathematically precise theory.
This paper describes an algorithm for selecting a consistent set within the consistent histories approach to quantum mechanics and investigates its properties. The algorithm uses a maximum information principle to select from among the consistent sets formed by projections defined by the Schmidt decomposition. The algorithm unconditionally predicts the possible events in closed quantum systems and ascribes probabilities to these events. A simple spin model is described and a complete classification of all exactly consistent sets of histories formed from Schmidt projections in the model is proved. This result is used to show that for this example the algorithm selects a physically realistic set. Other tentative suggestions in the literature for set selection algorithms using ideas from information theory are discussed.
This paper describes an algorithm for selecting a consistent set within the consistent histories approach to quantum mechanics and investigates its properties. The algorithm select from among the consistent sets formed by projections defined by the Schmidt decomposition by making projections at the earliest possible time. The algorithm unconditionally predicts the possible events in closed quantum systems and ascribes probabilities to these events. A simple random Hamiltonian model is described and the results of applying the algorithm to this model using computer programs are discussed and compared with approximate analytic calculations.
There are good motivations for considering some type of quantum histories formalism. Several possible formalisms are known, defined by different definitions of event and by different selection criteria for sets of histories. These formalisms have a natural interpretation, according to which nature somehow chooses one set of histories from among those allowed, and then randomly chooses to realise one history from that set; other interpretations are possible, but their scientific implications are essentially the same. The selection criteria proposed to date are reasonably natural, and certainly raise new questions. For example, the validity of ordering inferences which we normally take for granted – such as that a particle in one region is necessarily in a larger region containing it – depends on whether or not our history respects the criterion of ordered consistency, or merely consistency. However, the known selection criteria, including consistency and medium decoherence, are very weak. It is not possible to derive the predictions of classical mechanics or Copenhagen quantum mechanics from the theories they define, even given observational data in an extended time interval. Attempts to refine the consistent histories approach so as to solve this problem by finding a definition of quasiclassicality have so far not succeeded. On the other hand, it is shown that dynamical collapse models, of the type originally proposed by Ghirardi-Rimini-Weber, can be re-interpreted as set selection criteria within a quantum histories framework, in which context they appear as candidate solutions to the set selection problem. This suggests a new route to relativistic generalisation of these models, since covariant definitions of a quantum event are known.
Classical mechanics and standard Copenhagen quantum mechanics respect subspace implications. For example, if a particle is confined in a particular region RRR of space, then in these theories we can deduce that it is confined in regions containing RRR. However, subspace implications are generally violated by versions of quantum theory that assign probabilities to histories, such as the consistent histories approach. I define here a new criterion, ordered consistency, which refines the criterion of consistency and has the property that inferences made by ordered consistent sets do not violate subspace relations. This raises the question: do the operators defining our observations form an ordered consistent history? If so, ordered consistency defines a version of quantum theory with greater predictive power than the consistent histories formalism. If not, and our observations are defined by a non-ordered consistent quantum history, then subspace implications are not generally valid.