In the consistent histories approach to quantum theory probabilities are assigned to histories subject to a consistency condition of negligible interference. The approach has the feature that a given physical situation admits multiple sets of consistent histories that cannot in general be united into a single consistent set, leading to a number of counterintuitive or contrary properties if propositions from different consistent sets are combined indiscriminately. An alternative viewpoint is proposed in which multiple consistent sets are classified according to whether or not there exists any unifying probability for combinations of incompatible sets which replicates the consistent histories result when restricted to a single consistent set. A number of examples are exhibited in which this classification can be made, in some cases with the assistance of the Bell, Clauser-Horne-Shimony-Holt, or Leggett-Garg inequalities together with Fine's theorem. When a unifying probability exists logical deductions in different consistent sets can in fact be combined, an extension of the ``single framework rule.'' It is argued that this classification coincides with intuitive notions of the boundary between classical and quantum regimes and in particular, the absence of a unifying probability for certain combinations of consistent sets is regarded as a measure of the ``quantumness'' of the system. The proposed approach and results are closely related to recent work on the classification of quasiprobabilities and this connection is discussed.
In the consistent histories approach to quantum theory probabilities are assigned to histories subject to a consistency condition of negligible interference. The approach has the feature that a given physical situation admits multiple sets of consistent histories that cannot in general be united into a single consistent set, leading to a number of counterintuitive or contrary properties if propositions from different consistent sets are combined indiscriminately. An alternative viewpoint is proposed in which multiple consistent sets are classified according to whether or not there exists any unifying probability for combinations of incompatible sets which replicates the consistent histories result when restricted to a single consistent set. A number of examples are exhibited in which this classification can be made, in some cases with the assistance of the Bell, Clauser-Horne-Shimony-Holt, or Leggett-Garg inequalities together with Fine's theorem. When a unifying probability exists logical deductions in different consistent sets can in fact be combined, an extension of the single framework rule.'' It is argued that this classification coincides with intuitive notions of the boundary between classical and quantum regimes and in particular, the absence of a unifying probability for certain combinations of consistent sets is regarded as a measure of the quantumness'' of the system. The proposed approach and results are closely related to recent work on the classification of quasiprobabilities and this connection is discussed.
In a recent paper Kent has pointed out that in consistent histories quantum theory it is possible, given initial and final states, to construct two different consistent families of histories, in each of which there is a proposition that can be inferred with probability one, and such that the projectors representing these two propositions are mutually orthogonal. In this note we stress that, according to the rules of consistent history reasoning two such propositions are not contrary in the usual logical sense namely, that one can infer that if one is true then the other is false, and both could be false. No single consistent family contains both propositions, together with the initial and final states, and hence the propositions cannot be logically compared. Consistent histories quantum theory is logically consistent, consistent with experiment as far as is known, consistent with the usual quantum predictions for measurements, and applicable to the most general physical systems. It may not be the only theory with these properties, but in our opinion, it is the most promising among present possibilities.
In the histories formulation of quantum theory, sets of coarse-grained histories that are consistent obey the classical probability rules. It has been argued that these sets can describe the quasi-classical behaviour of closed quantum systems, e.g. Omnès (Rev. Mod. Phys. 64(2), 339, 1992) and Hartle (Les Houches1992). Most physical scenarios admit multiple different consistent sets and one can view each of these as a separate context. Using propositions from different consistent sets to make inferences leads to paradoxes such as contrary inferences, first noted by Kent (Phys. Rev. Lett. 78(15), 2874, 1997). In this contribution, we use the consistent histories to describe a quasi-classical and macroscopic system to show that paradoxes involving contextuality persist even in the quasi-classical limit. This is distinctively different from the contextuality of standard quantum theory, where the contextuality paradoxes do not persist in the quasi-classical limit. Specifically, we consider different consistent sets for the arrival time problem of a (quasi-classical) ball in an infinite square well. For this setting, we construct two different consistent sets. We find the probabilities that each consistent set assigns to the simple question of whether the ball ever crossed the middle of the interval. We show that one consistent set concludes with certainty that the ball crossed it while the other consistent set concludes with certainty that it did not. Our results point to the need for constraints on the histories sets, additional to the consistency condition, to recover the correct quasi-classical limit in this formalism and lead to the motto ‘all consistent sets are equal’, but ‘some consistent sets are more equal than others’.
A system of quantum reasoning for a closed system is developed by treating nonrelativistic quantum mechanics as a stochastic theory. The sample space corresponds to a decomposition, as a sum of orthogonal projectors, of the identity operator on a Hilbert space of histories. Provided a consistency condition is satisfied, the corresponding Boolean algebra of histories, called a framework, can be assigned probabilities in the usual way, and within a single framework quantum reasoning is identical to ordinary probabilistic reasoning. A refinement rule, which allows a probability distribution to be extended from one framework to a larger (refined) framework, incorporates the dynamical laws of quantum theory. Two or more frameworks which are incompatible because they possess no common refinement cannot be simultaneously employed to describe a single physical system. Logical reasoning is a special case of probabilistic reasoning in which (conditional) probabilities are 1 (true) or 0 (false). As probabilities are only meaningful relative to some framework, the same is true of the truth or falsity of a quantum description. The formalism is illustrated using simple examples, and the physical considerations which determine the choice of a framework are discussed. \textcopyright{} 1996 The American Physical Society.
In consistent-history quantum theory, a description of the time development of a quantum system requires choosing a framework or consistent family, and then calculating probabilities for the different histories that it contains. It is argued that the framework is chosen by the physicist constructing a description of a quantum system on the basis of questions he wishes to address, in a manner analogous to choosing a coarse-graining of the phase space in classical statistical mechanics. The choice of framework is not determined by some law of nature, though it is limited by quantum incompatibility, a concept that is discussed using a two-dimensional Hilbert space (spin half particle). Thus certain questions of physical interest can only be addressed using frameworks in which they make (quantum-mechanical) sense. The physicist's choice does not influence reality, nor does the presence of choices render the theory subjective. On the contrary, predictions of the theory can, in principle, be verified by experimental measurements. These considerations are used to address various criticisms and possible misunderstandings of the consistent-history approach, including its predictive power, whether it requires a new logic, whether it can be interpreted realistically, the nature of ``quasiclassicality,'' and the possibility of ``contrary'' inferences.
The decoherent (consistent) histories formalism has been proposed as a means of eliminating measurements as a fundamental concept in quantum mechanics. In this formalism, probabilities can be assigned to any description which satisfies a particular consistency condition. The formalism, however, admits incompatible descriptions which cannot be combined, unlike classical physics. This seems to leave an ambiguity in the choice of the description. I argue that this ambiguity is removed by considering the observer as a physical system.
This paper presents an elementary introduction to consistent quantum theory, as developed by Griffiths and others over the past 25years. The theory is a version of orthodox (Copenhagen) quantum mechanics based on the notion that the unique and mysterious feature of quantum, as opposed to classical systems, is the simultaneous existence of multiple incompatible representations of reality, referred to as ``frameworks.'' A framework is a maximal set of properties of a system for which probabilities can be consistently defined. This notion is expressed by stating that a framework provides an exhaustive set of exclusive alternatives, but no single framework suffices to fully characterize a quantum system. Any prediction of the theory must be confined to a single framework and combining elements from different frameworks leads to quantum mechanically meaningless statements. This ``single-framework rule'' is the precise mathematical statement of Bohr's complementarity. It is shown that if the microscopic description is assumed to incorporate these elements in a local setting, then distant entanglements, macroscopic measurements, wave-function collapse, and other features of quantum behavior follow in a logical manner. The essential elements of the theory are first explained using the simplest quantum system, a single spin-21 degree of freedom at one time.