The Brown Rules develops a framework for deciding whether a sequence of probability transitions can arise from one consistent underlying unitary history. The central idea is that individual probability tables can each be valid on their own while still being incompatible when required to describe successive stages of the same process. The paper identifies probability-only tests that can rule out incompatible histories, shows how hidden phase information constrains composition, and demonstrates that in important cases the set of compatible outcomes occupies only a restricted portion of the possible probability space. It also extends the framework to longer histories and outlines computational approaches for testing compatibility. The results apply specifically to finite-dimensional unitary models and provide consistency tests within that setting rather than proposing a modification of quantum mechanics.
The Brown Rules develops a framework for deciding whether a sequence of probability transitions can arise from one consistent underlying unitary history. The central idea is that individual probability tables can each be valid on their own while still being incompatible when required to describe successive stages of the same process. The paper identifies probability-only tests that can rule out incompatible histories, shows how hidden phase information constrains composition, and demonstrates that in important cases the set of compatible outcomes occupies only a restricted portion of the possible probability space. It also extends the framework to longer histories and outlines computational approaches for testing compatibility. The results apply specifically to finite-dimensional unitary models and provide consistency tests within that setting rather than proposing a modification of quantum mechanics.
Precise rules are developed in order to formalize the reasoning processes involved in standard non-relativistic quantum mechanics, with the help of analogies from classical physics. A classical or quantum description of a mechanical system involves a {\it framework}, often chosen implicitly, and a {\it statement} or assertion about the system which is either true or false within the framework with which it is associated. Quantum descriptions are no less ``objective'' than their classical counterparts, but differ from the latter in the following respects: (i) The framework employs a Hilbert space rather than a classical phase space. (ii) The rules for constructing meaningful statements require that the associated projectors commute with each other and, in the case of time-dependent quantum histories, that consistency conditions be satisfied. (iii) There are incompatible frameworks which cannot be combined, either in constructing descriptions or in making logical inferences about them, even though any one of these frameworks may be used separately for describing a particular physical system. A new type of ``generalized history'' is introduced which extends previous proposals by Omnès, and Gell-Mann and Hartle, and a corresponding consistency condition which does not involve density matrices or single out a direction of time. Applications which illustrate the formalism include: measurements of spin, two-slit diffraction, and the emergence of the classical world from a fully quantum description.
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 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.
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.
This paper investigates whether locally consistent quantum probability histories can always be combined into a single globally consistent unitary history. It demonstrates, through an explicit and independently verified counterexample in the simplest nontrivial setting, that every three-time subset of a multi-time history can be individually realizable even though no single sequence of unitary transformations can reproduce the complete collection of transition probabilities. The paper identifies shared connecting phases as the mechanism behind this local-to-global obstruction, develops a general parameter-counting condition for when global realization may be possible, and relates the result to broader quantum compatibility and marginal problems as well as the independently developed Quantum Composition Paradox.