In the consistent histories formulation of quantum theory, the probabilistic predictions and retrodictions made from observed data depend on the choice of a consistent set. We show that this freedom allows the formalism to retrodict contrary propositions which correspond to orthogonal commuting projections and which each have probability one. We also show that the formalism makes contrary probability one predictions when applied to Gell-Mann and Hartle's generalized time-neutral quantum mechanics.
It was pointed out recently [A. Kent, Phys. Rev. Lett. 78 (1997) 2874] that the consistent histories approach allows contrary inferences to be made from the same data. These inferences correspond to projections
P and
Q, belonging to different consistent sets, with the properties that
PQ=QP=0 and
P=1−Q. To many, this seems undesirable in a theory of physical inferences. It also raises a specific problem for the consistent histories formalism, since that formalism is set up so as to eliminate contradictory inferences, i.e. inferences
P and
Q where
P=1−Q. Yet there seems to be no sensible physical distinction between contradictory and contrary inferences. It seems particularly hard to defend the asymmetry, since (i) there is a well-defined quantum histories formalisms which admits both contradictory and contrary inferences, and (ii) there is also a well-defined formalism, based on ordered consistent sets of histories, which excludes both. In a recent comment, Griffiths and Hartle, while accepting the validity of the examples given in the above paper, restate their own preference for the consistent histories formalism. As this brief reply explains, in so doing, they fail to address the arguments made against their approach to quantum theory.
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.
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.
From data in the present we can predict the future and retrodict the past. These predictions and retrodictions are for histories – most simply time sequences of events. Quantum mechanics gives probabilities for individual histories in a decoherent set of alternative histories. This paper discusses several issues connected with the distinction between prediction and retrodiction in quantum cosmology: the difference between classical and quantum retrodiction, the permanence of the past, why we predict the future but remember the past, the nature and utility of reconstructing the past(s), and information theoretic measures of the utility of history.