1,323 research outputs found
L\"uders' and quantum Jeffrey's rules as entropic projections
We prove that the standard quantum mechanical description of a quantum state
change due to measurement, given by Lueders' rules, is a special case of the
constrained maximisation of a quantum relative entropy functional. This result
is a quantum analogue of the derivation of the Bayes--Laplace rule as a special
case of the constrained maximisation of relative entropy. The proof is provided
for the Umegaki relative entropy of density operators over a Hilbert space as
well as for the Araki relative entropy of normal states over a W*-algebra. We
also introduce a quantum analogue of Jeffrey's rule, derive it in the same way
as above, and discuss the meaning of these results for quantum bayesianism
Laboratory Games and Quantum Behaviour: The Normal Form with a Separable State Space
The subjective expected utility (SEU) criterion is formulated for a particular four-person âlaboratory gameâ that a Bayesian rational decision maker plays with Nature, Chance, and an Experimenter who influences what quantum behaviour is observable by choosing an orthonormal basis in a separable complex Hilbert space of latent variables. Nature chooses a state in this basis, along with an observed data series governing Chance's random choice of consequence. When Gleason's theorem holds, imposing quantum equivalence implies that the expected likelihood of any data series w.r.t. prior beliefs equals the trace of the product of appropriate subjective density and likelihood operators.
Quantum Cognition based on an Ambiguous Representation Derived from a Rough Set Approximation
Over the last years, in a series papers by Arrechi and others, a model for
the cognitive processes involved in decision making has been proposed and
investigated. The key element of this model is the expression of apprehension
and judgement, basic cognitive process of decision making, as an inverse Bayes
inference classifying the information content of neuron spike trains. For
successive plural stimuli, it has been shown that this inference, equipped with
basic non-algorithmic jumps, is affected by quantum-like characteristics. We
show here that such a decision making process is related consistently with
ambiguous representation by an observer within a universe of discourse. In our
work ambiguous representation of an object or a stimuli is defined by a pair of
maps from objects of a set to their representations, where these two maps are
interrelated in a particular structure. The a priori and a posteriori
hypotheses in Bayes inference are replaced by the upper and lower
approximation, correspondingly, for the initial data sets each derived with
respect to a map. We show further that due to the particular structural
relation between the two maps, the logical structure of such combined
approximations can only be expressed as an orthomodular lattice and therefore
can be represented by a quantum rather than a Boolean logic. To our knowledge,
this is the first investigation aiming to reveal the concrete logic structure
of inverse Bayes inference in cognitive processes.Comment: 23 pages, 8 figures, original research pape
Quantum mechanics as a theory of probability
We develop and defend the thesis that the Hilbert space formalism of quantum
mechanics is a new theory of probability. The theory, like its classical
counterpart, consists of an algebra of events, and the probability measures
defined on it. The construction proceeds in the following steps: (a) Axioms for
the algebra of events are introduced following Birkhoff and von Neumann. All
axioms, except the one that expresses the uncertainty principle, are shared
with the classical event space. The only models for the set of axioms are
lattices of subspaces of inner product spaces over a field K. (b) Another axiom
due to Soler forces K to be the field of real, or complex numbers, or the
quaternions. We suggest a probabilistic reading of Soler's axiom. (c) Gleason's
theorem fully characterizes the probability measures on the algebra of events,
so that Born's rule is derived. (d) Gleason's theorem is equivalent to the
existence of a certain finite set of rays, with a particular orthogonality
graph (Wondergraph). Consequently, all aspects of quantum probability can be
derived from rational probability assignments to finite "quantum gambles". We
apply the approach to the analysis of entanglement, Bell inequalities, and the
quantum theory of macroscopic objects. We also discuss the relation of the
present approach to quantum logic, realism and truth, and the measurement
problem.Comment: 37 pages, 3 figures. Forthcoming in a Festschrift for Jeffrey Bub,
ed. W. Demopoulos and the author, Springer (Kluwer): University of Western
Ontario Series in Philosophy of Scienc
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