Quantum conditional probability

American Journal of Physics

Conditional probabilities in quantum systems which have both initial and final boundary conditions are commonly evaluated using the Aharonov-Bergmann-Lebowitz rule. In this short note we present a seemingly disturbing paradox that appears when applying the rule to systems with slightly broken degeneracies. In these cases we encounter a singular limit-the probability "jumps" when going from perfect degeneracy to negligibly broken one. We trace the origin of the paradox and solve it from both traditional and modern perspectives in order to highlight the physics behind it: the necessity to take into account the finite resolution of the measuring device. As a practical example, we study the application of the rule to the Zeeman effect. The analysis presented here may stress the general need to first consider the governing physical principles before heading to the mathematical formalism, in particular when exploring puzzling quantum phenomena.

New Journal of Physics, 2014

Buschʼs theorem deriving the standard quantum probability rule can be regarded as a more general form of Gleasonʼs theorem. Here we show that a further generalization is possible by reducing the number of quantum postulates used by Busch. We do not assume that the positive measurement outcome operators are effects or that they form a probability operator measure. We derive a more general probability rule from which the standard rule can be obtained from the normal laws of probability when there is no measurement outcome information available, without the need for further quantum postulates. Our general probability rule has prediction-retrodiction symmetry and we show how it may be applied in quantum communications and in retrodictive quantum theory.

International Journal of Theoretical Physics, 1986

We analyze two approaches to conditional probability, The first approach follows Gudder and Marchand, M~czyfisky, Cassinelli and Beltrametti, Cassinelli and Truini. The second approach follows R6nyi and Kalmfir. The main result is a characterization of the first approach with the help of a function, similarly as in the second approach.

Physical Review A, 1998

We discuss quantum conditional probability and its applications to deterministic hidden-variable models. We derive empirical tests corresponding to mathematical no-go proofs, providing rigorous statistical tests based on experimental outcomes. Evidently, it now possible to examine the statistical power of the empirical tests, and place confidence intervals on the parameters that precisely measure the departure of hidden-variable models from quantum experimental outcomes. Moreover, reinterpretation of well-known results in the light of quantum conditional probability provides other experimental demonstrations and no-go proofs: outcomes for the familiar Young double-slit experiment show that there are no deterministic hidden-variable models of the type considered by Kochen and Specker ͓J.

Axioms, 2014

There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.

We study the origin of quantum probabilities as arising from non-boolean propositionaloperational structures. We apply the method developed by Cox to non distributive lattices and develop an alternative formulation of non-Kolmogorvian probability measures for quantum mechanics. By generalizing the method presented in previous works, we outline a general framework for the deduction of probabilities in general propositional structures represented by lattices (including the non-distributive case).

Foundations of Physics, 1988

We show that the quantum mechanical rules for manipulating probabilities follow naturally from standard probability theory. We do this by generalizing a result of Khinchin regarding characteristic functions. From standard probability theory we obtain the methods" usually assoeiated with quantum theory, that is', the operator method, eigenvalues, the Born rule, and the fact that only the eigenvalues of the operator have nonzero probability. We discuss the general question as to why quantum mechanics seemingly necessitates different methods than standard probability theory and argue that the quantum mechanical method is much richer in its ability to generate a wide variety of probability distributions which are inaccessibe by way of standard probability theory.

Journal of Mathematical Physics

The aim of this paper is to analyze the reconstructability of quantum mechanics from classical conditional probabilities representing measurement outcomes conditioned on measurement choices. We will investigate how the quantum mechanical representation of classical conditional probabilities is situated within the broader frame of noncommutative representations. To this goal, we adopt some parts of the quantum formalism and ask whether empirical data can constrain the rest of the representation to conform to quantum mechanics. We will show that as the set of empirical data grows conventional elements in the representation gradually shrink and the noncommutative representations narrow down to the unique quantum mechanical representation.