Quantum Probability and Applications III

Patrick Suppes, Mario Zanotti

Part I. Foundations of Probability: 1. Necessary and sufficient conditions for existence of a unique measure strictly agreeing with a qualititative probability ordering 2. Necessary and sufficient qualitative axioms for conditional probability 3. On using random relations to generate upper and lower probabilities 4. Conditions on upper and lower probabilities to imply probabilities 5. Qualitative axioms for random-variable representation of extensive quantifiers Part II. Causality and Quantum Mechanics: 6. Stochastic incompleteness of quantum mechanics 7. On the determinism of hidden variable theories with strict correlation and conditional statistical independence of observables 8. A new proof of the impossibility of hidden variables using the principles of exchangeability and identity of conditional distribution 9. When are probabilistic explanations possible? 10. Causality and symmetry 11. New Bell-type inequalities for N>4 necessary for existence of a hidden variable 12. Existence of hidden variables having only upper probabilities Part III. Applications in Education: 13. Mastery learning of elementary mathematics: theory and data.

Luigi Accardi, Wilhelm von Waldenfels

Luigi Accardi, Wilhelm von Waldenfels

Maik Reddiger

Abstract By formulating the axioms of quantum mechanics, von Neumann also laid the foundations of a “quantum probability theory”. As such, it is regarded a generalization of the “classical probability theory” due to Kolmogorov. Outside of quantum physics, however, Kolmogorov’s axioms enjoy universal applicability. This raises the question of whether quantum physics indeed requires such a generalization of our conception of probability or if von Neumann’s axiomatization of quantum mechanics was contingent on the absence of a general theory of probability in the 1920s. This work argues in favor of the latter position. In particular, it shows how to construct a mathematically rigorous theory for non-relativistic N -body quantum systems subject to a time-independent scalar potential, which is based on Kolmogorov’s axioms and physically natural random variables. Though this theory is provably distinct from its quantum-mechanical analog, it nonetheless reproduces central predictions of the latter. Further work may make an empirical comparison possible. Moreover, the approach can in principle be adapted to other classes of quantum-mechanical models. Part II of this series discusses the empirical violation of Bell inequalities in the context of this approach. Part III addresses the projection postulate and the question of measurement.

Accardi, Luigi, Waldenfels, Wilhelm

H. Scutaru

We obtain a general formula for the transition probabilities between any state of the C* algebra of the canonical commutation relations (CCR-algebra) and a squeezed quasifree state (Theorem III.1). Applications of this formula are made for the case of multimode thermal squeezed states of quantum optics using a general canonical decomposition of the correlation matrix valid for any quasifree state. In the particular case of a one-mode CCR-algebra we show that the transition probability between two quasifree squeezed states is a decreasing function of the geodesic distance between the points of the upper half-plane representing these states. In the special case of the purification map it is shown that the transition probability between the state of the enlarged system and the product state of real and fictitious subsystems can be a measure for the entanglement.