Quantum conditional and quantum conditional probability

Miklós Rédel

Robert Furber, Bart Jacobs

This paper presents a categorical account of conditional probability, covering both the classical and the quantum case. Classical conditional probabilities are expressed as a certain "triangle-fill-in" condition, connecting marginal and joint probabilities, in the Kleisli category of the distribution monad. The conditional probabilities are induced by a map together with a predicate (the condition). The latter is a predicate in the logic of effect modules on this Kleisli category. This same approach can be transferred to the category of C*-algebras (with positive unital maps), whose predicate logic is also expressed in terms of effect modules. Conditional probabilities can again be expressed via a triangle-fill-in property. In the literature, there are several proposals for what quantum conditional probability should be, and also there are extra difficulties not present in the classical case. At this stage, we only describe quantum systems with classical parametrization.

Sylvia Pulmannová

Dėnes Petz

Ignacio Pérez, Alfredo Luis

Abstract We investigate the consistency of conditional quantum probabilities. This is whether there is compatibility between the Kolmogorov–Bayes conditional probabilities and the Born rule. We show that they are not compatible in the sense that there are situations where there is no legitimate density matrix that may reproduce the conditional statistics of the other observable via the Born rule. This is to say that the Gleason theorem does not apply to conditional probabilities. Moreover, we show that when this occurs the joint statistics is nonclassical. We show that conditional probabilities are not equivalent to state reduction, so these results do not affect the validity of the Lüders expression.

K Ivana

Kpka’s D-poset is a very important notion in quantum structures. In this paper the conditional probability on the Kpka’s D-posets is studied. The notion of conditional probability is introduced and the basic properties of conditional probability are proved.