Jean-Paul Marchand

Noncommutative Probability on von Neumann Algebras

We generalize ordinary probability theory to those von Neumann algebras A, for which Dyes generalized version of the Radon‐Nikodym theorem holds. This includes the classical case in which A is an Abelian von Neumann algebra generated by an observable or complete set of commuting observables. Via Gleasons theorem, this also includes the case of ordinary quantum mechanics, in which A=B(H) is the von Neumann algebra of all bounded operators on a separable Hilbert space H. Particular consideration is given to the concepts of conditioning, sufficient statistics, coarse‐graining, and filtering.

The Inverse Decay Problem

Let UK(t) be a one‐parameter operator family of positive type in a Hilbert space K and U(t) its minimal unitary dilation with infinitesimal generator H. If UK(t) is a contractive semigroup, then H is not positive. If in addition UK(t)→0 for t → ∞, then there exists a state φ∈K on which H is not defined. We interpret these and other results in the context of the quantum‐mechanical theory of unstable particles and the scattering theory of Lax and Phillips.

Bures distance and relative entropy

We have previously constructed an entropy functional which characterizes statistical inference from partial measurement by maximum relative entropy. Here we discuss the mathematical properties of this concept in greater detail and establish its relation to the Bures distance and the Uhlmann transition probability.

Relative coarse-graining

The problem of statistical inference based on a partial measurement (“coarse-graining”) requires the specification of an a priori distribution. We reformulate the ordinary theory such as to encompass systematically a wide range of a priori distributions (“relative coarse-graining”). This is done in a mathematical setting which admits an interpretation in both classical probability and quantum mechanics. The formalism is illustrated in a few simple examples, such as the die whose geometrical shape is known, the spin in thermal equilibrium with an unknown reservoir, and the position measurement of a one-dimensional particle. It is shown that some of the limitations of the usual theory are a consequence of the fact that it is restricted to “equidistributed” (symmetric) a priori states.

Conditional expectations on von Neumann algebras: A new approach

Abstract In the framework of non-commutative probability theory on von Neumann algebras the concept of conditional expectation is redefined in such a way that it exists relative to arbitrary a priori distributions and subalgebras. We derive explicit expressions for the expectations relative to Abelian subalgebras and establish their connection with the concept of coarse-graining.

Statistical Inference and Entropy

We construct an entropy function such that statistical inference with respect to a partial measurement and a given a priori distribution is characterized by maximal entropy.

Statistical Inference and Quantum Mechanical Measurement

We analyze the quantum mechanical measuring process from the standpoint of information theory. Statistical inference is used in order to define the most likely state of the measured system that is compatible with the readings of the measuring instrument and the a priori information about the correlations between the system and the instrument. This approach has the advantage that no reference to the time evolution of the combined system need be made. It must, however, be emphasized that the result is to be interpreted as the statistically inferred state of the original system rather than the state of the system after measurement. The phenomenon of “reduction of states” appears in this light as a consequence of incomplete information rather than the physical interaction between measured system and measuring instrument.

A note on relative entropy

The quantum-mechanical concept of relative entropy is discussed from an information-theoretic point of view. We show that not all definitions found in the recent literature are equally suitable for the purpose of statistical inference by entropy maximization.

Statistical inference in coupled quantum systems

We discuss the problem of statistical inference in a coupled system. The method is based on the concept of inference introduced in [1].

Statistical Inference in Quantum Mechanics

This lecture is a brief account of a new theory of statistical inference which is applicable to classical and quantum physics and generalizes the concept of coarse-graining. The mathematical setting is non-commutative probability theory on von Neumann algebras.