Infinite Dimensional Analysis, Quantum Probability and Related Topics

Quantum entanglements, describing truly quantum couplings, are studied and classified for discrete compound states. We show that classical-quantum correspondences such as quantum encodings can be treated as d-entanglements leading to a special class of separable compound states. The mutual information for the d-compound and for q-compound (entangled) states leads to two different types of entropies for a given quantum state. The first one is the von Neumann entropy, which is achieved as the supremum of the information over all d-entanglements, and the second one is the dimensional entropy, which is achieved at the standard entanglement, the true quantum entanglement, coinciding with a d-entanglement only in the commutative case. The q-conditional entropy and q-capacity of a quantum noiseless channel, defined as the supremum over all entanglements, is given as the logarithm of the dimensionality of the input von Neumann algebra. It can double the classical capacity, achieved as the supremum over all semiquantum couplings (d-entanglements, or encodings), which is bounded by the logarithm of the dimensionality of a maximal Abelian subalgebra. The entropic measure for essential entanglement is introduced.

We show that, in finite dimensions, the only monotone metrics on the space of invertible density matrices for which the (+1) and (-1) affine connections are mutually dual are constant multiples of the Bogoliubov–Kubo–Mori metric.

While briefly reviewing results about (nets of) generalized conditional expectations and martingale-type convergences, we present an unified version of the topic and prove martingale-type convergence results of monotone nets of generalized conditional expectations in the full generality. In so doing, we fill some gaps.

We show a simple way to generate a Markovian cocycle on H ⊗ Γ (L²(ℝ₊, K)) from a semigroup of bounded operators on (H ⊗ L²(ℝ₊, K)) ⊕ H ⊕ (H ⊗ L²(ℝ_-, K) ⊗ L²(ℝ₊, K))⊕ (H ⊗ L²(ℝ_-, K)).

This paper, based on results obtained in recent years with various coauthors,^1–3,13,53 presents a proposal to extend some classical geometric concepts to a class of infinite-dimensional manifolds such as current groups and to a class of infinite-dimensional bundles including the ones arising in the family index theorem.

The basic idea is to extend the notion of trace underlying many geometric concepts using renormalized traces which are linear functionals on pseudo-differential operators. The definition of "renormalized traces" involves extra data on the manifolds or vector bundles, namely a weight given by a field of elliptic operators which becomes part of the geometric data, leading to the notion of weighted manifold and weighted vector bundle. This weight is a source of anomaly arising typically as a Wodzicki residue of some pseudo-differential operator.

We investigate the anomalies that arise when trying to extend to the infinite-dimensional setting classical results of finite-dimensional geometry such as a Weitzenböck formula, Chern–Weil invariants or the relation between the first Chern form on a complex vector bundle and the curvature on the associated determinant bundle. When comparable, we relate our approach to the one adopted for similar problems in noncommutative geometry.

We introduce a kind of quantum decomposition of generators of a discrete group with a suitable length function and construct a sequence of one-mode interacting Fock spaces associated with a filtration of the group. We show stochastic convergence of such a sequence of interacting Fock spaces and obtain a general stochastic limit theorems on discrete groups. As an application to free groups, we see that the Haagerup states give rise to a Gaussian–Poisson transform through its coherent expression.