Geometry of the state spaces in quantum probability

Attila Andai, Attila Lovas

Multiplying a likelihood function with a positive number makes no difference in Bayesian statistical inference, therefore after normalization the likelihood function in many cases can be considered as probability distribution. This idea led Aitchison to define a vector space structure on the probability simplex in 1986. Pawlowsky-Glahn and Egozcue gave a statistically relevant scalar product on this space in 2001, endowing the probability simplex with a Hilbert space structure. In this paper, we present the noncommutative counterpart of this geometry. We introduce a real Hilbert space structure on the quantum mechanical finite dimensional state space. We show that the scalar product in quantum setting respects the tensor product structure and can be expressed in terms of modular operators and Hamilton operators. Using the quantum analogue of the log-ratio transformation, it turns out that all the newly introduced operations emerge naturally in the language of Gibbs states. We show an orthonormal basis in the state space and study the introduced geometry on the space of qubits in details.

Karl Gustafson

Dorje C. Brody, Lane P. Hughston

Abstract Our purpose here is to draw attention to an interesting relationship that can be shown to hold between (a) information geometry, and (b) quantum geometry. By ‘information geometry’, we mean the natural geometry associated with families of probability distributions. This is a subject that goes back more than half a century to the pioneering work of Rao (1945), who showed that the Fisher information matrix associated with a parametrised family of probability dis tributions gives rise to a natural Riemannian metric on the parameter space. This then allows one to speak in a precise way of the ‘distance’ between two probability distributions, an idea that turns out to be useful in problems of statistical inference. An extensive literature has thus developed, following this line of enquiry, on applications of differential geometry to statistics (see, e.g., Amari (1985), Barndorff-Nielsen et al. (1986), Murray and Rice (1993)). By ‘quantum geometry’, on the other hand, we mean the geometry of the manifold of states associated with a given quantum mechanical system. In particular, we consider the manifold of ‘pure’ states associated with a given complex Hilbert space. This is the space of ‘rays’ through the origin in the Hilbert space, which has the structure of a complex projective space. It is known that the usual rules of quantum mechanics allow one to construct a natural metric on this space, namely the Fubini-Study metric, the specification of which is equivalent to all the familiar structure associated with ordinary quantum mechanics.

Pavel Bóna

Prasenjit Deb

Johan G. F. Belinfante

The set of pure quantum states is described as an abstract space with a geometry determined by transition probabilities. We describe all possible structures for three-dimensional transition probability spaces with less than ten states, as well as some even larger spaces of a certain symmetric type. It is shown that the orthoclosed subspaces of a transition probability space form an atomistic orthomodular poset.