Quantum Statistical Manifolds: The Finite-Dimensional Case

Jan Naudts

R. F. Streater

We show that a choice of Young function, for quantum states given by density operators, leads to an Orlicz norm such that the set of states of Cramér class becomes a Banach manifold. A comparison is made with the case studied by Pistone and Sempi, which arises in the theory of non-parametric estimation in classical statistics. The work of Pistone and Sempi The work of (Pistone and Sempi 1995) arises as a generalisation to infinitely many parameters of the theory of the best estimation of parameters of a probability distribution, using the data obtained by sampling. It is also sometimes called ‘nonparametric estimation’. In 1995, Pistone and Sempi obtained a notable formalism, making use of an Orlicz space. From the point of view of quantum mechanics, the classical case corresponds to the special case where all observables generate an abelian algebra. The quantum case of a finite-dimensional Hilbert space leads to the theory of quantum information, but does not involve delicate questions of topology; this is because all norms on a space of finite dimension are equivalent. The question arises, whether we can imitate the use of an Orlicz norm in the infinite-dimensional case. We here show that this is possible, by completing the outline made earlier (Streater 2004a). We must start with a brief review of the classical case. We follow (Streater 2004a), with minor corrections. (Pistone and Sempi 1995) develop a theory of best estimators (of minimum variance) among all locally unbiased estimators, in classical statistical theory.

Manuel Hohmann

Anna Jenčová

In the finite dimensional case, we introduce a family of torsion-free affine connections in the manifold of all invertible density matrices. These are the quantum version of Amari’s α-connections, based on the α-representation of the tangent space. The dual connections with respect to a monotone metric are, in general, not torsion-free. We compute the torsion and the Riemannian curvature. Finally, geodesics are used to define a divergence.

Jan Naudts

Quantum information geometry studies families of quantum states by means of differential geometry. A new approach is followed with the intention to facilitate the introduction of a more general theory in subsequent work. To this purpose, the emphasis is shifted from a manifold of strictly positive density matrices to a manifold of faithful quantum states on the C*-algebra of bounded linear operators. In addition, ideas from the parameter-free approach to information geometry are adopted. The underlying Hilbert space is assumed to be finite-dimensional. In this way, technicalities are avoided so that strong results are obtained, which one can hope to prove later on in a more general context. Two different atlases are introduced, one in which it is straightforward to show that the quantum states form a Banach manifold, the other which is compatible with the inner product of Bogoliubov and which yields affine coordinates for the exponential connection.

Carlos Alberto Alcalde