ar X iv : m at h - ph / 0 60 40 31 v 1 1 4 A pr 2 00 6 On the curvature of the quantum state space with pull - back metrics ∗

The aim of the paper is to extend the notion of α-geometry in the classical and in the noncommutative case by introducing a more general class of pull-back metrics and to give concrete formulas for the scalar curvature of these Riemannian manifolds. We introduce a more general class of pull-back metrics of the noncommutative state spaces, we pull back the Euclidean Riemannian metric of the space of self-adjoint matrices with functions which have an analytic extension to a neighborhood of the interval ]0, 1[ and whose derivative are nowhere zero. We compute the scalar curvature in this setting, and as a corollary we have the scalar curvature of the classical probability space when it is endowed with such a general pull-back metric. In the noncommutative setting we consider real and complex state spaces too. We give a simplification of Gibilisco’s and Isola’s conjecture for the first nontrivial classical probability space and we present the result of a numerical computation which indicate that the conjecture may be true for the space of real and complex qubits. Introduction The idea that the space of probability distributions can be endowed with Riemannian metric is due to Rao [20], and it was developed by Cencov [6], Amari and Nagaoka [1, 2] and Streater [21] among others. Cencov and Morozova [15] were the first to study the monotone metrics on classical statistical manifolds. They proved that such a metric is unique, up to normalization. The counterpart of this theorem in quantum setting was given by Petz [17], who showed that monotone metrics can be labeled by special operator monotone functions. Some differential geometrical quantities were computed for these manifolds with monotone metrics, one of them is the scalar curvature [8, 9, 14, 19]. The scalar curvature at every state measures the average statistical uncertainty of the state [16, 18]. This is one of the basic ideas of Petz’s conjecture [18], which is about the monotonicity of the scalar curvature with respect to the majorization relation keywords: state space, monotone statistical metric, scalar curvature; MSC: 53C20, 81Q99 [email protected] 1 when the state space is endowed with the Kubo–Mori metric. This conjecture is still unsolved, partial results can be found in [3, 4, 8, 10, 13, 18]. Cencov introduced the α-connections and α-geometry on the space of classical probability distributions [6], it was developed by Amari and Nagaoka [2], Gibilisco and Pistone [12]. Gibilisco and Isola showed that the idea of Petz’s conjecture can be extended to α-geometries. They also have another conjecture about the monotonicity of the scalar curvature when the classical and the noncommutative probability spaces are endowed with α-geometry [10]. This conjecture was proved for the space of probability distributions on a set which consists of two elements. They used the classical curvature formula for this one dimensional manifold, since its scalar curvature is zero. The aim of the paper is to extend the notion of α-geometry in the classical and in the noncommutative case by introducing a more general class of pull-back metrics and to give concrete formulas for the scalar curvature of these Riemannian manifolds. The classical and noncommutative probability spaces are one codimensional submanifolds of flat spaces. We use a general formula from differential geometry to compute the curvature of these one codimensional submanifolds. In the first section we do this computation for classical probability spaces when they are endowed with a special pull-back metric, with α-geometry. In the second section we introduce a more general class of pull-back metrics of the noncommutative state spaces, we pull back the Euclidean Riemannian metric of the space of self-adjoint matrices with functions which have analytic extension to a neighborhood of the interval ]0, 1[ and whose derivative are nowhere zero. We compute the scalar curvature in this setting, and as a corollary we have the scalar curvature of the classical probability space when it is endowed with such a general pull-back metric. In the noncommutative setting we consider real and complex state spaces too. We check the theorems in some special cases when the result is known from somewhere else, since the computation is a bit lengthy. Finally in the third section we give a simplification of the conjecture for the first nontrivial classical probability space and we present the result of a numerical computation which indicate that the conjecture may be true for the space of real and complex qubits. 1 Classical α-geometries We work on a special classical statistical manifold, on the space of probability distributions on a finite set. Definition 1 For every number n ∈ N let Pn denote the open set of the probability distributions on a space which consists of n points, that is Pn = { (θ1, . . . , θn) ∈ R ∣ ∣ ∣ ∀k ∈ {1, . . . , n} : θk > 0, n ∑