Hamilton-Jacobi theory and Information Geometry

07/11/2017
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Recently, a method to dynamically de ne a divergence function D for a given statistical manifold (M; g ; T) by means of the Hamilton-Jacobi theory associated with a suitable Lagrangian function L on TM has been proposed. Here we will review this construction and lay the basis for an inverse problem where we assume the divergence function D to be known and we look for a Lagrangian function L for which D is a complete solution of the associated Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to replace probability distributions with probability amplitudes.

Hamilton-Jacobi theory and Information Geometry

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application/pdf Hamilton-Jacobi theory and Information Geometry Florio M. Ciaglia, Fabio Di Cosmo, Giuseppe Marmo
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contenu protégé  Document accessible sous conditions - vous devez vous connecter ou vous enregistrer pour accéder à ou acquérir ce document.
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Recently, a method to dynamically de ne a divergence function D for a given statistical manifold (M; g ; T) by means of the Hamilton-Jacobi theory associated with a suitable Lagrangian function L on TM has been proposed. Here we will review this construction and lay the basis for an inverse problem where we assume the divergence function D to be known and we look for a Lagrangian function L for which D is a complete solution of the associated Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to replace probability distributions with probability amplitudes.
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Hamilton-Jacobi theory and Information Geometry Florio M. Ciaglia12 , Fabio Di Cosmo12 , and Giuseppe Marmo12 1 Dipartimento di Fisica, Università di Napoli “Federico II”, Via Cinthia Edificio 6, I-80126 Napoli, Italy 2 INFN-Sezione di Napoli, Via Cinthia Edificio 6, I-80126 Napoli, Italy Abstract. Recently, a method to dynamically define a divergence func- tion D for a given statistical manifold (M , g , T) by means of the Hamilton- Jacobi theory associated with a suitable Lagrangian function L on TM has been proposed. Here we will review this construction and lay the ba- sis for an inverse problem where we assume the divergence function D to be known and we look for a Lagrangian function L for which D is a com- plete solution of the associated Hamilton-Jacobi theory. To apply these ideas to quantum systems, we have to replace probability distributions with probability amplitudes. 1 Introduction In the field of information geometry, divergence functions are ubiquitous objects. A divergence function D is a positive semi-definite two-point function defined on M × M, where M is the manifold underlying the statistical model (M , g , T) under study (see [1,2,3]), such that D(m1 , m2) = 0 if and only if m1 = m2. Roughly speaking, the value D(m1 , m2) is interpreted as a “measure of differ- ence” between the probability distributions parametrized by m1 and m2. The exact meaning of this difference depends on the explicit model considered. If we imbed classical probabilities in the space of quantum systems, i.e., we re- place probabilities with probability amplitudes, it is still possible to define di- vergence functions and derive metric tensors for quantum states. For instance, when M = P(H) is the space of pure states of a quantum system with Hilbert space H, Wootter has shown (see [21]) that a divergence function D providing a meaningful notion of statistical distance between pure states may be intro- duced by means of the concepts of distinguishability and statistical fluctuations in the outcomes of measurements. It turns out that this statistical distance co- incides with the Riemannian geodesic distance associated with the Fubini-Study metric on the complex projective space. On the other hand, when M = Pn is the manifold of positive probability measure on χ = {1, ..., n}, and D is the Kullback-Leibler divergence function (see [1,2,3]), then the meaning of the “dif- ference” between m1 and m2 as measured by D is related with the asymptotic estimation theory for an empirical probability distribution extracted from inde- pendent samples associated with a given probability distribution (see [1]). One of the main features of a divergence function D is the possibility to extract from it a metric tensor g, and a skewness tensor T on M using an algorithm involving iterated derivatives of D and the restriction to the diagonal of M × M (see [1,2,3]). Given a statistical model (M , g , T) there is always a divergence func- tion whose associated tensors are precisely g and T (see [17]), and, what is more, there is always an infinite number of such divergence functions. In the context of classical information geometry, all statistical models share the “same” metric g, called the Fisher-Rao metric. This metric arise naturally when we consider M as immersed in the space P(χ) of probability distributions on the measure space χ, and, provided some additional requirements on symmetries are satisfied, it is essentially unique (see [3,9]). This means that, once the statistical manifold M ⊂ P(χ) is chosen, all the admissible divergence functions must give back the Fisher-Rao metric g. On the other hand, different admissible divergence func- tions lead to different third order symmetric tensors T. Quite interestingly, the metric tensor g is no longer unique in the quantum context (see [20]). In a recent work ([10]), a dynamical approach to divergence functions has been proposed. The main idea is to read a divergence function D, or more gen- erally, a potential function for a given statistical model (M , g , T), as the Hamil- ton principal function associated with a suitably defined Lagrangian function L on TM by means of the Hamilton-Jacobi theory (see [7,11]). From this point of view, a divergence function D becomes a dynamical object, that is, the function D is no more thought of as some fixed kinematical function on the double of the manifold of the statistical model, but, rather, it becomes the Hamilton prin- cipal function associated with a Lagrangian dynamical system on the tangent bundle of the manifold of the statistical model. In the variational formulation of dynamics [11], the solutions of the equations of motion are expressed as the critical points of the action functional: I (γ) = Z tfin tin L (γ , γ̇) dt , (1) where γ are curves on M with fixed extreme points m(tin) = min and m(tfin) = mfin, and L is the Lagrangian function of the system. In order to avoid tech- nical details, we will always assume that L is a regular Lagrangian (see [18]). The evaluation of the action functional on a critical point γc gives a two-point function3 : S (min , mfin) = I(γc) , (2) which is known in the literature as the Hamilton principal function. When a given dynamics admits of alternative Lagrangian description, it is possible to integrate alternative Lagrangians along the same integral curves and get different potential functions. If the determinant of the matrix of the mixed partial derivatives of S is different from zero, then it is possible to prove (see [11]) that S is a complete 3 In general, this function depends on the additional parameters tin and tfin, however we will always take tin = 0 and tfin = 1. solution of the Hamilton-Jacobi equation for the dynamics: H  x , ∂S ∂x , t  + ∂S ∂t = 0 , (3) where H is the Hamiltonian function ([8]) associated with the Lagrangian L. In this case, S(min , mfin) is called a complete solution for the Hamilton-Jacobi the- ory. It turns out that the existence of a complete solution S forces the dynamical system associated with the Lagrangian function L to be completely integrable, that is, to adimit n = dim(M) functionally independent constants of the motion which are transversal to the fibre of TM (see [7]). The main result of [10] is to prove that, given any statistical model (M , g , T), the Lagrangian functions: Lα = 1 2 gjk(x)vj vk + α 6 Tjkl(x)vj vk vl , (4) labelled by the one-dimensional real parameter α, are such that their associated Hamilton principal functions are potential functions for (M , g , T) in the sense that they allow to recover g and T as follows: ∂2 Sα ∂xj fin∂xk in xin