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International Journal of Engineering Science

Information geometry in functional spaces of classical and quantum finite statistical systems

Roman S. Ingarden

Published Jan 1, 1981• 42 citations
Publisher
DOI
10.1016/0020-7225(81)90153-1

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Information geometry, dynamics and discrete quantum mechanics

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We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the picture. We do this in three steps. Our starting point is information geometry, the natural geometry of the space of probability distributions. Dynamics requires additional structure. To evolve the Pk, we introduce coordinates Sk canonically conjugate to the Pk and a symplectic structure. We then seek to extend the metric structure of information geometry, to define a geometry over the full space of the Pk and Sk. Consistency between the metric tensor and the symplectic form forces us to introduce a Kähler geometry. The construction has notable features. A complex structure is obtained in a natural way. The canonical coordinates of the ψk = PkeiSk Kähler space are precisely the wave functions of quantum mechanics. The full group of unitary transformations is obtained. Finally, one may associate a Hilbert space with the Kähler space, which leads to the standard version of quantum theory. We also show that the metric that we derive here using purely geometrical arguments is precisely the one that leads to Wootters’ expression for the statistical distance for quantum systems.

arXiv:1207.6718•AIP conference proceedings•Jan 1, 2013•18 citations
A Riemannian metric is defined on the state space of a finite quantum system by the canonical correlation (or Kubo–Mori/Bogoliubov scalar product). This metric is infinitesimally induced by the (nonsymmetric) relative entropy functional or the von Neumann entropy of density matrices. Hence its geometry expresses maximal uncertainty. It is proven that the metric is monotone under stochastic mappings, however, an example shows that it is not the only such Riemannian metric. This fact is remarkable because in the probabilistic case, the Markovian monotonicity property characterizes the Fisher information metric. The essential difference appears in the curvatures of a classical state space and a quantum one. A conjecture is made that the scalar curvature is monotone with respect to the ‘‘more mixed’’ (statistical) partial order of density matrices. Furthermore, an information inequality resembling the Cramér–Rao inequality of classical statistics is established. The inequality provides a lower bound for the canonical correlation matrix (of an unbiased observable) and it is saturated when a (partial) observation level and the corresponding family of coarse-grained states are considered.
We formulate scattering experiments in terms of parameterized finite measures on outcome spaces and develop the associated information geometry defined by the Fisher metric. The decomposition of a scattering measure into total mass and normalized shape separates Fisher information into contributions from changes in overall scattering strength and redistribution among outcomes. Measurement processes are represented by probability or sub-probability kernels acting on the scattering measure. Such kernels contract Fisher information, placing a broad class of experimental operations within a common information-theoretic framework. This formulation connects scattering experiments to the theory of optimal experimental design through the Fisher geometry of scattering measures. The same finite-measure structure arises naturally in quantum measurements with incomplete detection, clarifying its place in the hierarchy from quantum measurement to classical outcome statistics.
We consider a system with a discrete configuration space. We show that the geometrical structures associated with such a system provide the tools necessary for a reconstruction of discrete quantum mechanics once dynamics is brought into the picture. We do this in three steps. Our starting point is information geometry, the natural geometry of the space of probability distributions. Dynamics requires additional structure. To evolve the Pk, we introduce coordinates Sk canonically conjugate to the Pk and a symplectic structure. We then seek to extend the metric structure of information geometry, to define a geometry over the full space of the Pk and Sk. Consistency between the metric tensor and the symplectic form forces us to introduce a Kähler geometry. The construction has notable features. A complex structure is obtained in a natural way. The canonical coordinates of the ψk = PkeiSk Kähler space are precisely the wave functions of quantum mechanics. The full group of unitary transformations is obtained. Finally, one may associate a Hilbert space with the Kähler space, which leads to the standard version of quantum theory. We also show that the metric that we derive here using purely geometrical arguments is precisely the one that leads to Wootters’ expression for the statistical distance for quantum systems.