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arXiv:2209.12761 · Frontiers in Physics

Exponential arcs in manifolds of quantum states

Jan Naudts

Published Feb 7, 2023• 3 citations
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Abstract

The manifold under consideration consists of the faithful normal states on a sigma-finite von Neumann algebra in standard form. Tangent planes and approximate tangent planes are discussed. A relative entropy/divergence function is assumed to be given. It is used to generalize the notion of an exponential arc connecting one state to another. The generator of the exponential arc is shown to be unique up to an additive constant. In the case of Araki’s relative entropy, every self-adjoint element of the von Neumann algebra generates an exponential arc. The generators of the composed exponential arcs are shown to add up. The metric derived from Araki’s relative entropy is shown to reproduce the Kubo–Mori metric. The latter is the metric used in linear response theory. The e- and m-connections describe a dual pair of geometries. Any finite number of linearly independent generators determines a submanifold of states connected to a given reference state by an exponential arc. Such a submanifold is a quantum generalization of a dually flat statistical manifold.

DOI
10.3389/fphy.2023.1042257

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The manifold under consideration consists of the faithful normal states on a sigma-finite von Neumann algebra in standard form. Tangent planes and approximate tangent planes are discussed. A relative entropy/divergence function is assumed to be given. It is used to generalize the notion of an exponential arc connecting one state to another. The generator of the exponential arc is shown to be unique up to an additive constant. In the case of Araki’s relative entropy, every self-adjoint element of the von Neumann algebra generates an exponential arc. The generators of the composed exponential arcs are shown to add up. The metric derived from Araki’s relative entropy is shown to reproduce the Kubo–Mori metric. The latter is the metric used in linear response theory. The e- and m-connections describe a dual pair of geometries. Any finite number of linearly independent generators determines a submanifold of states connected to a given reference state by an exponential arc. Such a submanifold is a quantum generalization of a dually flat statistical manifold.
Introduction The aim of information geometry is to introduce a suitable geometrical structure on families of probability distributions or quantum states. For parametrised statistical models, such structure is based on two fundamental notions: the Fisher information and the exponential family with its dual mixed parametrisation, see for example (Amari 1985, Amari and Nagaoka 2000). For the non-parametric situation, the solution was given by Pistone and Sempi (Pistone and Sempi 1995, Pistone and Rogantin 1999), who introduced a Banach manifold structure on the set P of probability distributions, equivalent to a given one. For each µ ∈ ρ, the authors considered the non-parametric exponential family at µ. As it turned out, this provides a C ∞ -atlas on ρ, with the exponential Orlicz spaces Lφ(µ) as the underlying Banach spaces, here φ is the Young function of the form φ(x) = cosh(x) — 1. The present contribution deals with the case of quantum states: we want to introduce a similar manifold structure on the set of faithful normal states of a von Neumann algebra M. Since there is no suitable definition of a non-commutative Orlicz space with respect to a state φ, it is not clear how to choose the Banach space for the manifold. Of course, there is a natural Banach space structure, inherited from the predualM. But, as it was already pointed out in (Streater 2004), this structure is not suitable to define the geometry of states: for example, any neighbourhood of a state φ contains states such that the relative entropy with respect to φ is infinite.
Abstract The space of quantum states can be endowed with a metric structure using the second order derivatives of the relative entropy, giving rise to the so-called Kubo–Mori–Bogoliubov inner product. We explore its geometric properties on the submanifold of faithful, zero-displacement Gaussian states parameterised by their covariance matrices, deriving expressions for the geodesic equations, curvature tensors and scalar curvature. Our analysis suggests that the curvature of the manifold is strictly monotonic with respect to the von Neumann entropy, and thus can be interpreted as a measure of state uncertainty. This provides supporting evidence for the Petz conjecture in continuous variable systems.
We unify two emergent aspects of quantum theory within a single information-geometricframework based on the Araki relative entropy and the associated Bogoliubov–Kubo–Mori(BKM) metric on a von Neumann algebra state manifold.First, we show that in the commutative reduction to a spatial density subalgebra, theBKM metric becomes the classical Fisher information metric. Its trace gives, after a singlenormalization choice, the expectation value of the Bohmian quantum potential, and thecorresponding Bohmian trajectories arise as Jacobi geodesics of a conformally related metric.The guidance equation is then recovered from the polar decomposition of states satisfyingthe emergent Schr¨odinger equation.Second, we show that the modular flow of the reference state yields the Heisenbergand von Neumann equations, and that the physical Schr¨odinger equation follows under astandard unitary state implementation. In the finite regulated K¨ahler sector, geometricprequantization of the BKM state manifold leads to a discrete horizon area spectrum and touncertainty relations, with an induced gravitational coupling derived from the Seeley–DeWittcoefficient [1].No continuum limit is taken, and no identification with the observed Newton constantis assumed. The results exhibit a common information-geometric framework for Bohmianmechanics, quantum dynamics, and emergent gravitational structures.
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.