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Cambridge University Press eBooks

On quantum information manifolds

Anna Jenčová

Published Oct 22, 2009• 3 citations
Publisher

Abstract

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.

DOI
10.1017/cbo9780511642401.017

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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.
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.
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.
Let H 0 be a selfadjoint operator such that Tr e −βH 0 is of trace class for some β < 1, and let χ ɛ denote the set of ɛ-bounded forms, i.e., ∥(H 0 +C) −1/2−ɛ X(H 0 +C) −1/2+ɛ ∥ < C for some C > 0. Let χ := Span ∪ ɛ∈(0,1/2] χ ɛ . Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρ x = e −H 0 −X−ψx , X ∈ χ. We show that if Tr e −H 0 = 1. 1. the map Φ, [Formula: see text] is a quantum Young function defined on χ 2. The Orlicz space defined by Φ is the tangent space of [Formula: see text] at ρ 0 ; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ 0 , consisting of p-nearby states (those [Formula: see text] obeying C −1 ρ 1+p ≤ σ ≤ Cρ 1 − p for some C > 1) admits a flat affine connection known as the (−1) connection, and the span of this set is part of the cotangent space of [Formula: see text] 4. These dual structures extend to the completions in the Luxemburg norms.
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.
Abstract The manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of C∗C^{*}C∗ C∗ -algebras and actions of Banach-Lie groups. Specificaly, classical probability distributions and quantum density operators may be both described as states (in the functional analytic sense) on a given C∗C^{*}C∗ C∗ -algebra A\mathscr {A}A A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states S\mathscr {S}S S of a possibly infinite-dimensional, unital C∗C^{*}C∗ C∗ -algebra A\mathscr {A}A A is partitioned into the disjoint union of the orbits of an action of the group G\mathscr {G}G G of invertible elements of A\mathscr {A}A A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space H\mathcal {H}H H are smooth, homogeneous Banach manifolds of G=GL(H)\mathscr {G}=\mathcal {GL}(\mathcal {H})G=GL(H) G=GL(H) , and, when A\mathscr {A}A A admits a faithful tracial state τ\tau τ τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through τ\tau τ τ is a smooth, homogeneous Banach manifold for G\mathscr {G}G G .
Abstract One of the key features of information geometry in the classical setting is the existence of a metric structure and a family of connections on the space of probability distributions. The uniqueness of the Fisher–Rao metric and the duality of these connections is at the heart of classical information geometry. However, these features do not carry over straightforwardly to quantum systems, where a Hermitian inner product structure on the Hilbert space induces a metric on the complex projective space of pure states, the Fubini–Study tensor, which is preserved under unitary evolution. In this work, we explore how the modification of the Hermitian tensor structure on the projective space may affect the geometry of pure quantum states, and whether such generalisations can be used to define dual connections with a direct correspondence to classical probability distribution functions, modified by the presence of a non-trivial phase. We show that it is indeed possible to construct a family of connections that are dual to each other in a generalised sense with respect to the real-valued sector of the Fubini–Study tensor. Using this biorthogonal formalism, we systematically classify the four types of tensors that can arise when the dynamics of a quantum system are governed by a non-Hermitian Hamiltonian, identifying both the complex-valued metric and the Berry curvature. Finally, we elucidate the role of the metric in a quantum natural gradient descent optimisation problem, generalised to the non-Hermitian case for a suitable choice of cost function.