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.
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.
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 manifold structure of subsets of classical probability distributions and quantum density operators in infinite dimensions is investigated in the context of 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∗ -algebra A A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states S S of a possibly infinite-dimensional, unital C∗ C∗ -algebra A A is partitioned into the disjoint union of the orbits of an action of the group G G of invertible elements of A A . Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space H H are smooth, homogeneous Banach manifolds of G=GL(H) G=GL(H) , and, when A A admits a faithful tracial state τ τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through τ τ is a smooth, homogeneous Banach manifold for G G .
A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square–integrable functions. More precisely, by consideration of the square–root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H. Therefore, H embodies the ‘state space’ of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S. The geometry in question is characterised by a natural Riemannian metric (the Fisher–Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramér–Rao and Bhattacharyya inequalities, described in terms of the geometry of the underlying real Hilbert space. As a comprehensive illustration of the utility of the geometric framework, the statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.
A statistical model M is a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square–integrable functions. More precisely, by consideration of the square–root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H. Therefore, H embodies the ‘state space’ of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S. The geometry in question is characterised by a natural Riemannian metric (the Fisher–Rao metric), thus allowing us to formulate the principles of classical statistical inference in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramér–Rao and Bhattacharyya inequalities, described in terms of the geometry of the underlying real Hilbert space. As a comprehensive illustration of the utility of the geometric framework, the statistical model M is then specialised to the case of a submanifold of the state space of a quantum mechanical system. This is pursued by introducing a compatible complex structure on the underlying real Hilbert space, which allows the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds in the case of quantum statistical estimation leads to a set of higher order corrections to the Heisenberg uncertainty relations for canonically conjugate observables.
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.
A statistical model M is specified by a family of probability distributions, characterised by a set of continuous parameters known as the parameter space. This possesses natural geometrical properties induced by the embedding of the family of probability distributions into the space of all square-integrable functions. More precisely, by consideration of the square-root density function we can regard M as a submanifold of the unit sphere S in a real Hilbert space H. Therefore, H effectively embodies the `state space' of the probability distributions, and the geometry of the given statistical model can be described in terms of the embedding of M in S. The geometry in question is characterised by a natural Riemannian metric (the Fisher-Rao metric), and as a consequence various aspects of classical statistical inference can be formulated in a natural geometric setting. In particular, we focus attention on the variance lower bounds for statistical estimation, and establish generalisations of the classical Cramér-Rao and Bhattacharyya bounds, described in terms of the geometry of the underlying real Hilbert space. The statistical model M can then be specialised to the case of a submanifold of the state space of a quantum mechanical system. This can be pursued by introducing a compatible complex structure on the underlying real Hilbert space, thus allowing the operations of ordinary quantum mechanics to be reinterpreted in the language of real Hilbert space geometry. The application of generalised variance bounds to quantum statistical estimation is shown to lead to higher order corrections to the Heisenberg uncertainty relations.