Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

Open Systems & Information Dynamics

Duality in Quantum Information Geometry

R. F. Streater

Published Mar 1, 2004• 14 citations
Publisher

Abstract

Let [Formula: see text] be a separable Hilbert space. We consider the manifold [Formula: see text] consisting of density operators ρ on [Formula: see text] such that ρp is of trace class for some p ɛ (0, 1). We say [Formula: see text] is nearby ρ if there exists C > 1 such that C−1 ρ < σ < Cρ. We show that the space of nearby points to ρ can be furnished with the two flat connections known as the (±)-affine structures, which are dual relative to the BKM metric. We furnish [Formula: see text] with a norm making it into a Banach manifold.

DOI
10.1023/b:opsy.0000024757.25401.db

Related papers

Quantum Orlicz Spaces in Information Geometry

R. F. Streater

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.

Open Systems & Information Dynamics•Dec 1, 2004•52 citations

Duality in quantum information manifolds

R. F. Streater

We consider the quantum information manifold whose underlying set M consists of density operators rho with the extra property that some fractional power of rho is of trace class. The topology is defined by defining a neighbourhood of a point rho to be all density operators that dominate rho and are dominated by rho. We show that this is the same set as that of all states whose relative Hamiltonian X in the sense of Araki is bounded, and such that X(t) is holomorphic in the circle |t| less than 1/2. Here, X(t) is the time evolution of X determined by the modular automorphism defined by rho. We show that M is a Banach manifold in Araki's norm, and that both the canonical and the mixture affine connections can be defined. These are dual relative to the Kubo-Mori metric, and so generalise Amari's dual theory to quantum theory in infinite dimensions.

arXiv:math-ph/0308037•Aug 28, 2003•1 citation

Duality versus dual flatness in quantum information geometry

Nihat Ay, Wilderich Tuschmann

We investigate questions in quantum information geometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finite-dimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal. We prove, moreover, that there is never a dually flat structure on the set of pure states. Thus any general theory of quantum information geometry that involves duality concepts must inevitably be based on dual structures which are nonflat.

Journal of Mathematical Physics•Mar 20, 2003•18 citations

On quantum information manifolds

Anna Jenčová

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.

Cambridge University Press eBooks•Oct 22, 2009•3 citations

Infinite dimensional quantum information geometry

Peter Sollich A, A C C Coolen, Lane P. Hughston +1 more

We present the construction of an infinite dimensional Banach manifold of quantum mechanical states on a Hilbert space H using different types of small perturbations of a given Hamiltonian H-0. We provide the manifold with a flat connection, called the exponential connection, and comment on the possibility of introducing the dual mixture connection.

•Jan 1, 2001•0 citations

Infinite dimensional quantum information geometry

Matheus R. Grasselli

We present the construction of an infinite dimensional Banach manifold of quantum mechanical states on a Hilbert space H using different types of small perturbations of a given Hamiltonian H0. We provide the manifold with a flat connection, called the exponential connection, and comment on the possibility of introducing the dual mixture connection

arXiv:math-ph/0007039•AIP conference proceedings•Jan 1, 2001•0 citations
Let [Formula: see text] be a separable Hilbert space. We consider the manifold [Formula: see text] consisting of density operators ρ on [Formula: see text] such that ρp is of trace class for some p ɛ (0, 1). We say [Formula: see text] is nearby ρ if there exists C > 1 such that C−1 ρ < σ < Cρ. We show that the space of nearby points to ρ can be furnished with the two flat connections known as the (±)-affine structures, which are dual relative to the BKM metric. We furnish [Formula: see text] with a norm making it into a Banach manifold.
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.
We consider the quantum information manifold whose underlying set M consists of density operators rho with the extra property that some fractional power of rho is of trace class. The topology is defined by defining a neighbourhood of a point rho to be all density operators that dominate rho and are dominated by rho. We show that this is the same set as that of all states whose relative Hamiltonian X in the sense of Araki is bounded, and such that X(t) is holomorphic in the circle |t| less than 1/2. Here, X(t) is the time evolution of X determined by the modular automorphism defined by rho. We show that M is a Banach manifold in Araki's norm, and that both the canonical and the mixture affine connections can be defined. These are dual relative to the Kubo-Mori metric, and so generalise Amari's dual theory to quantum theory in infinite dimensions.
We investigate questions in quantum information geometry which concern the existence and nonexistence of dual and dually flat structures on stratified sets of density operators on finite-dimensional Hilbert spaces. We show that the set of density operators of a given rank admits dually flat connections for which one connection is complete if and only if this rank is maximal. We prove, moreover, that there is never a dually flat structure on the set of pure states. Thus any general theory of quantum information geometry that involves duality concepts must inevitably be based on dual structures which are nonflat.
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.