Quantum Information Geometry and Quantum Estimation

2006 ◽  
pp. 145-184
Keyword(s):  
Quantum Information ◽  
Information Geometry

scholarly journals Quantum Orlicz Spaces in Information Geometry

2004 ◽  
Vol 11 (04) ◽  
pp. 359-375 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Quantum Information ◽  
Tangent Space ◽  
Selfadjoint Operator ◽  
Affine Connection ◽  
Orlicz Spaces ◽  
Trace Class ◽  
Information Geometry ◽  
Young Function ◽  
Affine Structure ◽  
Cotangent Space

Let H0 be a selfadjoint operator such that Tr e−βH0 is of trace class for some β < 1, and let χɛ denote the set of ɛ-bounded forms, i.e., ∥(H0+C)−1/2−ɛX(H0+C)−1/2+ɛ∥ < C for some C > 0. Let χ := Span ∪ɛ∈(0,1/2]χɛ. Let [Formula: see text] denote the underlying set of the quantum information manifold of states of the form ρx = e−H0−X−ψx, X ∈ χ. We show that if Tr e−H0 = 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 ρ0; its affine structure is defined by the (+1)-connection of Amari 3. The subset of a ‘hood of ρ0, consisting of p-nearby states (those [Formula: see text] obeying C−1ρ1+p ≤ σ ≤ Cρ1 − 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.

scholarly journals ON THE UNIQUENESS OF THE CHENTSOV METRIC IN QUANTUM INFORMATION GEOMETRY

2001 ◽  
Vol 04 (02) ◽  
pp. 173-182 ◽  
Author(s):  
M. R. GRASSELLI ◽  
R. F. STREATER
Keyword(s):  
Quantum Information ◽  
Information Geometry ◽  
Density Matrices ◽  
Affine Connections

We show that, in finite dimensions, the only monotone metrics on the space of invertible density matrices for which the (+1) and (-1) affine connections are mutually dual are constant multiples of the Bogoliubov–Kubo–Mori metric.

QUANTUM INFORMATION GEOMETRY AND NONCOMMUTATIVE Lp-SPACES

2005 ◽  
Vol 08 (02) ◽  
pp. 215-233 ◽  
Author(s):  
ANNA JENČOVÁ
Keyword(s):  
Quantum Information ◽  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Classical Case ◽  
Information Geometry ◽  
Positive Cone ◽  
Uniformly Convex ◽  
Von Neumann ◽  
Lp Spaces ◽  
Uniformly Convex Banach Spaces

Let M be a von Neumann algebra. We define the noncommutative extension of information geometry by embeddings of M into noncommutative Lp-spaces. Using the geometry of uniformly convex Banach spaces and duality of the Lp and Lq spaces for 1/p +1/q =1, we show that we can introduce the α-divergence, for α∈(-1, 1), in a similar manner as Amari in the classical case. If restricted to the positive cone, the α-divergence belongs to the class of quasi-entropies, defined by Petz.

scholarly journals Focus point: classical and quantum information geometry

2021 ◽  
Vol 136 (5) ◽  
Author(s):  
F. M. Ciaglia ◽  
S. Mancini ◽  
M. Ha Quang
Keyword(s):  
Quantum Information ◽  
Information Geometry ◽  
Focus Point

scholarly journals Quantum Natural Gradient

Quantum ◽  
2020 ◽  
Vol 4 ◽  
pp. 269 ◽  
Author(s):  
James Stokes ◽  
Josh Izaac ◽  
Nathan Killoran ◽  
Giuseppe Carleo
Keyword(s):  
Quantum Information ◽  
Gradient Descent ◽  
Information Geometry ◽  
General Purpose ◽  
Quantum Circuits ◽  
Descent Direction ◽  
Natural Gradient ◽  
Metric Tensor ◽  
Optimization Framework ◽  
Diagonal Approximation

A quantum generalization of Natural Gradient Descent is presented as part of a general-purpose optimization framework for variational quantum circuits. The optimization dynamics is interpreted as moving in the steepest descent direction with respect to the Quantum Information Geometry, corresponding to the real part of the Quantum Geometric Tensor (QGT), also known as the Fubini-Study metric tensor. An efficient algorithm is presented for computing a block-diagonal approximation to the Fubini-Study metric tensor for parametrized quantum circuits, which may be of independent interest.

scholarly journals Duality in Quantum Information Geometry

2004 ◽  
Vol 11 (01) ◽  
pp. 71-77 ◽  
Author(s):  
R. F. Streater
Keyword(s):  
Hilbert Space ◽  
Quantum Information ◽  
Separable Hilbert Space ◽  
Trace Class ◽  
Information Geometry ◽  
Banach Manifold ◽  
Flat Connections ◽  
Density Operators ◽  
Affine Structures

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.

scholarly journals Lie Group Cohomology and (Multi)Symplectic Integrators: New Geometric Tools for Lie Group Machine Learning Based on Souriau Geometric Statistical Mechanics

Entropy ◽  
2020 ◽  
Vol 22 (5) ◽  
pp. 498 ◽  
Author(s):  
Frédéric Barbaresco ◽  
François Gay-Balmaz
Keyword(s):  
Machine Learning ◽  
Quantum Information ◽  
Statistical Mechanics ◽  
Lie Group ◽  
Information Geometry ◽  
Symplectic Integrators ◽  
Coadjoint Orbits ◽  
Probability Densities ◽  
Fisher Metric

In this paper, we describe and exploit a geometric framework for Gibbs probability densities and the associated concepts in statistical mechanics, which unifies several earlier works on the subject, including Souriau’s symplectic model of statistical mechanics, its polysymplectic extension, Koszul model, and approaches developed in quantum information geometry. We emphasize the role of equivariance with respect to Lie group actions and the role of several concepts from geometric mechanics, such as momentum maps, Casimir functions, coadjoint orbits, and Lie-Poisson brackets with cocycles, as unifying structures appearing in various applications of this framework to information geometry and machine learning. For instance, we discuss the expression of the Fisher metric in presence of equivariance and we exploit the property of the entropy of the Souriau model as a Casimir function to apply a geometric model for energy preserving entropy production. We illustrate this framework with several examples including multivariate Gaussian probability densities, and the Bogoliubov-Kubo-Mori metric as a quantum version of the Fisher metric for quantum information on coadjoint orbits. We exploit this geometric setting and Lie group equivariance to present symplectic and multisymplectic variational Lie group integration schemes for some of the equations associated with Souriau symplectic and polysymplectic models, such as the Lie-Poisson equation with cocycle.

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