A Von Neumann Algebra Approach to Quantum Metrics: Quantum Relations

In a unital C*-algebra with a faithful trace functional $τ$ , the set $D_τ(A)$ of positive $ρ\in A$ of trace τ(ρ)=1 is an algebraic analogue of the space of density matrices (the set of all positive matrices of a fixed dimension of unit trace). Motivated by the literature concerning the metric properties of the space of density matrices, the present paper studies the density space $D_τ(A)$ in terms of the Bures metric. Linear maps on A that map $D_τ(A)$ back into itself are positive and trace preserving, hence, they may be viewed as an algebraic analogue of a quantum channel, which are studied intensely in the literature on quantum computing and quantum information theory. The main results in this paper are: (i) to establish that the Bures metric is indeed a metric, (ii) to prove that channels induce nonexpansive maps of the density space $D_τ(A)$ , (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space $D_τ(A)$ , and (iv) to analyse Bures contractions from the point of view of the Frobenius theory of cone preserving linear maps. Although the focus is on unital C*-algebras, an important class of examples is furnished by finite von Neumann algebras. Indeed, several of the C*-algebra results are established by first proving them for finite von Neumann algebras and then proving them for C*-algebras by embedding a C*-algebra A into its enveloping von Neumann algebra $A^{**}$ .In a unital C*-algebra with a faithful trace functional $τ$, the set $D_τ(A)$ of positive $ρ\in A$ of trace τ(ρ)=1 is an algebraic analogue of the space of density matrices (the set of all positive matrices of a fixed dimension of unit trace). Motivated by the literature concerning the metric properties of the space of density matrices, the present paper studies the density space $D_τ(A)$ in terms of the Bures metric. Linear maps on A that map $D_τ(A)$ back into itself are positive and trace preserving, hence, they may be viewed as an algebraic analogue of a quantum channel, which are studied intensely in the literature on quantum computing and quantum information theory. The main results in this paper are: (i) to establish that the Bures metric is indeed a metric, (ii) to prove that channels induce nonexpansive maps of the density space $D_τ(A)$, (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space $D_τ(A)$, and (iv) to analyse Bures contractions from the point of view of the Frobenius theory of cone preserving linear maps. Although the focus is on unital C*-algebras, an important class of examples is furnished by finite von Neumann algebras. Indeed, several of the C*-algebra results are established by first proving them for finite von Neumann algebras and then proving them for C*-algebras by embedding a C*-algebra A into its enveloping von Neumann algebra $A^{**}$.

arXiv:1704.00376Apr 2, 20178 citations

Metric and group structures from quantum physics

J. G. Provost, Gérard Vallée

Lecture notes in physicsJan 1, 19800 citations

Exponential arcs in manifolds of quantum states

Jan Naudts

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.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.

arXiv:2209.12761Frontiers in PhysicsFeb 7, 20233 citations

Exact exponents for smoothing the max-relative entropy and quantum information decoupling in von Neumann algebras

Zhiwen Lin, Hongsen Qiu, Xinyu Zhang

We establish the exact exponent for smoothing the max-relative entropy on arbitrary von Neumann algebras. The proof first develops the required large-deviation and smoothing estimates in the semifinite setting and then passes to general von Neumann algebras through Haagerup-Junge-Xu reduction. We next study catalytic quantum information decoupling when the reference system is described by an arbitrary von Neumann algebra. A key ingredient is an operator layer-cake formula valid on arbitrary von Neumann algebras. This formula leads to a convex-split estimate for normal states with a general von Neumann algebraic reference system. Combining this estimate with the smoothing exponent result, we obtain lower and upper bounds on the catalytic-decoupling reliability function. These bounds coincide below the corresponding critical rate, yielding the exact decoupling reliability function in that regime.We establish the exact exponent for smoothing the max-relative entropy on arbitrary von Neumann algebras. The proof first develops the required large-deviation and smoothing estimates in the semifinite setting and then passes to general von Neumann algebras through Haagerup-Junge-Xu reduction. We next study catalytic quantum information decoupling when the reference system is described by an arbitrary von Neumann algebra. A key ingredient is an operator layer-cake formula valid on arbitrary von Neumann algebras. This formula leads to a convex-split estimate for normal states with a general von Neumann algebraic reference system. Combining this estimate with the smoothing exponent result, we obtain lower and upper bounds on the catalytic-decoupling reliability function. These bounds coincide below the corresponding critical rate, yielding the exact decoupling reliability function in that regime.

cs.ITCross-listedarXiv:2607.07997Jul 10, 20260 citations

Indefinite metric and its application to quantum mechanics

Miyatake, Osamu

Dec 1, 19610 citations

Quantum Fidelity and the Bures Metric in Operator Algebras

Mizanur Rahaman

This dissertation undertakes the study of quantum delity, a distinguishability measure in the context of quantum mechanics, from the operator algebraic viewpoint. The notion of delity provides a quantitative measure of how close one state of a quantum system is to another state. High delity occurs when the two states are very close to each other. Evidently, this concept is closely related to a metric on the quantum states which is known as the Bures metric. In this thesis, delity and the Bures metric have been studied in the context of (i) unital C -algebras that possess a faithful positive trace functional and (ii) semi nite von Neumann algebras. In addition, these notions have been analysed in the matrix algebras in an e ort to relate to the quantum information theory literature.This dissertation undertakes the study of quantum delity, a distinguishability measure in the context of quantum mechanics, from the operator algebraic viewpoint. The notion of delity provides a quantitative measure of how close one state of a quantum system is to another state. High delity occurs when the two states are very close to each other. Evidently, this concept is closely related to a metric on the quantum states which is known as the Bures metric. In this thesis, delity and the Bures metric have been studied in the context of (i) unital C -algebras that possess a faithful positive trace functional and (ii) semi nite von Neumann algebras. In addition, these notions have been analysed in the matrix algebras in an e ort to relate to the quantum information theory literature.

Jun 1, 20170 citations

In a unital C*-algebra with a faithful trace functional

τ

, the set

D_τ(A)

of positive

ρ\in A

of trace τ(ρ)=1 is an algebraic analogue of the space of density matrices (the set of all positive matrices of a fixed dimension of unit trace). Motivated by the literature concerning the metric properties of the space of density matrices, the present paper studies the density space

D_τ(A)

in terms of the Bures metric. Linear maps on A that map

D_τ(A)

back into itself are positive and trace preserving, hence, they may be viewed as an algebraic analogue of a quantum channel, which are studied intensely in the literature on quantum computing and quantum information theory. The main results in this paper are: (i) to establish that the Bures metric is indeed a metric, (ii) to prove that channels induce nonexpansive maps of the density space

D_τ(A)

, (iii) to introduce and study channels on A that are locally contractive maps (which we call Bures contractions) on the metric space

D_τ(A)

, and (iv) to analyse Bures contractions from the point of view of the Frobenius theory of cone preserving linear maps. Although the focus is on unital C*-algebras, an important class of examples is furnished by finite von Neumann algebras. Indeed, several of the C*-algebra results are established by first proving them for finite von Neumann algebras and then proving them for C*-algebras by embedding a C*-algebra A into its enveloping von Neumann algebra

A^{**}