Spectral representations and density operators for infinite-dimensional homogeneous random fields

1976 ◽  
Vol 35 (4) ◽  
pp. 323-336 ◽  
Author(s):  
Dag Tj�stheim
Keyword(s):  
Random Fields ◽  
Infinite Dimensional ◽  
Density Operators ◽  
Spectral Representations

Infinitely entangled states

2003 ◽  
Vol 3 (4) ◽  
pp. 281-306
Author(s):  
M. Keyl ◽  
D. Schlingemann ◽  
R.F. Werner
Keyword(s):  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Entangled States ◽  
Entangled State ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Density Operators ◽  
Fundamental Paper ◽  
Extended Notion

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

INFINITE DIMENSIONAL STATIONARY RANDOM FIELDS OVER A LOCALLY COMPACT ABELIAN GROUP

2012 ◽  
Vol 23 (04) ◽  
pp. 1250029
Author(s):  
HERBERT HEYER ◽  
M. M. RAO
Keyword(s):  
Random Fields ◽  
Markov Random Fields ◽  
Locally Compact Abelian Group ◽  
Wide Sense ◽  
Euclidean Group ◽  
Locally Compact ◽  
Infinite Dimensional ◽  
Group Context ◽  
Stationary Random Fields ◽  
Lca Group

In this paper we consider spectral representation of infinite dimensional stationary random fields over an abelian locally compact (or LCA) group, and then extend the results of earlier authors who consider wide sense Markov random fields over a Euclidean group ℝn to the general LCA group context and obtain minimality properties. We also indicate possibilities of some extensions of these results to certain nonstationary classes.

scholarly journals SPHARMA approximations for stationary functional time series on the sphere

2021 ◽  
Author(s):  
Alessia Caponera
Keyword(s):  
Random Fields ◽  
Spectral Representation ◽  
Moving Average ◽  
Autoregressive Moving Average ◽  
Stationary Processes ◽  
Representation Theorems ◽  
Infinite Dimensional ◽  
Functional Time Series ◽  
Moving Average Processes ◽  
Natural Way

AbstractIn this paper, we focus on isotropic and stationary sphere-cross-time random fields. We first introduce the class of spherical functional autoregressive-moving average processes (SPHARMA), which extend in a natural way the spherical functional autoregressions (SPHAR) recently studied in Caponera and Marinucci (Ann Stat 49(1):346–369, 2021) and Caponera et al. (Stoch Process Appl 137:167–199, 2021); more importantly, we then show that SPHAR and SPHARMA processes of sufficiently large order can be exploited to approximate every isotropic and stationary sphere-cross-time random field, thus generalizing to this infinite-dimensional framework some classical results on real-valued stationary processes. Further characterizations in terms of functional spectral representation theorems and Wold-like decompositions are also established.

Sandwiched Rényi relative entropy on density operators

2020 ◽  
Vol 18 (08) ◽  
pp. 2150003
Author(s):  
Ting Zhang ◽  
Xiaofei Qi
Keyword(s):  
Relative Entropy ◽  
Unitary Operator ◽  
Trace Class ◽  
Quantum Information Theory ◽  
Complex Hilbert Space ◽  
Quantum States ◽  
Infinite Dimensional ◽  
Density Operators ◽  
Trace Class Operators ◽  
Rényi Relative Entropy

Relative entropies play important roles in classical and quantum information theory. In this paper, we discuss the sandwiched Rényi relative entropy for [Formula: see text] on [Formula: see text] (the cone of positive trace-class operators acting on an infinite-dimensional complex Hilbert space [Formula: see text]) and characterize all surjective maps preserving the sandwiched Rényi relative entropy on [Formula: see text]. Such transformations have the form [Formula: see text] for each [Formula: see text], where [Formula: see text] and [Formula: see text] is either a unitary or an anti-unitary operator on [Formula: see text]. Particularly, all surjective maps preserving sandwiched Rényi relative entropy on [Formula: see text] (the set of all quantum states on [Formula: see text]) are necessarily implemented by either a unitary or an anti-unitary operator.

scholarly journals Manifolds of classical probability distributions and quantum density operators in infinite dimensions

2019 ◽  
Vol 2 (2) ◽  
pp. 231-271 ◽  
Author(s):  
F. M. Ciaglia ◽  
A. Ibort ◽  
J. Jost ◽  
G. Marmo
Keyword(s):  
Disjoint Union ◽  
Separable Hilbert Space ◽  
Probability Distributions ◽  
Classical Case ◽  
Infinite Dimensions ◽  
Classical Probability ◽  
Infinite Dimensional ◽  
Density Operators ◽  
Banach Manifolds ◽  
Invertible Elements

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 $$\mathscr {A}$$A which is Abelian for Classical states, and non-Abelian for Quantum states. In this contribution, the space of states $$\mathscr {S}$$S of a possibly infinite-dimensional, unital $$C^{*}$$C∗-algebra $$\mathscr {A}$$A is partitioned into the disjoint union of the orbits of an action of the group $$\mathscr {G}$$G of invertible elements of $$\mathscr {A}$$A. Then, we prove that the orbits through density operators on an infinite-dimensional, separable Hilbert space $$\mathcal {H}$$H are smooth, homogeneous Banach manifolds of $$\mathscr {G}=\mathcal {GL}(\mathcal {H})$$G=GL(H), and, when $$\mathscr {A}$$A admits a faithful tracial state $$\tau $$τ like it happens in the Classical case when we consider probability distributions with full support, we prove that the orbit through $$\tau $$τ is a smooth, homogeneous Banach manifold for $$\mathscr {G}$$G.

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