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

arXiv:quant-ph/0504204

Separability and Entanglement-Breaking in Infinite Dimensions

Alexander Semenovich Holevo, Максим Евгеньевич Широков, Reinhard F. Werner

Published Apr 27, 2005• 28 citations
arXivHTMLPDF

Abstract

In this paper we give a general integral representation for separable states in the tensor product of infinite dimensional Hilbert spaces and provide the first example of separable states that are not countably decomposable. We also prove the structure theorem for the quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of M. Horodecki, Ruskai and Shor. In the finite dimensional case such channels can be characterized as having the Kraus representation with operators of rank 1. The above example implies existence of infinite-dimensional entanglement-breaking channels having no such representation.

DOI
10.48550/arxiv.quant-ph/0504204

Related papers

On the notion of entanglement in Hilbert spaces

Александр Семенович Холево, Максим Евгеньевич Широков, Reinhard F. Werner

A density operator (state) on a tensor product H ⊗ K of Hilbert spaces is separable if it is in the convex closure of the subset of all tensor product states. Non-separable states are called entangled. These concepts are of great importance in quantum information theory, but they have been studied in depth only in the finite-dimensional context [1]. In this note we give a general integral representation for separable states and provide the first example of separable states that are not countably decomposable. We also prove a structure theorem for quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of [2]. In the finite-dimensional case such channels can be characterized as having a Stinespring–Kraus representation (3) with operators Vj of rank 1. The above example implies the existence of infinite-dimensional entanglement-breaking channels having no such representation. In what follows, H, K ,... are separable Hilbert spaces, T(H) is the Banach space of trace-class operators and S(H) is the convex subset of all density operators on H. For brevity we shall also call them states, having in mind that a density operator ρ uniquely determines a normal state on the algebra B(H) of all bounded operators on H. Equipped with the trace-norm topology, S(H) is a complete separable metric space. If π is a Borel probability measure on S(H), then the Bochner integral

Russian Mathematical Surveys•Apr 30, 2005•47 citations

Entanglement-breaking channels in infinite dimensions

Alexander Semenovich Holevo

We give a representation for entanglement-breaking channels in separable Hilbert space that generalizes the "Kraus decomposition with rank one operators" and use it to describe the complementary channels. We also give necessary and sufficient condition of entanglement-breaking for a general quantum Gaussian channel. Application of this condition to one-mode channels provides several new cases where the additivity conjecture holds in full generality.

arXiv:0802.0235•Feb 3, 2008•3 citations

The Schmidt number and partially entanglement breaking channels in infinite dimensions

Максим Евгеньевич Широков

A definition of the Schmidt number of a state of an infinite dimensional bipartite quantum system is given and properties of the corresponding family of Schmidt classes are considered. The existence of states with a given Schmidt number such that any their countable convex decomposition does not contain pure states with finite Schmidt rank is established. Partially entanglement breaking channels in infinite dimensions are studied. Several properties of these channels well known in finite dimensions are generalized to the infinite dimensional case. At the same time, the existence of partially entanglement breaking channels (in particular, entanglement breaking channels) such that all operators in any their Kraus representations have infinite rank is proved.

arXiv:1110.4363•Oct 19, 2011•5 citations

Schmidt number and partially entanglement-breaking channels in infinite-dimensional quantum systems

Максим Евгеньевич Широков

The Schmidt number of a state of an infinite-dimensional composite quantum system is defined and several properties of the corresponding Schmidt classes are considered. It is shown that there are states with given Schmidt number such that any of their countable convex decompositions does not contain pure states of finite Schmidt rank. The classes of infinite-dimensional partially entanglement-breaking channels are considered, and generalizations of several properties of such channels, which were obtained earlier in the finite-dimensional case, are proved. At the same time, it is shown that there are partially entanglement-breaking channels (in particular, entanglement-breaking channels) such that all the operators in any of their Kraus representations are of infinite rank.

Mathematical Notes•May 1, 2013•6 citations

Entanglement-breaking channels in infinite dimensions

Alexander Semenovich Holevo

In the first part of the paper we give a representation for entanglement-breaking channels in separable Hilbert space that generalizes the “Kraus decomposition with rank-one operators” and use it to describe complementary channels. We also note that coherent information for antidegradable channel is always nonpositive. In the second part, we give necessary and sufficient condition for entanglement breaking for the general quantum Gaussian channel. Application of this condition to one-mode channels provides several new cases where the additivity conjecture holds in the strongest form.

Problems of Information Transmission•Sep 1, 2008•139 citations

The cross states of a composite quantum system: separability and entanglement in any Hilbert space dimension

Paolo Aniello

We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.

arXiv:2602.18125•Feb 20, 2026•0 citations
In this paper we give a general integral representation for separable states in the tensor product of infinite dimensional Hilbert spaces and provide the first example of separable states that are not countably decomposable. We also prove the structure theorem for the quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of M. Horodecki, Ruskai and Shor. In the finite dimensional case such channels can be characterized as having the Kraus representation with operators of rank 1. The above example implies existence of infinite-dimensional entanglement-breaking channels having no such representation.
A density operator (state) on a tensor product H ⊗ K of Hilbert spaces is separable if it is in the convex closure of the subset of all tensor product states. Non-separable states are called entangled. These concepts are of great importance in quantum information theory, but they have been studied in depth only in the finite-dimensional context [1]. In this note we give a general integral representation for separable states and provide the first example of separable states that are not countably decomposable. We also prove a structure theorem for quantum communication channels that are entanglement-breaking, generalizing the finite-dimensional result of [2]. In the finite-dimensional case such channels can be characterized as having a Stinespring–Kraus representation (3) with operators Vj of rank 1. The above example implies the existence of infinite-dimensional entanglement-breaking channels having no such representation. In what follows, H, K ,... are separable Hilbert spaces, T(H) is the Banach space of trace-class operators and S(H) is the convex subset of all density operators on H. For brevity we shall also call them states, having in mind that a density operator ρ uniquely determines a normal state on the algebra B(H) of all bounded operators on H. Equipped with the trace-norm topology, S(H) is a complete separable metric space. If π is a Borel probability measure on S(H), then the Bochner integral
A definition of the Schmidt number of a state of an infinite dimensional bipartite quantum system is given and properties of the corresponding family of Schmidt classes are considered. The existence of states with a given Schmidt number such that any their countable convex decomposition does not contain pure states with finite Schmidt rank is established. Partially entanglement breaking channels in infinite dimensions are studied. Several properties of these channels well known in finite dimensions are generalized to the infinite dimensional case. At the same time, the existence of partially entanglement breaking channels (in particular, entanglement breaking channels) such that all operators in any their Kraus representations have infinite rank is proved.
The Schmidt number of a state of an infinite-dimensional composite quantum system is defined and several properties of the corresponding Schmidt classes are considered. It is shown that there are states with given Schmidt number such that any of their countable convex decompositions does not contain pure states of finite Schmidt rank. The classes of infinite-dimensional partially entanglement-breaking channels are considered, and generalizations of several properties of such channels, which were obtained earlier in the finite-dimensional case, are proved. At the same time, it is shown that there are partially entanglement-breaking channels (in particular, entanglement-breaking channels) such that all the operators in any of their Kraus representations are of infinite rank.
In the first part of the paper we give a representation for entanglement-breaking channels in separable Hilbert space that generalizes the “Kraus decomposition with rank-one operators” and use it to describe complementary channels. We also note that coherent information for antidegradable channel is always nonpositive. In the second part, we give necessary and sufficient condition for entanglement breaking for the general quantum Gaussian channel. Application of this condition to one-mode channels provides several new cases where the additivity conjecture holds in the strongest form.
We introduce a class of states of a composite quantum system, the so-called cross states, that turn out to play a major role in the theory of entanglement for a genuinely infinite-dimensional bipartite system. In the case where at least one of the Hilbert spaces of the bipartition is finite-dimensional, all states are cross states, whereas, in the genuinely infinite-dimensional setting where the dimension of both Hilbert spaces is not finite, the cross states form a trace-norm dense, convex, proper subset of the set of all states. In the latter case, the cross states can be regarded as those physical states that possess a finite amount of entanglement; accordingly, all separable states are of this kind. We prove that, for any Hilbert space dimension, the separable states can be characterized as those cross states that minimize a suitable norm, i.e., the projective norm associated with the projective tensor product of two trace classes; all other cross states are density operators belonging to the projective tensor product space. This is a generalization of the classical cross norm criterion of separability. Finally, we define an extended real-valued entanglement function and study its main properties. Coherently with the interpretation of cross states as finitely entangled states, this function is finite, and coincides with the projective norm, precisely on the cross states of the system.