arXiv:0803.4417

Separable states and positive maps II

Erling Størmer

Published Mar 31, 2008• 0 citations

Abstract

Using the natural duality between linear functionals on tensor products of C*-algebras with the trace class operators on a Hilbert space H and linear maps of the C*-algebra into B(H), we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF C*-algebras

DOI: 10.48550/arxiv.0803.4417

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arXiv:0803.4417

Separable states and positive maps II

Erling Størmer

Published Mar 31, 2008• 0 citations

arXiv HTML PDF

Abstract

DOI: 10.48550/arxiv.0803.4417

Related papers

Separable states and positive maps II

Erling Størmer

Using the natural duality between linear functionals on tensor products of $C^*$ -algebras with the trace class operators on a Hilbert space $H$ and linear maps of the $C^*$ -algebra into $B(H)$ , we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF $C^*$ -algebras.Using the natural duality between linear functionals on tensor products of $C^*$-algebras with the trace class operators on a Hilbert space $H$ and linear maps of the $C^*$-algebra into $B(H)$, we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF $C^*$-algebras.

MATHEMATICA SCANDINAVICADec 1, 200915 citations

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Using the natural duality between linear functionals on tensor products of C*-algebras with the trace class operators on a Hilbert space H and linear maps of the C*-algebra into B(H), we study the relationship between separability, entanglement and the Peres condition of states and positivity properties of the linear maps.

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On Extension of Quantum Channels and Operations to the Space of Relatively Bounded Operators

Максим Евгеньевич Широков

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra $\mathfrak{B}(\mathcal{H})$ of bounded operators on a separable Hilbert space $\mathcal{H}$ ) to the space of all operators on $\mathcal{H}$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $\Phi$ can be uniquely extended to a bounded linear operator on the Banach space of all $\sqrt{G}$ -bounded operators on $\mathcal{H}$ provided that the operation $\Phi$ is $G$ -limited: the predual operation $\Phi_{*}$ maps the set of positive trace class operators $\rho$ with finite value of $\textrm{Tr}\rho G$ into itself. Assuming that $G$ has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the $G$ -limited property. Applications to the theory of Bosonic Gaussian channels are considered.We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra $$\mathfrak{B}(\mathcal{H})$$ of bounded operators on a separable Hilbert space $$\mathcal{H}$$ ) to the space of all operators on $$\mathcal{H}$$ relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation $$\Phi$$ can be uniquely extended to a bounded linear operator on the Banach space of all $$\sqrt{G}$$ -bounded operators on $$\mathcal{H}$$ provided that the operation $$\Phi$$ is $$G$$ -limited: the predual operation $$\Phi_{*}$$ maps the set of positive trace class operators $$\rho$$ with finite value of $$\textrm{Tr}\rho G$$ into itself. Assuming that $$G$$ has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the $$G$$ -limited property. Applications to the theory of Bosonic Gaussian channels are considered.

arXiv:1903.06086Lobachevskii Journal of MathematicsApr 1, 20207 citations

Using the natural duality between linear functionals on tensor products of

C^*

-algebras with the trace class operators on a Hilbert space

H

and linear maps of the

C^*

-algebra into

B(H)

, we give two characterizations of separability, one relating it to abelianness of the definite set of the map, and one on tensor products of nuclear and UHF

C^*

-algebras.

Let

H

and

K

be (finite or infinite dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from

{\mathcal B}(H)

into

{\mathcal B}(K)

is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is give which shows that a state

\rho

H\otimes K

is separable if and only if

(\Phi\otimes I)\rho\geq 0

for all positive finite rank elementary operators

\Phi

. Examples of NCP and indecomposable positive linear maps are given and are used to recognize some entangled states that cannot be recognized by the PPT criterion and the realignment criterion.

p

-summing operators in algebraic quantum field theory in the sense of Araki, Haag and Kastler (

p=2

) and more recently in the framework of general probabilistic spaces (

p=1

), we will outline that Banach operator ideals in the sense of Pietsch, or equivalently tensor products of Banach spaces in the sense of Grothendieck are even lurking in the foundations and philosophy of quantum physics and quantum information theory. In particular, we concentrate on their importance in algebraic quantum field theory. In doing so, we establish a canonical isometric isomorphism between the Hilbert spaces

H\otimes_2 (K \otimes_2 L)

and

(H \otimes_2 K) \otimes_2 L

(Theorem 3.8) and revisit the role of trace class operators. A few applications are specified, including the appropriateness of the class of Hilbert-Schmidt operators and an implied Banach operator ideal representation of the tensor product of two complex Hilbert spaces

H \otimes_2 K

(Proposition 3.4) and a purely linear algebraic description of the quantum teleportation process (Example 3.10).

We analyse possibility to extend a quantum operation (sub-unital normal completely positive linear map on the algebra

\mathfrak{B}(\mathcal{H})

of bounded operators on a separable Hilbert space

\mathcal{H}

) to the space of all operators on

\mathcal{H}

relatively bounded w.r.t. a given positive unbounded operator. We show that a quantum operation

\Phi

can be uniquely extended to a bounded linear operator on the Banach space of all

\sqrt{G}

-bounded operators on

\mathcal{H}

provided that the operation

\Phi

G

-limited: the predual operation

\Phi_{*}

maps the set of positive trace class operators

\rho

with finite value of

\textrm{Tr}\rho G

into itself. Assuming that

G

has discrete spectrum of finite multiplicity we prove that for a wide class of quantum operations the existence of the above extension implies the

G

-limited property. Applications to the theory of Bosonic Gaussian channels are considered.