MATHEMATICA SCANDINAVICA

Separable states and positive maps II

Erling Størmer

Published Dec 1, 2009• 15 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: 10.7146/math.scand.a-15114

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Separable states and positive maps II

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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 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

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MATHEMATICA SCANDINAVICA

Separable states and positive maps II

Erling Størmer

Published Dec 1, 2009• 15 citations

Publisher

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: 10.7146/math.scand.a-15114

Related papers

Separable states and positive maps II

Erling Størmer

arXiv:0803.4417Mar 31, 20080 citations

Separable states and positive maps

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 study the relationship between separability, entanglement and the Peres condition of states and positivity properties of the linear maps.

arXiv:0710.3071Oct 16, 20072 citations

0 A CHARACTERIZATION OF POSITIVE LINEAR MAPS AND CRITERIA OF ENTANGLEMENT FOR QUANTUM STATES

Jinchuan Hou

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 $\rho$ on $H\otimes K$ 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.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 $\rho$ on $H\otimes K$ 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.

Aug 16, 201641 citations

On a characterization of positive maps

W. A. Majewski, Marcin Marciniak

Drawing on results of Choi, Størmer and Woronowicz, we present a nearly complete characterization of certain important classes of positive maps. In particular, we construct a general class of positive linear maps acting between two matrix algebras ℬ( ) and ℬ( ), where and are finite-dimensional Hilbert spaces. It turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on ℬ( ⊗ ). Subsequently, the relation between entanglements and positive maps is described. Finally, a new characterization of the cone ℬ( ) + ⊗ℬ( ) + is given.Drawing on results of Choi, Størmer and Woronowicz, we present a nearly complete characterization of certain important classes of positive maps. In particular, we construct a general class of positive linear maps acting between two matrix algebras ℬ( ) and ℬ( ), where and are finite-dimensional Hilbert spaces. It turns out that elements of this class are characterized by operators from the dual cone of the set of all separable states on ℬ( ⊗ ). Subsequently, the relation between entanglements and positive maps is described. Finally, a new characterization of the cone ℬ( ) + ⊗ℬ( ) + is given.

Journal of Physics A Mathematical and GeneralJul 16, 200160 citations

Reduction criterion for separability

Nicolas J. Cerf, Christoph Adami, Robert M. Gingrich

We introduce a separability criterion based on the positive map $\ensuremath{\Gamma}:\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}(\mathrm{Tr}\ensuremath{\rho})\ensuremath{-}\ensuremath{\rho},$ where $\ensuremath{\rho}$ is a trace-class Hermitian operator. Any separable state is mapped by the tensor product of $\ensuremath{\Gamma}$ and the identity into a non-negative operator, which provides a simple necessary condition for separability. This condition is generally not sufficient because it is vulnerable to the dilution of entanglement. In the special case where one subsystem is a quantum bit, $\ensuremath{\Gamma}$ reduces to time reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\ifmmode\times\else\texttimes\fi{}2$ and $2\ifmmode\times\else\texttimes\fi{}3$ systems. Finally, a simple connection between this map for two qubits and complex conjugation in the ``magic'' basis [Phys. Rev. Lett. 78, 5022 (1997)] is displayed.We introduce a separability criterion based on the positive map $\ensuremath{\Gamma}:\stackrel{\ensuremath{\rightarrow}}{\ensuremath{\rho}}(\mathrm{Tr}\ensuremath{\rho})\ensuremath{-}\ensuremath{\rho},$ where $\ensuremath{\rho}$ is a trace-class Hermitian operator. Any separable state is mapped by the tensor product of $\ensuremath{\Gamma}$ and the identity into a non-negative operator, which provides a simple necessary condition for separability. This condition is generally not sufficient because it is vulnerable to the dilution of entanglement. In the special case where one subsystem is a quantum bit, $\ensuremath{\Gamma}$ reduces to time reversal, so that this separability condition is equivalent to partial transposition. It is therefore also sufficient for $2\ifmmode\times\else\texttimes\fi{}2$ and $2\ifmmode\times\else\texttimes\fi{}3$ systems. Finally, a simple connection between this map for two qubits and complex conjugation in the ``magic'' basis [Phys. Rev. Lett. 78, 5022 (1997)] is displayed.

Physical Review AAug 1, 1999173 citations

A factorization property of positive maps on $C^*$ -algebrasA factorization property of positive maps on $C^*$-algebras

B. V. Rajarama Bhat, Hiroyuki Osaka

The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let $A_i$ , $C_i$ be unital C*-algebras and let $α_i$ be positive linear maps from $A_i$ to $C_i,$ $i=1,2$ . We obtain conditions under which any positive map $β$ from the minimal C*-tensor product $A_1 \otimes_{min} A_2$ to $C_1 \otimes_{min} C_2$ , such that $ α_1 \otimes α_2 \geq β$, factorizes as $β= γ\otimes α_2$ for some positive map $γ$ . In particular we show that when $α_i \colon A_i \rightarrow B(\mathcal H_i)$ are completely positive (CP) maps for some Hilbert spaces $\mathcal H_i$ $(i=1,2)$ , and $α_2$ is a pure CP map and $β$ is a CP map so that $α_1 \otimes α_2 - β$ is also CP, then $β= γ\otimes α_2$ for some CP map $γ$ . We show that a similar result holds in the context of positive linear maps when $A_2 = C_2 = B(\mathcal H)$ and $α_2 = id$ . As an application we extend \cite[IX Theorem]{PM}( revisited recently by Huber et al in \cite{HLLM}) to show that for any linear map $τ$ from a unital C*-algebra $A$ to a C*-algebra $C$ , if $τ\otimes id_k$ is decomposable for some $k \geq 2$ , where $id_k$ is the identity map on the algebra $M_k(\mathbb {C} )$ of $k\times k$ matrices, then $τ$ is completely positive.The purpose of this short note is to clarify and present a general version of an interesting observation by Piani and Mora (Physic. Rev. A 75, 012305 (2007)), linking complete positivity of linear maps on matrix algebras to decomposability of their ampliations. Let $A_i$, $C_i$ be unital C*-algebras and let $α_i$ be positive linear maps from $A_i$ to $C_i,$ $i=1,2$. We obtain conditions under which any positive map $β$ from the minimal C*-tensor product $A_1 \otimes_{min} A_2$ to $C_1 \otimes_{min} C_2$, such that $ α_1 \otimes α_2 \geq β$, factorizes as $β= γ\otimes α_2$ for some positive map $γ$. In particular we show that when $α_i \colon A_i \rightarrow B(\mathcal H_i)$ are completely positive (CP) maps for some Hilbert spaces $\mathcal H_i$ $(i=1,2)$, and $α_2$ is a pure CP map and $β$ is a CP map so that $α_1 \otimes α_2 - β$ is also CP, then $β= γ\otimes α_2$ for some CP map $γ$. We show that a similar result holds in the context of positive linear maps when $A_2 = C_2 = B(\mathcal H)$ and $α_2 = id$. As an application we extend \cite[IX Theorem]{PM}( revisited recently by Huber et al in \cite{HLLM}) to show that for any linear map $τ$ from a unital C*-algebra $A$ to a C*-algebra $C$, if $τ\otimes id_k$ is decomposable for some $k \geq 2$, where $id_k$ is the identity map on the algebra $M_k(\mathbb {C} )$ of $k\times k$ matrices, then $τ$ is completely positive.

arXiv:1912.02381Dec 5, 20192 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.

A factorization property of positive maps on

C^*

-algebras

A_i

C_i

be unital C*-algebras and let

α_i

be positive linear maps from

A_i

C_i,

i=1,2

. We obtain conditions under which any positive map

β

from the minimal C*-tensor product

A_1 \otimes_{min} A_2

C_1 \otimes_{min} C_2

, such that $ α_1 \otimes α_2 \geq β$, factorizes as

β= γ\otimes α_2

for some positive map

γ

. In particular we show that when

α_i \colon A_i \rightarrow B(\mathcal H_i)

are completely positive (CP) maps for some Hilbert spaces

\mathcal H_i

(i=1,2)

, and

α_2

is a pure CP map and

β

is a CP map so that

α_1 \otimes α_2 - β

is also CP, then

β= γ\otimes α_2

for some CP map

γ

. We show that a similar result holds in the context of positive linear maps when

A_2 = C_2 = B(\mathcal H)

and

α_2 = id

. As an application we extend \cite[IX Theorem]{PM}( revisited recently by Huber et al in \cite{HLLM}) to show that for any linear map

τ

from a unital C*-algebra

A

to a C*-algebra

C

, if

τ\otimes id_k

is decomposable for some

k \geq 2

, where

id_k

is the identity map on the algebra

M_k(\mathbb {C} )

k\times k

matrices, then

τ

is completely positive.