Cones of positive maps and their duality relations

We study the so-called K-positive linear maps from B(L) into B(H) for finite dimensional Hilbert spaces L and H and give characterizations of the dual cone of the cone of K-positive maps. Applications are given to decomposable maps and PPT-states.

arXiv:0810.4253Oct 23, 200821 citations

Choi-Jamiołkowski isomorphism(s) and channel/state duality, naturally!

Paul E. Lammert

quant-pharXiv:2610.27394Oct 21, 20260 citations

Cones of positive maps and their duality relations

Erling Størmer, Karol Zyczkowski, et al.

Published Jan 1, 2009• 73 citations

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Abstract

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Cones of positive maps and their duality relations

Łukasz Skowronek, Erling Størmer, Karol Życzkowski

The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable, and k-entangled operators due to the Jamiołkowski–Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.The structure of cones of positive and k-positive maps acting on a finite-dimensional Hilbert space is investigated. Special emphasis is given to their duality relations to the sets of superpositive and k-superpositive maps. We characterize k-positive and k-superpositive maps with regard to their properties under taking compositions. A number of results obtained for maps are also rephrased for the corresponding cones of block positive, k-block positive, separable, and k-entangled operators due to the Jamiołkowski–Choi isomorphism. Generalizations to a situation where no such simple isomorphism is available are also made, employing the idea of mapping cones. As a side result to our discussion, we show that extreme entanglement witnesses, which are optimal, should be of special interest in entanglement studies.

arXiv:0902.4877Journal of Mathematical PhysicsJun 1, 200969 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

The Jamiołkowski Isomorphism and a Simplified Proof for the Correspondence Between Vectors Having Schmidt Number k and k-Positive Maps

Kedar S. Ranade, Mazhar Ali

Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and, more generally, as witnesses for states having Schmidt number not greater than k. Such maps and witnesses are related to k-positive maps, and their properties may be investigated by making use of the Jamiołkowski isomorphism. In this article we review the properties of this isomorphism, noting that there are actually two related mappings bearing that name. As a new result, we give a simplified proof for the correspondence between vectors having Schmidt number k and k-positive maps and thus for the Jamiołkowski criterion for complete positivity. Another consequence is a special case of a result by Choi, namely that k-positivity implies complete positivity, if k is the dimension of the smaller one of the Hilbert spaces on which the operators act.Positive maps which are not completely positive are used in quantum information theory as witnesses for convex sets of states, in particular as entanglement witnesses and, more generally, as witnesses for states having Schmidt number not greater than k. Such maps and witnesses are related to k-positive maps, and their properties may be investigated by making use of the Jamiołkowski isomorphism. In this article we review the properties of this isomorphism, noting that there are actually two related mappings bearing that name. As a new result, we give a simplified proof for the correspondence between vectors having Schmidt number k and k-positive maps and thus for the Jamiołkowski criterion for complete positivity. Another consequence is a special case of a result by Choi, namely that k-positivity implies complete positivity, if k is the dimension of the smaller one of the Hilbert spaces on which the operators act.

arXiv:quant-ph/0702255Open Systems & Information DynamicsDec 1, 200716 citations

Theory of Generalized Mapping Cones in the Finite-Dimensional Case

Łukasz Skowronek

In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Stormer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamiolkowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Stormer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.In the finite-dimensional case, we present a new approach to the theory of cones with a mapping cone symmetry, first introduced by Stormer. Our method is based on a definition of an inner product in the space of linear maps between two algebras of operators and the fact that the Jamiolkowski-Choi isomorphism is an isometry. We consider a slightly modified class of cones, although not substantially different from the original mapping cones by Stormer. Using the new approach, several known results are proved faster and often in more generality than before. For example, the dual of a mapping cone turns out to be a mapping cone as well, without any additional assumptions. The main result of the paper is a characterization of cones with a mapping cone symmetry, saying that a given map is an element of such cone if and only if the composition of the map with the conjugate of an arbitrary element in the dual cone is completely positive. A similar result was known in the case where the map goes from an algebra of operators into itself and the cone is a symmetric mapping cone. Our result is proved without the additional assumptions of symmetry and equality between the domain and the target space. We show how it gives a number of older results as a corollary, including an exemplary application.

arXiv:1008.3237Aug 19, 20101 citation

Duality of cones of positive maps

Erling Størmer

arXiv:0810.4253Oct 23, 200821 citations

Choi-Jamiołkowski isomorphism(s) and channel/state duality, naturally!

Paul E. Lammert

Provided a maximally entangled state vector in ${\mathcal{H}}\otimes{\mathcal{H}}$ , the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of ${\mathsf CP}({\mathcal{H}},{\mathcal{K}})$ (completely positive maps of density operators on ${\mathcal{H}}$ tothose on ${\mathcal{K}}$ ) onto the density operators on ${\mathcal{K}}\otimes{\mathcal{H}}$ , which extends to a unitary map. Here, ${\mathcal{H}}$ and ${\mathcal{K}}$ are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both ${\mathcal{H}}$ and ${\mathcal{K}}$ which produces density operators on either ${\mathsf L}({\mathcal{H}},{\mathcal{K}})$ (Hilbert-Schmidt inner product) or ${\mathcal{K}}\otimes{\mathcal{H}}^*$ instead of ${\mathcal{K}}\otimes{\mathcal{H}}$ . In particular, no entangled state input is required. There are no such binatural maps carried on ${\mathcal{K}}\otimes{\mathcal{H}}$ , but those natural in only ${\mathcal{K}}$ are canonically associated with unitary maps ${\mathcal{H}}\rightarrow{\mathcal{H}}^*$ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.Provided a maximally entangled state vector in ${\mathcal{H}}\otimes{\mathcal{H}}$, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of ${\mathsf CP}({\mathcal{H}},{\mathcal{K}})$ (completely positive maps of density operators on ${\mathcal{H}}$ tothose on ${\mathcal{K}}$) onto the density operators on ${\mathcal{K}}\otimes{\mathcal{H}}$, which extends to a unitary map. Here, ${\mathcal{H}}$ and ${\mathcal{K}}$ are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both ${\mathcal{H}}$ and ${\mathcal{K}}$ which produces density operators on either ${\mathsf L}({\mathcal{H}},{\mathcal{K}})$ (Hilbert-Schmidt inner product) or ${\mathcal{K}}\otimes{\mathcal{H}}^*$ instead of ${\mathcal{K}}\otimes{\mathcal{H}}$. In particular, no entangled state input is required. There are no such binatural maps carried on ${\mathcal{K}}\otimes{\mathcal{H}}$, but those natural in only ${\mathcal{K}}$ are canonically associated with unitary maps ${\mathcal{H}}\rightarrow{\mathcal{H}}^*$ and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.

quant-pharXiv:2610.27394Oct 21, 20260 citations

Provided a maximally entangled state vector in

{\mathcal{H}}\otimes{\mathcal{H}}

, the Choi-Jamiolkowski isomorphism of quantum information theory (or channel/state duality) produces a mapping of

{\mathsf CP}({\mathcal{H}},{\mathcal{K}})

(completely positive maps of density operators on

{\mathcal{H}}

tothose on

{\mathcal{K}}

) onto the density operators on

{\mathcal{K}}\otimes{\mathcal{H}}

, which extends to a unitary map. Here,

{\mathcal{H}}

and

{\mathcal{K}}

are {finite-dimensional} Hilbert spaces.The nature of this construction, the form of the associated formulas, and the possibility of versions which do not require an entangled vector seed, are illuminated from the perspective of {naturality}, in the sense of Eilenberg and MacLane.There is a unique such correspondence (``Choi map'') natural in both

{\mathcal{H}}

and

{\mathcal{K}}

which produces density operators on either

{\mathsf L}({\mathcal{H}},{\mathcal{K}})

(Hilbert-Schmidt inner product) or

{\mathcal{K}}\otimes{\mathcal{H}}^*

instead of

{\mathcal{K}}\otimes{\mathcal{H}}

. In particular, no entangled state input is required. There are no such binatural maps carried on

{\mathcal{K}}\otimes{\mathcal{H}}

, but those natural in only

{\mathcal{K}}

are canonically associated with unitary maps

{\mathcal{H}}\rightarrow{\mathcal{H}}^*

and derive from the binatural maps. The construction recovers the standard Choi-Jamiolkowski formulae and shows how they express naturality.