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arXiv:0904.3783 · Proceedings of the London Mathematical Society

Operator system structures on ordered spaces

Vern I. Paulsen, Ivan G. Todorov, Mark Tomforde

Published May 28, 2010• 92 citations
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Abstract

Given an Archimedean order unit space (V, V+, e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system 𝒮 is completely boundedly isomorphic to either OMIN(𝒮) or to OMAX(𝒮). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(Mn) to OMAX(Mm) if and only if it is entanglement breaking.

DOI
10.1112/plms/pdq011

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Given an Archimedean order unit space (V, V+, e), we construct a minimal operator system OMIN(V) and a maximal operator system OMAX(V), which are the analogues of the minimal and maximal operator spaces of a normed space. We develop some of the key properties of these operator systems and make some progress on characterizing when an operator system 𝒮 is completely boundedly isomorphic to either OMIN(𝒮) or to OMAX(𝒮). We then apply these concepts to the study of entanglement breaking maps. We prove that for matrix algebras a linear map is completely positive from OMIN(Mn) to OMAX(Mm) if and only if it is entanglement breaking.
Abstract This paper is devoted to the classification of quantum systems among the quantum spaces. In the normed case we obtain a complete solution to the problem when an operator space turns out to be an operator system. The min and max quantizations of a local order are described in terms of the min and max envelopes of the related state spaces. Finally, we characterize min-max-completely positive maps between Archimedean order unit spaces and investigate entanglement breaking maps in the general setting of quantum systems. Bibliography: 34 titles.
An s-entanglement breaking map between operator systems is a point-norm limit of entanglement breaking maps, which are a generalization of the corresponding notion in matrix algebras. We develop some of the key properties of this map, and obtain some conditions when the completely positive linear maps between operator systems coincide with the s-entanglement breaking maps. Especially we show that a linear map from a nuclear operator system to the maximal operator system structure of an Archimedean ordered * -vector space is completely positive if and only if it is s-entanglement breaking. We also discuss the relationships between s-entanglement breaking maps and weak * -entanglement breaking maps, and nuclear maps between operator systems.
There are various notions of positivity for matrices and linear matrix-valued maps that play important roles in quantum information theory. The cones of positive semidefinite matrices and completely positive linear maps, which represent quantum states and quantum channels respectively, are the most ubiquitous positive cones. There are also many natural cones that can been regarded as "more" or "less" positive than these standard examples. In particular, entanglement theory deals with the cones of separable operators and entanglement witnesses, which satisfy very strong and weak positivity properties respectively. Rather complementary to the various cones that arise in entanglement theory are norms. The trace and operator norms for operators and the diamond and completely bounded norms for superoperators are the typical norms that are seen throughout quantum information theory. In this work our main goal is to develop a family of norms that play a role analogous to the cone of entanglement witnesses. We investigate the basic mathematical properties of these norms, including their relationships with other well-known norms, their isometry groups, and their dual norms. We also make the place of these norms in entanglement theory rigorous by showing that entanglement witnesses arise from minimal operator systems, and analogously our norms arise from minimal operator spaces. Finally, we connect the various cones and norms considered here to several seemingly unrelated problems from other areas. We characterize the problem of whether or not non-positive partial transpose bound entangled states exist in terms of one of our norms, and provide evidence in favour of their existence. We also characterize the minimum gate fidelity of a quantum channel, the maximum output purity and its completely bounded counterpart, and the geometric measure of entanglement in terms of these norms.
We introduce a matricial analogue of an Archimedean order unit space, which we call a kkk-AOU space. We develop the category of kkk-AOU spaces and kkk-positive maps and exhibit functors from this category to the category of operator systems and completely positive maps. We also demonstrate the existence of injective envelopes and C*-envelopes in the category of kkk-AOU spaces. Finally, we show that finite-dimensional quantum correlations can be characterized in terms of states on finite-dimensional kkk-AOU spaces. Combined with previous work, this yields a reformulation of Tsirelson's conjecture in terms of operator systems and kkk-AOU spaces.
Linear maps between finite-dimensional ordered vector spaces with orders induced by proper cones C A and C B are called entanglement breaking if their partial application sends the maximal tensor product K ⊗ max C A into the minimal tensor product K ⊗ min C B for any proper cone K . We study the larger class of Lorentz-entanglement breaking maps where K is restricted to be a Lorentz cone of any dimension, i.e., any cone over a Euclidean ball. This class of maps appeared recently in the study of asymptotic entanglement annihilation and it is dual to the linear maps factoring through Lorentz cones. Our main results establish connections between these classes of maps and operator ideals studied in the theory of Banach spaces. For operators u : X → Y between finite-dimensional normed spaces X and Y we consider so-called central maps which are positive with respect to the cones C A = C X and C B = C Y . We show how to characterize when such a map factors through a Lorentz cone and when it is Lorentz-entanglement breaking by using the Hilbert-space factorization norm γ 2 and its dual ⁎ γ 2 ⁎ . We also study the class of Lorentz-entanglement annihilating maps whose local application sends the Lorentzian tensor product C A ⊗ L C A into the minimal tensor product C B ⊗ min C B . When C A is a cone over a finite-dimensional normed space and C B is a Lorentz cone itself, the central maps of this kind can be characterized by the 2-summing norm π 2 . Finally, we prove interesting connections between these classes of maps for general cones, and we identify examples with particular properties, e.g., cones with an analogue of the 2-summing property.
Let H and K be (finite- or infinite-dimensional) complex Hilbert spaces. A characterization of positive completely bounded normal linear maps from into is given, which particularly gives a characterization of positive elementary operators including all positive linear maps between matrix algebras. This characterization is then applied to give a representation of quantum channels (operations) between infinite-dimensional systems. A necessary and sufficient criterion of separability is given which shows that a state ρ on H ⊗ K is separable if and only if (Φ⊗ I )ρ ⩾ 0 for all positive finite-rank elementary operators Φ. 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.