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arXiv:2306.07642 · Linear Algebra and its Applications

C⁎-extreme entanglement breaking maps on operator systems

Sriram Balasubramanian, Neha Hotwani

Published Jan 9, 2024• 4 citations
arXivHTMLPDF
DOI
10.1016/j.laa.2024.01.004

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This paper studies the class of stochastic maps, or channels, for which (I⊗Φ)(Γ) is always separable (even for entangled Γ). Such maps are called entanglement breaking, and can always be written in the form Φ(ρ)=∑ k R k Tr F k ρ where each R k is a density matrix and F k >0. If, in addition, Φ is trace-preserving, the {F k } must form a positive operator valued measure (POVM). Some special classes of these maps are considered and other characterizations given. Since the set of entanglement-breaking trace-preserving maps is convex, it can be characterized by its extreme points. The only extreme points of the set of completely positive trace preserving maps which are also entanglement breaking are those known as classical-quantum or CQ. However, for d≥3, the set of entanglement breaking maps has additional extreme points which are not extreme CQ maps.
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.