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arXiv:1202.4255 · Reviews in Mathematical Physics

FACIAL STRUCTURES FOR VARIOUS NOTIONS OF POSITIVITY AND APPLICATIONS TO THE THEORY OF ENTANGLEMENT

SEUNG-HYEOK KYE

Published Feb 19, 2013• 49 citations
arXivHTMLPDF

Abstract

In this expository note, we explain facial structures for the convex cones consisting of positive linear maps, completely positive linear maps, and decomposable positive linear maps between matrix algebras, respectively. These will be applied to study the notions of entangled edge states with positive partial transposes and optimality of entanglement witnesses.

DOI
10.1142/s0129055x13300021

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In this expository note, we explain facial structures for the convex cones consisting of positive linear maps, completely positive linear maps, and decomposable positive linear maps between matrix algebras, respectively. These will be applied to study the notions of entangled edge states with positive partial transposes and optimality of entanglement witnesses.
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.
From the physical point of view entanglement witnesses (EWs) define a universal tool for analysis and classification of quantum entangled states. From the mathematical perspective bipartite EWs provide highly non-trivial generalization of positive operators and establish elegant correspondence with the theory of positive maps in matrix algebras. We concentrate on theoretical analysis of various important notions like decomposability, atomicity, optimality, extremality and exposedness. Several methods of construction are provided as well. Our discussion is illustrated by many examples enabling the reader to see the intricate structure of these objects. It is shown that the theory of EWs finds elegant geometric formulation in terms of convex cones and related geometric structures.
This paper investigates the properties of Choi polynomials and their fundamental role in the theory of positive linear maps between matrix algebras. By focusing on Hermitian symmetric biquadratic forms, we establish a connection between the positivity of these forms and the structure of positive maps. We specifically explore the construction of indecomposable positive maps in matrix algebras, and their application as entanglement witnesses. Our analysis extends to the detection of Positive Partial Transpose (PPT) entangled states and the classification of edge PPT states in Mm(C)⊗Mn(C)M_m(\mathbb{C}) \otimes M_n(\mathbb{C})Mm​(C)⊗Mn​(C). Our results provide a refined framework for identifying non-separable states that escape the standard PPT criterion, contributing to the broader understanding of entanglement distillation and quantum information theory.
We consider entanglement witnesses arising from positive linear maps which generate exposed extremal rays. We show that every entangled state can be detected by one of these witnesses, and this witness detects a unique set of entangled states among those. Therefore, they provide a minimal set of witnesses to detect any kind of entanglement in a sense. Furthermore, if those maps are indecomposable then they detect large classes of entangled states with positive partial transposes which have nonempty relative interiors in the cone generated by all PPT states. We also provide a one-parameter family of indecomposable positive linear maps which generate exposed extremal rays. This gives the first examples of such maps in three-dimensional matrix algebra.
The convex cone V1\mathbb{V}_1V1​ generated by separable states is contained in the cone T\mathbb{T}T of all positive semi-definite block matrices whose block transposes are also positive semi-definite. We characterize faces of the cone V1\mathbb{V}_1V1​ induced by faces of the cone T\mathbb{T}T which are determined by pairs of subspaces of matrices.