Minimal and Maximal Operator Spaces and Operator Systems in Entanglement Theory

Abstract

We examine k-minimal and k-maximal operator spaces and operator systems, and investigate their relationships with the separability problem in quantum information theory. We show that the matrix norms that define the k-minimal operator spaces are equal to a family of norms that have been studied independently as a tool for detecting k-positive linear maps and bound entanglement. Similarly, we investigate the k-super minimal and k-super maximal operator systems that were recently introduced and show that their cones of positive elements are exactly the cones of k-block positive operators and (unnormalized) states with Schmidt number no greater than k, respectively. We characterize a class of norms on the k-super minimal operator systems and show that the completely bounded versions of these norms provide a criterion for testing the Schmidt number of a quantum state that generalizes the recently-developed separability criterion based on trace-contractive maps.

17 pages, to appear in JFA

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Cited by in corpus (17)

• The automorphism group of separable states in quantum information theory
• On a measure of distance for quantum strategies
• Facial structures for various notions of positivity and applications to the theory of entanglement
• Eventually Entanglement Breaking Maps
• Characterizing Operations Preserving Separability Measures via Linear Preserver Problems
• Positive maps and entanglement in real Hilbert spaces
• Quantum Gate Fidelity in Terms of Choi Matrices
• The Super Operator System Structures and their applications in Quantum Entanglement Theory
• Three-qubit entanglement witnesses with the full spanning properties
• Various notions of positivity for bi-linear maps and applications to tri-partite entanglement
• Norms and Cones in the Theory of Quantum Entanglement
• There are many more positive maps than completely positive maps
• The classification problem for finitely generated operator systems and spaces
• Completely Bounded Norms of $k$-positive Maps
• Cones with a Mapping Cone Symmetry in the Finite-Dimensional Case
• Duality of Entanglement Norms
• Linear preservers and quantum information science