Absolutely dilatable bimodule maps

TL;DR

This paper characterizes absolutely dilatable completely positive bimodule maps on bounded operators, introduces various types based on ancilla, and links their properties to the Connes Embedding Problem, advancing understanding in quantum operator theory.

Contribution

It provides a complete characterization of absolutely dilatable bimodule maps and establishes their connection to the Connes Embedding Problem, introducing new classifications based on ancilla types.

Findings

Local absolutely dilatable maps admit an exact abelian ancilla factorization.

They are limits of unitary conjugations in the commutant of the von Neumann algebra.

The Connes Embedding Problem is equivalent to all such maps being approximately quantum.

Abstract

We characterise absolutely dilatable completely positive maps on the space of all bounded operators on a Hilbert space that are also bimodular over a given von Neumann algebra as rotations by a suitable unitary on a larger Hilbert space followed by slicing along the trace of an additional ancilla. We define the local, quantum and approximately quantum types of absolutely dilatable maps, according to the type of the admissible ancilla. We show that the local absolutely dilatable maps admit an exact factorisation through an abelian ancilla and show that they are limits in the point weak* topology of conjugations by unitaries in the commutant of the given von Neumann algebra. We show that the Connes Embedding Problem is equivalent to deciding if all absolutely dilatable maps are approximately quantum.