Composition of PPT Maps

TL;DR

This paper proves that iterating any unital or trace-preserving PPT map approaches entanglement breaking maps, supporting Christandl's conjecture asymptotically, and characterizes graph-based maps with respect to their positivity and entanglement properties.

Contribution

It demonstrates the asymptotic validity of Christandl's conjecture and characterizes graph-based maps' properties related to positivity and entanglement.

Findings

Iterates of PPT maps approach entanglement breaking maps asymptotically.

The least parameter value for graph-based maps to be entanglement breaking is determined.

Christandl's conjecture holds for certain graph-parameterized map families.

Abstract

M. Christandl conjectured that the composition of any trace preserving PPT map with itself is entanglement breaking. We prove that Christandl's conjecture holds asymptotically by showing that the distance between the iterates of any unital or trace preserving PPT map and the set of entanglement breaking maps tends to zero. Finally, for every graph we define a one-parameter family of maps on matrices and determine the least value of the parameter such that the map is variously, positive, completely positive, PPT and entanglement breaking in terms of properties of the graph. Our estimates are sharp enough to conclude that Christandl's conjecture holds for these families.