Decomposability of Linear Maps under Tensor Products

TL;DR

This paper investigates the stability of decomposability of linear maps under tensor powers, showing that only completely positive or copositive maps maintain this property indefinitely, and provides bounds for when other maps lose decomposability.

Contribution

It proves that non-CP and non-copositive decomposable maps become non-decomposable after enough tensor powers and introduces explicit bounds for this transition.

Findings

Only CP and copositive maps remain decomposable under tensor powers.

Non-CP and non-copositive decomposable maps lose decomposability after finite tensoring.

New examples of non-decomposable positive maps are constructed.

Abstract

Both completely positive and completely copositive maps stay decomposable under tensor powers, i.e under tensoring the linear map with itself. But are there other examples of maps with this property? We show that this is not the case: Any decomposable map, that is neither completely positive nor completely copositive, will lose decomposability eventually after taking enough tensor powers. Moreover, we establish explicit bounds to quantify when this happens. To prove these results we use a symmetrization technique from the theory of entanglement distillation, and analyze when certain symmetric maps become non-decomposable after taking tensor powers. Finally, we apply our results to construct new examples of non-decomposable positive maps, and establish a connection to the PPT squared conjecture.