TY - JOUR

T1 - Decomposability of linear maps under tensor powers

AU - Mueller-Hermes, Alexander

PY - 2018

Y1 - 2018

N2 - 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 positive partial transpose squared conjecture.

AB - 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 positive partial transpose squared conjecture.

U2 - 10.1063/1.5045559

DO - 10.1063/1.5045559

M3 - Journal article

VL - 59

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 10

M1 - 102203

ER -