Existence of non-trivial tensor-stable positive maps

Determine whether there exists a linear map P between finite-dimensional matrix algebras M_{d_1} → M_{d_2} that is tensor-stable positive (meaning P^{⊗n} is positive for every integer n ≥ 1) but is neither completely positive nor completely co-positive.

Background

The paper introduces tensor-stable positive maps, i.e., linear maps P for which all tensor powers P{⊗n} are positive. Completely positive and completely co-positive maps are trivial examples. The authors prove that for any fixed n there exist non-trivial maps that are n-tensor-stable positive, and they also show that no non-trivial tensor-stable positive maps exist in two-dimensional cases.

They reduce the general existence question to a one-parameter family built from Werner channels and show that an affirmative answer would imply the existence of NPPT bound-entangled states. Establishing existence (or non-existence) would have significant implications for quantum information theory, including bounds on quantum channel capacities.

References

Our construction used to obtain this theorem does not seem to suffice for constructing a non-trivial tensor-stable positive map (i.e.\ one for all $n\inN$), and at the time of writing we do not know whether such a map exists.

— Positivity of linear maps under tensor powers  (1502.05630 - Müller-Hermes et al., 2015) in Section 1 (Introduction and main results), after Theorem “Existence of n-tensor-stable positive maps”