Decomposability of Linear Maps under Tensor Products

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.

arXiv comment26 pages, 3 figures

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Cited by in corpus (6)

• When Do Composed Maps Become Entanglement Breaking?
• Positive maps and trace polynomials from the symmetric group
• Characterization of equivariant maps and application to entanglement detection
• Annihilating Entanglement Between Cones
• Halos and undecidability of tensor stable positive maps
• Decomposable Pauli diagonal maps and Tensor Squares of Qubit Maps