Decomposition of non-symmetric Kraus operators preserving extremality in the asymptotic limit

Determine whether there exists a decomposition method for non-symmetric or non-diagonalizable Kraus operators of a unital quantum channel that preserves extremality under tensor powers in the asymptotic limit, by transforming the operator products for each n-fold tensor power into a form that ensures the linear independence required by the Landau–Streater extremality criterion for unital channels.

Background

The paper studies whether copies of a symmetric unital quantum channel can be approximated by mixtures of unitarily implemented channels, and shows that n(n+1)/2 copies can be approximated arbitrarily well. A key step involves preserving extremal properties of channels under tensor powers.

Using the Landau–Streater theorem, extremality of a unital channel is characterized by the linear independence of the set {A_k^† A_l ⊕ A_l A_k^†}. For symmetric Kraus operators, the author employs Cholesky decomposition to transform operator products in tensor powers into the required form while preserving linear independence, thereby maintaining extremality.

However, for non-symmetric (or non-diagonalizable) Kraus operators, no analogous decomposition method is known to the author, leaving open whether extremality can be preserved in the asymptotic limit for such channels under tensor powers.