Decomposition method for non-symmetric Kraus operators that preserves extremality under tensor powers

Determine whether there exists a decomposition method, analogous to Cholesky decomposition, that applies to non-symmetric or non-diagonalizable Kraus operators of a unital quantum channel and ensures that, for any tensor power of the channel, the Landau–Streater linear-independence condition is satisfied so that extremality is preserved in the asymptotic limit (i.e., reproducing the form required in equation (8)).

Background

The paper examines whether extremal properties of unital quantum channels are preserved under tensor products. Using the Landau–Streater criterion, extremality requires the linear independence of the set {A_k^† A_l ⊕ A_l A_k^†}. For tensor powers of a channel, an analogous linear-independence requirement must hold for products of Kraus operators across copies.

For symmetric Kraus operators, the authors show that Cholesky decomposition can be used to obtain the necessary structure while preserving linear independence, thereby maintaining extremality for tensor powers. However, for non-symmetric (or non-diagonalizable) Kraus operators, the existence of a comparable decomposition that preserves extremality remains unresolved. Establishing such a method would generalize the results beyond symmetric channels.