2. Quantum Operations: The Hans-Jürgen algorithm

We are considering the Random Quantum Maps

,

mapping the set of N×N density matrices ρ in itself. The Map Γ can be represented by a N²×N² matrix Φ, the so-called Superoperator, which in turn can be reshuffled into a positive and Hermitian N²×N² dynamical matrix D

            that is                           (1)

Other that positivity and Hermiticity, another condition must be fulfilled by D, for being a dynamical matrix representing a quantum map Γ. This is the Trace Preserving (TP) condition, namely

                                         (2)

The Hans-Jürgen algorithm for producing a dynamical matrix D consists in:

• generate a CPXL N²×N² non Hermitian matrix X drawn according to the Ginibre Unitary Ensemble (the subroutine GinUE.f can be used for this purpouse);



• compute the N×N Hermitian matrix

  
  

  and diagonalize it, obtaining U_B and Λ_B such that

               and                 

• use the latter decomposition for fixing the Hermitian matrix

  

• compute the dynamical matrix

  

  that now is positive, hermitian and fulfill eq. (2) by construction.