 
Summary: Almost depolarizing channels with short Kraus
decompositions
Guillaume AUBRUN
Universit´e Lyon 1, France
Guillaume AUBRUN (Lyon) Short Kraus decompositions Luminy, May 2008 1 / 15
Completely positive maps
M(Cd ) : d × d complex matrices  A, B = Tr AB.
Definition (equivalent to the usual one)
A linear map : M(Cd ) M(Cd ) is completely positive (CP) if there
is a random matrix V : (, P) M(Cd ) so that
(X) = EVXV
.
Depends only on the covariance matrix of V ( M+(M(Cd )).
Therefore V can be chosen to be finitely supported.
Kraus decomposition. Any CP : M(Cd ) M(Cd ) can be
decomposed as a sum of Kraus operators
(X) =
N
i=1
Vi XV
