  On account of the Abel-Galois no-go theorem for the algebraic solution to
quintic and higher order polynomials, the eigenvalue problem and the associated
characteristic equation for a general noise dynamics in dimension $d$ via the
Choi-Jamiolkowski approach cannot be solved in general via radicals. We provide
a way around this impasse by decomposing the Choi matrix into simpler, not
necessarily positive, Hermitian operators that are diagonalizable via radicals,
which yield a set of `positive' and `negative' Kraus operators. The price to
pay is that the sufficient number of Kraus operators is $d^4$ instead of $d^2$,
sufficient in the Kraus representation. We consider various applications of the
formalism: the Kraus repesentation of the 2-qubit amplitude damping channel,
the noise resulting from a 2-qubit system interacting dissipatively with a
vacuum bath; defining the maximally dephasing and purely dephasing components
of the channel in the new representation, and studying their entanglement
breaking and broadcast properties.
