Returns a Kraus representation of a channel specified via the superoperator matrix.

Quantum channel E: L(H1) -> L(H2) may be described by a collection of operators A_i, called Kraus operators, such that

E(\rho) = \sum_i A_i \rho A_i^\dagger.

Kraus representation is not unique. Alternatively, E may be specified by its superoperator matrix K(E) defined so that

K(E) vec(\rho) = vec(E(\rho))

where the vectorization map \(vec\) rearranges d-by-d matrices into d**2-dimensional vectors. Superoperator matrix is unique for a given channel. It is also called the natural representation of a quantum channel.

The most expensive step in the computation of a Kraus representation from a superoperator matrix is eigendecomposition. Therefore, the cost of the conversion is O(d**6) where d is the dimension of the input and output Hilbert space.

superoperator Superoperator matrix specifying a quantum channel.
atol Tolerance used to check which Kraus operators to omit.

Sequence of Kraus operators of the channel specified by superoperator. Kraus operators with Frobenius norm smaller than atol are omitted.

ValueError If superoperator is not a valid superoperator matrix.