The operator-sum-difference representation of a quantum noise channel

When a model for quantum noise is exactly solvable, a Kraus (or operator-sum) representation can be derived from the spectral decomposition of the Choi matrix for the channel. More generally, a Kraus representation can be obtained from any positive-sum (or ensemble) decomposition of the matrix. Here we extend this idea to any Hermitian-sum decomposition. This yields what we call the “operator-sum-difference” (OSD) representation, in which the channel can be represented as the sum and difference of “subchannels.” As one application, the subchannels can be chosen to be analytically diagonalizable, even if the parent channel is not (on account of the Abel-Galois irreducibility theorem), though in this case the number of the OSD representation operators may exceed the channel rank. Our procedure is applicable to general Hermitian (completely positive or non-completely positive) maps and can be extended to the more general, linear maps. As an illustration of the application, we derive an OSD representation for a two-qubit amplitude-damping channel.