By Ralph Decker Bennett

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INTRODUCTION AND DEFINITION OF TERMS and the linearity of U immediately implies Eq. 6. Also notice that the linearity of U implies that Eq. 6 is true for both mixed and pure . In this discussion of the time evolution of , it is implicitly assumed that represents an isolated system. HSB /. e initial reduced system state is S D. t / D U SB SB U SB Á[ : It is assumed that the composite system is isolated and will obey unitary time evolution, but the reduced system dynamics are deﬁned through the ﬂat operator and, in general, will not be unitary.

1 yielding 1 5, which is false. Hence, S does not yield a valid output state for j ih j. is idea is encapsulated in the idea of positivity domains of the channel. S is not positive everywhere on the reduced system space, but only positive density matrices have a clear physical interpretation in the lab. S is, therefore, said to have a positivity domain, which is deﬁned such that the superoperator S will take every valid initial density matrix in the positivity domain to a valid ﬁnal density matrix.

E Kraus decomposition can be used to describe a completely positive map. However, the practicality of the decomposition is restricted to ﬁnite dimensional system-environment Hilbert spaces. For systems with high- (or inﬁnite-) dimensional Hilbert spaces, the Kraus decomposition will lead to an impractically large number of Kraus operators. , to model electromagnetic noise in transmission lines). In such cases, completely positive maps are usually dealt with using dynamical semigroup methods.

### An Attempt to Test the Quantum Theory of X-Ray Scattering by Ralph Decker Bennett

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