6. Even though the Kraus representation gives an internal view of the dynamics of the open...

 

 

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6. Even though the Kraus representation gives an "internal" view of the dynamics of the open system Q, the Kraus operators contain some information about the environment. Concretely, let & be an evolution map with Kraus operators A,..., An, and p a state of Q. Define numbers Wkj = Tr(APA). (a) Show that if the initial state of E is pure (that is, 10)(0] for some unit vector 0)) and U(QE) is as in question 4, then the matrix [W] is the final density operator of E. (b) Suppose that Q is a qubit and the the Kraus operators for are al and bZ, where a,b R with a +62 1. First confirm that maps density operators to density operators. Second, find the eigenvalues of the final state of E when the p is one of 10X0, +X+, and 1. 6. Even though the Kraus representation gives an "internal" view of the dynamics of the open system Q, the Kraus operators contain some information about the environment. Concretely, let & be an evolution map with Kraus operators A,..., An, and p a state of Q. Define numbers Wkj = Tr(APA). (a) Show that if the initial state of E is pure (that is, 10)(0] for some unit vector 0)) and U(QE) is as in question 4, then the matrix [W] is the final density operator of E. (b) Suppose that Q is a qubit and the the Kraus operators for are al and bZ, where a,b R with a +62 1. First confirm that maps density operators to density operators. Second, find the eigenvalues of the final state of E when the p is one of 10X0, +X+, and 1.