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**I have been hanging on this problem for a long time without a solution!!**

Suppose we have a single qubit principal system ,interacting with a single qubit environment through the transformation

U=P0[tex]\otimes[/tex]I+P1[tex]\otimes[/tex]X

where X is the usual Pauli matrix (acting on the enviornment)and P0=|0><0| ,P1=|1><1| are projectors (acting on the system ).Give the quantum operation for this process ,in the operator-sum representation ,assuming the environment starts in the state |0>

I have two problem about this exercise :

1.how to represent the principal system

2.how to find an orthonormal basis for the environment so that the operatior-sum representation can be arrived