Kraus Matrices and Unitary Matrix

[Henriamaa]

I am having trouble getting the kraus matrices(E_k)) from a unitary matrix. This task is trivial if one uses dirac notation. But supposing I was coding, I can't put in bras and kets in my code so I need a systematic way of getting kraus matrices from a unitary matrix(merely using matrices). So supposing, the environment was in the |0> and my unitary matrix was the controlled not gate. I expect my kraus matrices to be to the projection operators |0><0| and |1><1|. The system and the environment are of course 2 dimensional.

P.S I do know about this definition E_k = <e_k|U|0>. It is not really helpful. |e_k> is a basis for the environment.

 

[azntoon]

The problem of decomposing a unitary matrix into kraus matrices is known as the Kraus decomposition problem. Unfortunately, there is no known general solution to this problem. However, for some simple unitary matrices, such as the controlled-NOT gate, it is possible to find the Kraus matrices explicitly. For the controlled-NOT gate, the Kraus matrices can be obtained by first writing the unitary matrix in its Pauli decomposition form:U = (I × X) + (X × I)Where I is the identity matrix and X is the Pauli X matrix. The Kraus matrices can then be easily obtained as:E1 = (I × |0><0|) + (X × |1><1|)E2 = (I × |1><1|) + (X × |0><0|)These two Kraus matrices are the projectors onto the |0> and |1> states of the environment, respectively.