What are CP maps in QM good for?

[mtak0114]

Hi
I am trying to teach myself quantum mechanics and I have heard a lot about Completely Positive maps but I haven't been able to find anything on them could someone please tell me what they are and what they are good fore?

cheers

Mark

 

[rsg]

From linear algebra point of view, a bounded operator A acting on a Hilbert space H is said to be positive (P), if for all |x\rangle\in H, \langle x|A|x\rangle\geq0.
An operator E which maps density operators of a space H_1 to H_2 is called completely positive (CP). (Now you understand why they are impotent).
Equivalently, E is completely positive, if and only if I_n\otimes E is a positive operator for all n\geq0. I_n is the identity operator. Testing an operator is CP or not is a difficult problem. The operators which are P but not CP can be used as entanglement witnesses.

For more details read Nielsen and Chuang.