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.