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.

 

[Entropia]

Hello Mark,

Completely Positive (CP) maps are an important concept in quantum mechanics. They are mathematical operations that describe how a quantum system evolves over time. These maps are used to model the behavior of quantum systems and predict their future states.

One of the main uses of CP maps is in the study of open quantum systems. These are quantum systems that interact with their environment and lose information to it. CP maps help us understand how this information loss affects the evolution of the system.

CP maps are also essential in quantum information theory, where they are used to describe the transfer of quantum information between different systems. This has applications in quantum communication and quantum computing.

Furthermore, CP maps are crucial in the study of quantum entanglement, which is a fundamental property of quantum systems. They allow us to quantify and manipulate entanglement, which has applications in fields such as quantum cryptography and teleportation.

In summary, CP maps are an essential tool in the study of quantum mechanics and have various applications in different areas of research. I hope this helps clarify their purpose and importance in the field. Keep up the self-teaching, and best of luck on your journey through quantum mechanics.