Mathematically, completely positive maps on C\*-algebras generalize positive linear functionals in that every positive linear functional on a C\*-algebra $A$ is a completely positive map of $A$ into $\mathbb{C}$. Furthermore, we have the Stinespring construction as a powerful generalization of the GNS construction. 


Certainly, the relationship between completely positive maps and positive linear functionals can only go so far. I am curious about what physics has to say about this analogy/generalization. It seems that completely positive maps should serve as generalized states of a quantum system, but I've mostly seen cp maps arise in the discussion of quantum channels and quantum operations. I'd like to know precisely in what sense a completely positive map can be viewed as a generalized physical state.



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> **Question:** What is a completely positive map, physically? Particularly, in what precise sense can a completely positive map be regarded as a generalized (physical) state?
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If there are nice survey papers discussing the above relationship, such a reference may serve as an answer to my question.



What is the physical difference between states and unital completely positive maps?
The main interpretation, which is fundamental in quantum information theory, is that the transpose of a UCP map $E$ is a linear map on quantum states that represents a realistic information channel. This is the correct generalization of a Markov map or a stochastic map in classical probability theory. Such a map $E^T$ has to be linear, or otherwise it violates superposition of classical probabilities. $E$ has to be unital, or otherwise $E^T$ does not conserve total probability. It has to be positive, or otherwise negative probabilities can be created. It has to be completely positive, or otherwise $E$ tensored with doing nothing on a companion system is not positive. Then Stinespring's theorem says that the necessary conditions can be interpreted as sufficient for $E^T$ to be realistic.


There is also the Choi isomorphism theorem that identifies CP (but not UCP) maps between two systems with positive states on one system. This is also important, but much more secondary. The category of $C^\*$-algebras with UCP maps as the morphisms, and with the arrows reversed to express the transpose, is a realistic category of quantum probability.


