# What is the physical difference between states and unital completely positive maps?

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.

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?

If there are nice survey papers discussing the above relationship, such a reference may serve as an answer to my question.

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As a consequence of Choi's theorem on completely positive maps, they can be physically interpreted as follows: In the context of quantum information theory, the operators ${V_j}$ are Kraus operators, which are not necessarily unique in terms of the states. Any square root factorization of the Choi matrix $B^*B$ gives such a matrix. Because of the Kraus operators, the linear functionals are restricted to the eigenstates of the Choi matrix. Here is source that continues to discuss this: http://en.wikipedia.org/wiki/Completely_positive

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There is an isomorphism between quantum channels and states known as the Jamiołkowski isomorphism, and so you can link properties of one with properties of the other.

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I've somehow been reincarnated as a new me...but I wanted to thank you for this answer. It's precisely the sort of thing I've been looking for. –  Jon Bannon Apr 13 '12 at 18:38
No problem. Hope it helps. –  Joe Fitzsimons Apr 14 '12 at 17:11