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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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3 Answers 3

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

If you pass one side of a maximally entangled state into the channel, you get the state defined by the Jamiołkowski isomorphism (neglecting normalization). I think this is as close as you can get to a physical interpretation.

Note that if you pass the maximally entangled state through the channel and keep (don't trace out) the channel's environment, you get the channel ket, which is a tripartite pure state. The three subsystems correspond to channel input, channel output, and channel environment output (a.k.a. the indices on Kraus operators). Tracing out the environment on the channel ket gives the Jamiołkowski state, while tracing out the channel output part of the channel ket gives the Jamiołkowski state for the complimentary channel. These statements are all true regardless of whether the channel is unital. If the channel is unital then the subsystem of the channel ket corresponding to channel output will be fully entangled. The subsystem corresponding to channel input is always fully entangled for a trace preserving channel.

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Thank you for this useful answer. –  Jon Bannon Apr 13 '12 at 18:39
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There is a good blog post by Nathaniel Johnston which outlines the parallels between various types of channels and the corresponding states. I link to part 2 because that one in turn links to part 1. –  Dan Stahlke Apr 14 '12 at 2:02

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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