Hi again,(adsbygoogle = window.adsbygoogle || []).push({});

Another, possibly trivial, question.

In quantum dynamics we consider maps containing the evolution of a system.

Suppose we have a completely positive (CP henceforth) map

(1)[itex]\Gamma:\mathcal{M}_k\rightarrow \mathcal{M}_n[/itex]

This map has following properties:

- Trace-preserving
- Complex linear
- Continuous in time (not important here)

This map is CP if it is d-positive for all d. I gather from this it should be possible to trivially extend the system we are interested in to the system+an environment 'of dimension d'.

More usable is the fact that the extended map [itex]id_d\otimes \Gamma[/itex] is positive.

This map is defined for the Schrödinger picture, acting on density matrices. There exists a dual map [itex]\Gamma^*[/itex] in the Heisenberg picture, i.e. acting on observables. (this dual is unity/unital preserving)

In various texts I've found that it should be easy to show that

[itex]\Gamma[/itex] is CP iff [itex]\Gamma^*[/itex] is CP

Now some class mates used some really fishy manipulation (perhaps bad mathematical notation/practice) which seems to have gotten me completely lost.

Here's what they did.

in the schrödinger picture the map is trace-preserving thus

[tex]tr \rho X = tr \Gamma(\rho)X = tr \rho\Gamma^*(X) [/tex]

So far it seems ok to agree except for the first equality. In general the dimensions k and n in(1)aren't the same so the matrix product in one of the sides will not be defined.

Then they extend the map to use d-positivity of [itex]\Gamma[/itex]. From this they infer that the dual map is also d-positive. (symbolic, no explanation)

I'm not sure how they justified all this hand-wavy(at best) stuff. Another problem is that they don't appreciate attaining maximal rigour in this kind of things so they might have overlooked these problems.

Looking forward to some suggestions,

Joris

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# Complete positivity and quantum dynamics

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