Complete positivity and quantum dynamics

[JorisL]

Hi again,

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) \Gamma:\mathcal{M}_k\rightarrow \mathcal{M}_n

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 into the system+an environment 'of dimension d'.
More usable is the fact that the extended map id_d\otimes \Gamma is positive.

This map is defined for the Schrödinger picture, acting on density matrices. There exists a dual map \Gamma^* 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
\Gamma is CP iff \Gamma^* 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
tr \rho X = tr \Gamma(\rho)X = tr \rho\Gamma^*(X)
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 \Gamma. 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

 

[kith]

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.

What are M_k and M_n? Some kind of state space? If the CP map acts on the density matrix of a certain open system, the dimensions of these spaces must be the same.

in the schrödinger picture the map is trace-preserving thus
tr \rho X = tr \Gamma(\rho)X

This identity not only uses the trace-preserving property of Γ but implies that ρ and Γ(ρ) are the same state. This is certainly wrong in general. If Γ is the time evolution of the open system your equation states that the expectation value of all observables X is constant.

 

[JorisL]

You are right, the spaces M_k are the observable on a k-dimensional system.
The identity should in my opinion hold if we don't have the observable X in there since the trace of a d.m. should be normalised (in our convention, one can use a normalisation constant).

I'll try to figure some more of this out and give an expanded explanation of the context. Although it's only 2 pages long in my notes (a little bit short if you ask me). Some reference texts would be nice.

 

[bhobba]

Two things that may interest you in this context is Gleason's Theorem:
http://kof.physto.se/cond_mat_page/theses/helena-master.pdf

And especially Wigner's theorem:
http://arxiv.org/abs/0808.0779

Thanks
Bill