Why does the partial trace give the reduced density matrix?

[naima]

Hi Pf
I am looking for the reason why after an interaction "tracing out" the external dof of the system gives the density matrix of the particle.
I looked in Ballentine if it comes from an axiom but i did not find (one occurrence for partial trace)
I have the tool but i have not the reason why it is a good tool.

 

[atyy]

Let the whole system S be divided into A and B, where we observe A and trace out B. You write the observable on the whole system $O_{S} = O_{A} \otimes I_{B}$, and apply the Born rule to the whole system, and the reduced density matrix should pop out.

 

[Demystifier]

Hi Pf
I am looking for the reason why after an interaction "tracing out" the external dof of the system gives the density matrix of the particle.
I looked in Ballentine if it comes from an axiom but i did not find (one occurrence for partial trace)
I have the tool but i have not the reason why it is a good tool.

Think of tracing out as averaging over unobserved degrees of freedom, which is akin to the concept of marginal probability:
http://en.wikipedia.org/wiki/Marginal_distribution