Why does tracing out help in averaging over unobserved degrees of freedom?

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Tracing out external degrees of freedom (dof) in quantum systems provides the reduced density matrix for the observed subsystem. This process involves applying the Born rule to the entire system, represented as ##O_{S} = O_{A} \otimes I_{B}##, where A is the observed subsystem and B is the traced-out subsystem. The concept is closely related to marginal probability, allowing for the averaging over unobserved dof. The discussion highlights the lack of axiomatic justification in Ballentine's text for this procedure, despite its utility in quantum mechanics.

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naima
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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.
 
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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.
 
naima said:
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
 

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