Averaging Initial & Endstates in Trace Formula

[jk22]

Basically the question is : since the experiments are repeated and results averaged, should not initial and endstates be mixed states ?

So now we should give two density matrices, so how do we average like in the trace formula : $\langle A\rangle=tr(\rho A)$ ?

Is it $\rho=\sum_i p_i|\phi_i\rangle\langle\phi_i|$ with $p_i=\langle\phi_i\underbrace{|\Psi\rangle\langle\Psi|}_{\rho_{init}?}\phi_i\rangle$

So that now $\rho_{init}$ were the initial density matrix and hence shall be written as one of a mixed state too with now unknow probabilities ?

 

[mfb]

If your initial state changes from repetition to repetition that sounds like a possible approach.
You average over the initial state and sum over all final states you consider to be the same outcome.

 

[jk22]

Hence if we consider a Bell type experiment and we apply the Copenhagen interpretation, then $\langle\Psi|A\otimes B|\Psi\rangle$ is for 1 photon output for Psi the singlet state.

But the formula $\sum_i A(a,\lambda_i)B(b,\lambda_i)$ would in fact mean an average about several photon output. Thus the quantum system has a pure state initially and the measured end statistics would correspond to a mixed state ?