One-side measurement

[A23]

Let a 2 particle quantum system defined by a wavefunction

[math]\sum_{n=1}^N \mid\chi_{i_n}\rangle\mid\psi_{j_n}\rangle[/math]

 

[math] i_n, j_n [/math] are application from [math]\mathbb{N}_N[/math] to [math]\mathbb{N}[/math]

 

Sum out of basis of eigenvector

[math]\hat{A}\mid\chi_n\rangle=a_n\mid\chi_n\rangle[/math]

[math]\hat{B}\mid\psi_n\rangle=b_n\mid\psi_n\rangle[/math]

 

My question is about a partial measurement process:

 

suppose A only were measured. Could we say, since the second operator is here unkown, we take hence an unkown basis for the second space [math]\{\mid u_n\rangle\}[/math].

 

Now let the measured value for A was a_n, the endstate has to be [math]\sum_k\mid\chi_n\rangle\mid u_k\rangle[/math] ?