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I was looking at a problem, where the state (a pure state) of a two-particle system was given as:

[tex] | \Psi> = A_{11}| a_1 b_1 > + A_{12} | a_1 b_2> + A_{21} | a_2 b_1> + A_{22} | a_2 b_2> [/tex]

In the system particle a can only be in the states a_1 or a_2 and particle b in b_1 or b_2.

The density operator should be

[tex] \rho = |\Psi><\Psi| [/tex]

The question is, what happens if we make a measurement that determines the state of particle a? Will the collapsed state still be pure, or can it somehow fall into a mixed state? The second alternative sounds unintiutive to me, but I have been surprised many times enough...

Assuming that the measurement yielded the result a_1, then I think the density operator should collaps into

[tex]\rho_1' = p_1 ~\rho ~p_1[/tex]

where p_1 the projection operator

[tex]p_1 = |a_1><a_1|[/tex]

is. After making sure that the norm is OK, I then find that

[tex] Trace(\rho_1^2) = 1 [/tex],

which means that the state after the measurement is still pure. Is this correct? I don't feel confident with this, and I sense that I might be missing something.