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|psi> = |up> |down> + |down> |up>

So far, so good. Now let us measure the spin of the particles, and let the outcome of the measurement be that the first particle is in the state |up> and the second particle is in the state |down>. If measurement induces the collapse, then after the measurement the wave function is

|psi'> = |up> |down>

However, |psi'> is NOT symmetric under the exchange of the two particles.

Does it mean that the collapse violates the exchange symmetry?