Density matrices & spin correlation

[caimzzz]

Homework Statement

So I have two 1/2 spin systems A and B in a singlet state $$|\psi > = \frac{1}{\sqrt{2}} ( |+-> - |-+> )$$. The question is: If I measured B and got $$S_{Bz} = 1/2$$. What will I measure on state A on z axis?

Homework Equations

The Attempt at a Solution

The answer I think is that I will measure spin -1/2 on z axis with probability 1.
My problem is that I tried to calculate the probability for measuring A on z axis with eigenvalue -1/2 and got probability 1/2. My attempt was:
$$\rho = |\psi><\psi|$$ which is density matrix for A and B. My probability is then $$P = tr ( \rho |-+><-+| )$$ $$\rho = 1/2 ( |+-><+-> - |+-><-+| - |-+><+-| + |-+><-+| )$$.
And then I get: $$P=<-+| \rho |-+> = \frac{1}{2}$$[/B]
I think that my equation for probability may me wrong. What should I do?

 

[mark726]

Hi there,

First of all, your density matrix for A and B is correct. However, your calculation for the probability of measuring A on the z axis with eigenvalue -1/2 is incorrect. The correct calculation is as follows:

P = Tr(\rho |+-><-+|) = <+-|\rho|-+> = 1/2

The reason for this is that when you measure B and obtain an eigenvalue of 1/2 for S_Bz, you are effectively collapsing the combined system A and B into the state |+->. This means that when you measure A on the z axis, you are actually measuring the z component of the spin of the state |+->, which is -1/2 with probability 1.

I hope this helps clarify things for you. Let me know if you have any further questions.