(thought) experiment re: Bell's Theorem and Schrodinger's cat

I apologize if this question is naive. I am wondering about what would happen with the following experiment.

Start with a standard Bell's Theorem setup: We have two quibits entangled in a particular way and sent in opposite directions, where they are measured by independent observers. The distribution of joint measurements is, Bell says, inconsistent with any theory of local variables.

But now, before we send the quibits out, a single central person measures both quibits. That person does not communicate with either observer.

My question is, does the distribution of measurements of the observers change?

This seems like a paradox to me for the following reason. The central person actually knows what the outcomes will be of the observers' measurements. Therefore, the central person can set or calculate "local variables" that always explain what measurements will occur. Yet, to the observers, this experiment seems indistinguishable from the standard Bell's Theorem experiment (they can't even tell whether the central person is there or not!). So they should see the same distribution, which no local variable theory can explain.

...

One possibility that I thought of is that the answer might depend on whether the light cones of the central person and the observers overlap, in some way, i.e. on whether the information can be transmitted at conventional speeds or not. However, we are assuming that the central person does not communicate with them in any way...(is this assumption even well-defined?).

But my guess is that the observers should see the same distribution of measurements. The resolution of the paradox would be that, from the observers' perspectives, the central person and her local variables are in a superposition, just like Schrodinger's cat. Could this be right?

No, the distribution does change.

If the 'central' person measures both the qubits in the $|0\rangle$, $|1\rangle$ basis, say, then the observers do not see the same distribution in general. Only their measurements along the z-axis still have the same distribution, but by tilting their measurement axis, they can conclude whether the qubit was measured or not.

More concretely, let the state being prepared be

$$|\psi \rangle = \frac{|01\rangle + |10 \rangle}{\sqrt{2}}$$

Then the density matrix corresponding to this (pure) state is

$$\rho = \frac{1}{2} \left( \begin{array}{ccc} 0 & 0 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right),$$ whereas the state of the system after measurement by the central person would become $$\rho_m = \frac{1}{2} \left( \begin{array}{ccc} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 \\ \end{array} \right),$$ which is not a pure state and is distinguishable from the previous state.

• Thanks, I didn't think about the different axes of measurement. – usul Jan 30 '14 at 14:20
• That's all right. Paradoxes are a nice way to really think about a system and understand what goes on properly. :) – Abhinav Jan 30 '14 at 16:24