Wave function collapse of entangled photons?

Question

If we have two entangled photons then they are described by the same wave function. But if I measured the position of the signal photon and it collapses into a position, would the idler photons position also collapse into a position?

Answer 1

"Entangled" is not a unique descriptor for the quantum state of two particles (i.e. there are many very different entangled states), and the position wavefunction of photons is not particularly well defined, but it is indeed reasonable to ask what happens when you take two particles in the entangled state $$ |\Psi\rangle=\int\mathrm dx \, \psi(x)|x\rangle\otimes|x\rangle, $$ i.e. a coherent superposition with some relatively flat coefficient $\psi(x)$ of states where both particles are at position $x$ with respect to some marker (possibly with different markers for the two particles, i.e. they could be very far away), and you measure the first one in the position basis.

In that case, you will detect the first particle with probability $|\psi(x)|^2$ at each position $x$, and if you condition on this detection then all position measurements of the second particle will collapse it to position $x$.

However:

• It is misleading to think (at least without reservations) that the measurement on the first particle has caused the wavefunction of the second particle to collapse. This is because

• when you measure the first particle, you have no control of what measurement result you'll get, so from the particle of the person measuring the second particle, all they see is an even spread; in fact,

• this is the case for all possible observables on the second particle: there is no measurement at all that you can do on particle 2 that will let you deduce whether the first particle was measured or not.

• To emphasize, this protocol cannot be used for anything resembling superluminal communication.

Because of the lack of control over the measurement result, there is really nothing you can do within this protocol that will let you even attempt to communicate. To try and overcome this, you might try to e.g. switch between position and momentum measurements, and try to communicate via the choice of observable; this is proposed endlessly and it also doesn't work: there is simply no way to use entanglement to communicate faster than light.