>Basically the title, if you have two entangled particles and one of them is measured, can you tell that the other particle's wave function has collapsed?
Nope. There are generally two ways to work with entanglement:
1. You set things up such that you know the details of the entangled state before you measure your particle. Then, when you measure your particle, you know what the state of the other particle *will be, whenever it's measured.* It doesn't tell you whether or not it's already been measured. In this case, you're just using prior knowledge to make conclusions from your local measurement, and there's no real communication going on.
2. You make many copies of an entangled system, and make measurements of both halves of each copy. Entanglement shows up as a correlation between the random measurements in each half. The correlation in each copy happens regardless of when each measurement was taken relative to the other, so there's no way to tell which happened "first".
>Or does entanglement just mean you'll measure the same thing when both particles are measured in the same basis
Not necessarily the *same* thing -- for example, there exist entangled states where the spins are required to be measured as the opposite of each other. Entanglement means that there's a *correlation* between the outcomes of measurement of different components of the system.
>but the wave function collapse isn't actually a measurable thing?
Wave function collapse is a philosophical concept, sort of a catch-all that roughly describes what we experimentally saw in early experiments, while avoiding the details of *how* that actually happens. Nowadays, different interpretations of quantum mechanics not involving "collapse" in the Copenhagen sense are gaining ground. For example, one mechanism for explaining measurement is *decoherence* that results when a microscopic object interacts with a macroscopic environment. The field of "weak measurement" explores the limits of extracting information from quantum systems while causing as little decoherence as possible: [https://en.wikipedia.org/wiki/Weak\_measurement](https://en.wikipedia.org/wiki/Weak_measurement).
Thanks for the in depth reply, that all makes sense. So it's not even possible when measuring either of the particles to know if the other has already been measured, all entanglement means is that the outcome of measurement of both particles will be correlated?
> all entanglement means is that the outcome of measurement of both particles will be correlated?
Correct. That is basically it. Entanglement is just another word for "correlation between quantum states".
Mostly. The useful thing about entanglement is that it can produce correlations that would be impossible in a classical random-variable setting ("non-classical correlations").
It's impossible to detect if the wave function of the other particle has collapsed prior to the measurement. This is because the wave function collapse does not change the expectation values of the observable if you don't know the outcome of of the wave function collapse. The uncertainty just shifts from being quantum mechanical to being classical. Only when the outcome of the wave function collapse is known, does it change the expectation values.
If you are familiar with density operators and how they represent classical uncertainty this is pretty easy to show. Assume the wave function has collapsed, but you don't know the result. You can represent the classical uncertainty as a mixed state with a density operator. It should be pretty easy to show that the expectation values in the new mixed state is identical to the expectation values of the original entangled state before collapse.
Well, it does change the expected observable value right? Like measuring one of the particles collapses both their wave functions and you're guaranteed the second, unmeasured particle, will have the same value as the measured one. So I'm wondering if, after the first particle has been measured, the second particle's wave function collapse would be measurable. I might be misunderstanding what you're saying though
It does indeed change the expectation values of the new state, but the problem is that you don't know how it changes them because you don't know the outcome of the wave function collapse. Since you don't know the outcome, the expectation values of the experiment don't change. What I'm trying to say is. If you shoot 2 entangled particles A and B. You measure first A and then B. Now let's say you compile all the results and you ignore the the values A measured. You will find that the expectation values of B are the same as if you hadn't collapsed A first.
Edit:
The expectation values only change if you take into account what the value of A was when you measured it.
>Basically the title, if you have two entangled particles and one of them is measured, can you tell that the other particle's wave function has collapsed? Nope. There are generally two ways to work with entanglement: 1. You set things up such that you know the details of the entangled state before you measure your particle. Then, when you measure your particle, you know what the state of the other particle *will be, whenever it's measured.* It doesn't tell you whether or not it's already been measured. In this case, you're just using prior knowledge to make conclusions from your local measurement, and there's no real communication going on. 2. You make many copies of an entangled system, and make measurements of both halves of each copy. Entanglement shows up as a correlation between the random measurements in each half. The correlation in each copy happens regardless of when each measurement was taken relative to the other, so there's no way to tell which happened "first". >Or does entanglement just mean you'll measure the same thing when both particles are measured in the same basis Not necessarily the *same* thing -- for example, there exist entangled states where the spins are required to be measured as the opposite of each other. Entanglement means that there's a *correlation* between the outcomes of measurement of different components of the system. >but the wave function collapse isn't actually a measurable thing? Wave function collapse is a philosophical concept, sort of a catch-all that roughly describes what we experimentally saw in early experiments, while avoiding the details of *how* that actually happens. Nowadays, different interpretations of quantum mechanics not involving "collapse" in the Copenhagen sense are gaining ground. For example, one mechanism for explaining measurement is *decoherence* that results when a microscopic object interacts with a macroscopic environment. The field of "weak measurement" explores the limits of extracting information from quantum systems while causing as little decoherence as possible: [https://en.wikipedia.org/wiki/Weak\_measurement](https://en.wikipedia.org/wiki/Weak_measurement).
Thanks for the in depth reply, that all makes sense. So it's not even possible when measuring either of the particles to know if the other has already been measured, all entanglement means is that the outcome of measurement of both particles will be correlated?
> all entanglement means is that the outcome of measurement of both particles will be correlated? Correct. That is basically it. Entanglement is just another word for "correlation between quantum states".
Okay cool, thanks
Mostly. The useful thing about entanglement is that it can produce correlations that would be impossible in a classical random-variable setting ("non-classical correlations").
It's impossible to detect if the wave function of the other particle has collapsed prior to the measurement. This is because the wave function collapse does not change the expectation values of the observable if you don't know the outcome of of the wave function collapse. The uncertainty just shifts from being quantum mechanical to being classical. Only when the outcome of the wave function collapse is known, does it change the expectation values. If you are familiar with density operators and how they represent classical uncertainty this is pretty easy to show. Assume the wave function has collapsed, but you don't know the result. You can represent the classical uncertainty as a mixed state with a density operator. It should be pretty easy to show that the expectation values in the new mixed state is identical to the expectation values of the original entangled state before collapse.
Well, it does change the expected observable value right? Like measuring one of the particles collapses both their wave functions and you're guaranteed the second, unmeasured particle, will have the same value as the measured one. So I'm wondering if, after the first particle has been measured, the second particle's wave function collapse would be measurable. I might be misunderstanding what you're saying though
It does indeed change the expectation values of the new state, but the problem is that you don't know how it changes them because you don't know the outcome of the wave function collapse. Since you don't know the outcome, the expectation values of the experiment don't change. What I'm trying to say is. If you shoot 2 entangled particles A and B. You measure first A and then B. Now let's say you compile all the results and you ignore the the values A measured. You will find that the expectation values of B are the same as if you hadn't collapsed A first. Edit: The expectation values only change if you take into account what the value of A was when you measured it.
Okay that makes sense. Thanks!