# Quantum Entanglement - Measuring Twice

In the answer here and on the wiki article and many other articles it is mentioned that if one of 2 entangled particles is measured their state collapses according to the Copenhagen interpretation.

Lets take the example from the EPR paradox article, which mentions a positron and an electron occupying quantum states and becoming entangled. There are two observers, Alice and Bob.

In state I, the electron has spin pointing upward along the z-axis (+z) and the positron has spin pointing downward along the z-axis (-z).

In state II they are opposite.

Alice now measures the spin along the z-axis. She can obtain one of two possible outcomes: +z or -z. Suppose she gets +z. According to the Copenhagen interpretation of quantum mechanics, the quantum state of the system collapses into state I.

Meaning if Bob measures the spin, he will get -z.

My question is, what happens if Bob or Alice measure the spin along the z-axis again, will it remain z+ for Alice and z- for Bob or can it change between measurements?

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## 1 Answer

Yes, if you measure the spin again and assuming the absence of a magnetic field, the measured value of $j_z$ of a particular electron will be the same as it was after the latest measurement of the same quantity – if nothing else was measured or happened in between.

This is true whether the observer is called Alice, Bob, or Barack. The reason why $j_z$ isn't changed is known as the angular momentum conservation law. So if Alice and Bob measure their electron's values of $j_z$ twice, the second measurements will be the same as the first ones, and they will obviously obey the same correlation as well.

However, if you measure e.g. $j_x$, the spin with respect to a perpendicular axis, in between, the final measurement of $j_z$ won't be correlated with the first one. In this perpendicular case, the final $j_z$ will have 50% vs 50% probability to be up and down, respectively, regardless of the value of $j_x$ we measured in between. The state of a spin-1/2 particle – i.e. all the predictions we can make for future measurements of the spin – are fully dictated by the last measurement we did.

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After you added: "if nothing else was measured or happened in between." it got confusing again, how can something be measured in between the first and the second measurement? –  Timo Huovinen May 21 '12 at 18:57
It would be in between the first and second measurements of $j_z$. In such a case the second measurement of $j_z$ would be the third measurement overall. –  David Z May 21 '12 at 20:38
"The reason why $j_z$ isn't changed is known as the angular momentum conservation law." I think this reason is wrong. By the same reasoning, $j_x$ cannot change as well. The real reason $j_z$ remains the same is because the state collapsed into an eigenstate of $j_z$ after the latest measurements. –  Siyuan Ren May 22 '12 at 1:51
Dear Karsus, indeed, by the same reasoning, $j_x$ cannot change, either. By rotational invariance, the behavior of $j_x$ and $j_z$ is obviously the same behavior. And indeed, it doesn't change. When you measure $j_x$ twice, you get the same result. If you measure $j_x$ and then $j_z$, the probability of $j_z=+1/2$ is 50% just like that of $j_z=-1/2$ after the first measurement. And $j_z$ doesn't change between the two measurements; the second measurement just determines which answer was right. The terminology of a "collapse" is always misleading. There is no "collapse" as a real process. –  Luboš Motl May 22 '12 at 4:35