I'm currently trying to understand why superluminal information transport is not possible.

Therefore, I would like to get some help concerning the definition of information or the general setting.

Imagine a system in the state $|\psi +>=1/\sqrt{2}(|01>-|10>)$ (with particles A & B). Now the two particles are seperated and person A measures the spin on particle A.

  1. question: If now, person B does not know the outcome of person's A measurement, then according to the reduced density matrix the probability for measuring spin up or down should be 50:50. But if person B knows the outcome of the Experiment, Person B will always measure the same (either spin up or down, depending on the outcome of the other measurement). Is this correct?

I don't think so, because if it was correct, then even though the total spin of the whole System is zero, it might be possible that person B who has a 50:50 possibility to measure spin up or down (if Person B does not know the outcome of the measurement), measures spin up and so the two of them measure spin up and the total spin would be 1. But this is not possible, since it is a singlet -state.

So what does person B really measure if he/she does not know the outcome of the measurement of person A?

  1. If person A & B measure simultaneously (in a relativistic meaning), person A & B know each others' results after the measurement. So, in a way there was information transfered (namely each other's state) to each other with a speed greater than light. But, I am pretty sure that even though they know now each other's state, this is not the kind of information one normally talks about when saying that information cannot be transfered at a speed greater than light. So what kind of information are we really talking about?

When looking at the teleportation protocol, the key is that the person who possesses the target qubit does not perform any measurement on it's qubit before this person gets the information that contains the measurement outcome of the other person. So does information mean "knowing the state of a system without measuring"?

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    $\begingroup$ First, that's not a singlet state. You need a $-$ sign for the singlet state. Second, in your state (and in the singlet state), if B measures spin down, A always measures spin up. $\endgroup$ – Peter Shor Jun 13 '16 at 14:13

Your question 1 has absolutely nothing to do with entanglement, and for that matter absolutely nothing to do with quantum mechanics. You can formulate exactly the same question this way:

You and I each have in our pockets a tennis ball that is either red or green. We somehow each know for certain that they are of the same color (and that either color is equally likely), but we've never looked to see which color. One day, when we are far apart, we each look at our tennis balls. You see that yours is red. What is the probability that I'll see that mine is red?

The answer is: For someone who knows that you saw the color red, the probability that I'll see the color red is 100%. For someone who knows the basic setup but doesn't know what color you saw, the setup is 50%. This is not a contradiction. This kind of probability measures a person's knowledge, and the probability can differ for people with different amounts of knowledge.

Likewise: Person A measures spin up. Then you talk about "the" density matrix for the spin of particle B. But there are two density matrices: One for the person who knows what Person A measured and one for the person who doesn't know. The probability that B will be spin down is 100% for the first person and 50% for the second person. Again, this is not a contradiction.

Entanglement raises some much more subtle and (if you're new to this) eye-popping issues, but all of those issues arise from the possibility that Persons A and B can make independent choices about the direction of their spin measurements. You've got them always both measuring in the same direction, so none of those issues arise.

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What kind of information are we talking about in superluminal information transport? It might be more intuitive to call it superluminal communication.

For the prohibited kind of superluminal information transport, you need to be able to achieve the following.

At the start of the protocol, A holds a classical bit which either has value 0 or 1, and which is uncorrelated with B's state. You can assume that each of the values has probability $\frac{1}{2}$. Assume A never changes the value of this bit.

At the end of the protocol, B holds a classical bit which is correlated with A's bit.

Any kind of superluminal information transport that permits A and B to achieve this is prohibited.

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  • $\begingroup$ Thanks for your answer. But still, I don't really get what kind of information we're talking about. "Any kind of superluminal information transport that permits A and B to achieve this is prohibited." What do you mean with that? Do you mean that there could be a way for them to communicate at a Speed faster than light so that it may seem that the two bits were correlated, e.g. assume there exists a way to communicate faster than light so that A & B can communicate with each other so that they can "influence" the outcome of the measurement and create some correlation? Or what do you mean? $\endgroup$ – anonymous Jun 13 '16 at 18:31
  • $\begingroup$ I mean that if A has a message that is unknown to everybody except A, she cannot send it to B at a speed faster than light. In particular, if A could take a bit that she knows, and arrange to have B know a bit that is correlated with A's bit, then A and B could use error-correcting codes to send a message faster than light. $\endgroup$ – Peter Shor Jun 13 '16 at 19:21
  • $\begingroup$ On the other hand, C could flip coins, and arrange to send the results to both A and B so they receive them simultaneously. This is fine, because A can't use the random bits from C coin to send a message to B faster than light. The results of measuring EPR pairs behave like this. $\endgroup$ – Peter Shor Jun 13 '16 at 19:26
  • $\begingroup$ ok, thanks a lot! I think that I have understood it :) $\endgroup$ – anonymous Jun 14 '16 at 21:01

Question 1 is based on the false premise that B measurement does not always obey a 50:50 probability. In fact persons A and B both always have a 50:50 probability of getting a spin up. That particles A and B are entangled means that there is a correlation between the measurements of A and those of B: when A is up, B is down; when B is down, A is up. But this correlation is not apparent to B until B knows the outcome from A, and this knowledge has to travel classically. From B local point of view, B always see 50:50 outcomes.

Question 2 is based on another false premise, that there exists a relativistic meaning of simultaneity. The relativistic view is that simultaneity cannot be defined absolutely for separated events. If A and B are not at the same place, even if they perform their measurements at the same time in the frame where they are both at rest (supposing there is one), still for some observers A will measure before B does while for some other observers B will measure before A does. This implies that no causal relationship can exist between A and B measurements.

So, no information can travel via entanglement from A to B.

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I would like to get some help concerning the definition of information [in the context of sending information]

Suppose Alice and Bob are playing a game. The communication game. The communication game goes as follows:

  1. A referee flips a coin, and tells Alice the result.
  2. (Alice and Bob apply some strategy X.)
  3. Elsewhere, Bob tells a second referee what he thinks the coin flip outcome was.
  4. If Bob guessed correctly, the game is won.

Communication is equivalent to winning that game more than 50% of time.

If Alice and Bob can use mechanism X to win the game more than 50% of the time, then they can use X to send arbitrary messages. (And, vice versa, being able to send arbitrary messages to each other is a way to win the game.)

But what about shared measurement results?

You can't use shared random bits to win the communication game.

There are coordination games where shared random bits are useful, such as the Two Generals' Problem where two attacks have to happen at the same time. But shared random bits aren't useful for the communication game, so it's misleading to think of them as transmitted information.

But what about entanglement?

You can't use entanglement to win the communication game. This has been proven mathematically.

However, entanglement is still useful for winning some other coordination games. In particular, there's a class of games where classical solutions that won with high enough probability would always hide a solution to the communication game but some entanglement-based solutions get over those probability thresholds without resorting to winning the communication game! We call such games "Bell tests".

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