Quantum solution to the two-army problem?

Question

The two-army problem can be described as follows:

Two blue armes is separated by the white army, they can win the battle only if they attack together. However, the messenger who send the attack signal may be caught by the white army and the sender won't know for sure whether or not his partner has received the message.In the classical case, the cooperation of the blue armies is never guaranteed.

Question is, is this problem become trivial in quantum regime?

My naive solution is that the blue #1 just prepare spin singlet state, send a spin off which entangles with the local one. The blue #2 just prepare a spin up filter, if the spin comes and pass the filter, the local spin of blue #1 will be spin down, thus blue #1 will know that blue #2 has received his attack message.

According to the two answers provided by @Martin and @OON, it seems that my solution is wrong. Can anyone came up with a solution that solve the two army problem in quantum mechanics?

Answer 1

The quantum entanglement works in the same way as the classical correlation. Imagine that the commander of the 1st army puts two socks - black and white in two boxes and send one of them to the 2nd army. Then he discovers that in his box the sock is white. What he knows is that if the 2nd commander opens his box he discovhes the black sock. What he doesn't know is whether the 2nd commander received the box and opened it.

The only difference in the quantum case is that after measurement of the spin you know not the measurement outcomes themselves but the quantum state that may give a certain probability distribution for the 2nd lab. You still don't know what happened with the 2nd particle and whether any measurement happened at all.

Moreso if you consider the case of the standard Bell state where both commanders measure spin along the same axis the situation becomes exactly the same as in case of boxes with socks.

So no. If the 1st commander measures spin down all he knows is that thr 2nd one would measure spin up IF he received the other particle. Not whether he received it at all.

Answer 2

There are multiple problems with this approach.

a) In your scenario we have two systems A and B and a maximally entangled state shared by them. Now B is send to the right and measured. But in any case, we can have two outcomes: Either B is spin up or B is spin down. You can only know that it is spin down if it was spin down beforehand. But then, A doesn't know anything they didn't know before.

In different words: Entanglement is only correlations. If I know my systems are correlated, I can predict part of the system knowing something about the other part, but if I change one part of the system in any way, I'll just destroy the correlation.

b) The entangled particles are sent before the battle. Now there is no reason to suppose they didn't get intercepted by the white army. Since A cannot know anything that B did (or did not do) to its part of the system, it cannot know whether the other party is not actually the white army instead of the blue.

c) There is never a backchannel. B never sends any signal to A. But then, A cannot know about anything that happened to B, because that would violate relativity. In order to send information, you have to send a signal with at most the speed of light (that's really the first problem from a different perspective).

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But could entanglement help in any way with that dilemma?

You have two problems to worry about: - Authentication of the recipient. You want to make sure that the other party is who they claim to be. - Knowing that the party received the signal.

That directly implies that the other party needs to be able to respond to you - if they can't, you can't know that they received their signal, because you'd get information without signals, which violates physics.

If there is no way for blue to respond, nothing will help.

How could entanglement help? Well, it could potentially help with the first problem (authentication), but you can also do it by classical means.

Now there are two scenarios worth talking about:

• Let's assume the entangled particles were distributed before the two parts of the armies split. In that case, you could use any (insecure) line of communication to establish a secure, i.e. encrypted, line of communication using the entangled particles (one pair won't be enough, though). Once you have a secure line of communication, you could just send the signal, wait for a response reading the signal back to you and then you can be sure that the other party received the signal.
• Now let's assume that entanglement is not distributed in advance. In that case, it won't help, because you can't know that it isn't white who caught the other parts of the entangled particles. You'd then have to establish that it was really blue who received the particles by some other means. But that's the same as establishing authenticity of your communication partner over an insecure channel - something that can be achieved by classical means, also (have a look at e.g. four-factor authentication). Just as an example, I could ask a few questions whose answer only blue knows and send the signal and blue would respond that they received the signal and also tell you the answers to your questions.