# What makes quantum-entanglement communication so surprising? [duplicate]

I don't understand why quantum entanglement seems so surprising to physicists in the case of communication.

Let us say we take a pair of shoes, and put the two shoes in two separate boxes, and shoot one of the box into a black hole while holding the other box at the other end of the universe. If I were to open the box inside the black hole and find the left shoe in it, it is obvious that the other shoe in the far end of the universe will be a right shoe. The two objects are pre-entangled pairs, there is no need of further communication to maintain their entanglement.

• Because quantum shoes come in more than two shapes... Jul 11 '17 at 15:58
• I perceive that the underlying question may be less about the instant "communication" at a distance, and perhaps more about how/why/whether entangled particles actually start out in an indeterminate state. In other words, can you imagine separating the pair of shoes so that neither shoe was definitely in one or there other box? AND that which shoe you discover at the end of the universe was not dependent upon how it was put in, but rather how you opened the box? Jul 11 '17 at 16:23
• Related/possible duplicate: physics.stackexchange.com/q/3158/50583 Jul 11 '17 at 22:16
• See this question and my accepted answer thereto: physics.stackexchange.com/questions/330554/… Jul 11 '17 at 22:54
• One effect is that one can use entanglement to send signals over channels that exceed their (classical) capacity, e.g. two bits of classical information can be sent using only one qubit, see superdense coding. Jul 12 '17 at 0:26

I like to teach undergraduates about what new mathematics you can do with entanglement, with a game that I like to call "Betrayal."

# Rules of Betrayal

Betrayal is a cooperative game for a team of 3 people. They are playing against and trying to beat the game. The game is usually going to force one member of this team to act at cross-purposes to the other two; I will call this person the "traitor" for short. (Of course since they're cooperating with the others they don't want to betray their friends!)

The whole game consists of some large number $N$ of similar rounds. The team is allowed to prepare however they want between each of these rounds.

In each round, all three teammates are put into boxes through which no known communication can travel.1 They can bring whatever they want to into that box. Each box has a timer, a screen onto which the game will display a message, and two buttons marked $0$ and $1$. After we display the message on the screen, the timer counts 5 minutes which they have to press exactly one button once, locking in their numbers. We then bring everyone back together and sum up these three numbers, and they win based on whether the sum is even or odd. Tampering with the box in any way other than pressing one button once during a round is, of course, against the rules.

Getting more in-depth, 25% of our tests are control rounds. We display on all three of their screens, "Make the sum of your three numbers even." When we get back their chosen numbers and sum them together, the team beats that round only if this sum is even. Simple enough, right?

In 75% of our rounds we randomly choose one of the teammates to be the traitor, and we lie to them by broadcasting that same message onto their screen: "Make the sum of your three numbers even." As for the other two, we broadcast the true objective onto their screen: "Make the sum of your three numbers odd." When we get back their chosen numbers and sum them together, the team beats that round only if this sum is odd. Everybody is, of course, informed in advance about these rules.

Now in this hypothetical future, the team plays this for something like 3-4 weeks straight, doing 300 rounds total, and they win if they beat 270 or more of these rounds. To give some financial incentive, let's say the team gets to split a trillion credits if they win, but they have to pay ten thousand credits if they lose, where a credit in the future is equal to the present value of a dollar.

(Footnote 1. Technically, this is achieved by those boxes being inside of spaceships which fly until they're more than 5 light-minutes apart from each other, but that's not super-important.)

# Classical limits

It turns out in classical probability the above game is exactly identical to a slightly revised game: we similarly separate the players and then we ask all of the players "what would you do if you told us to make the sum even?" and they press one button; then we ask all of them "what would you do if you told us to make the sum odd?". Doing so we get 6 total answers, of course. Let's call our players Alice, Bob, and Carol and denote their answers to the question $(A,B,C)_{o,e}$ where $A_e$ is "Alice's answer if we ask her to help make the sum even."