According to some textbooks, the light-speed communication barrier is impossible to break even though entangled particles can seemingly send information back and forth instantaneously. We will explore why this light-speed limit is thought to be true, then we will debunk it. (No doubt Einstein will roll over in his grave.)
If you are familiar with the Star Trek series, you may have wondered how Captain Kirk can instantaneously communicate with Starfleet Command which is light-years away. As it turns out, the twenty-third century has an Alice and Bob. Alice is a Starfleet admiral. Bob is the captain of the USS Schrodinger Cat. Using quantum-entanglement communication, Alice will attempt to warn Bob that the Bohr (who are far worse than the Borg) are attacking a nearby Federation colony: Einstein IV. To accomplish this, Alice must send Bob the following quantum state:
The entangled system that is essential to their communication is mathematically described below:
Equation 2 above shows the first 0 and first 1 belong to Alice (A) and the second 0 and second 1 belong to Bob (B). Alice's first step is to calculate the tensor product of equations 1 and 2:
Alice then applies a CNOT gate to equation 3:
Alice uses a Hadamard gate (H) on equation 4. Several steps ensue. Feel free to skip to the final result: equation 9.
At equation 9 we end up with four entangled states, each with a probability of 0.25. The table below summarizes which states belong to Alice (A) and which states belong to Bob (B):
Alice does a measurement, so only one of the four entangled states will be received by Bob. If Alice measures |00>, Bob will receive the quantum state Alice intended to send, but if Alice measures any of the other states, Bob will need to apply an X-gate and/or a Z-gate to get the right state. How does Bob know which gates to apply? Alice gives him a call and tells him! Once Bob receives Alice's call (no faster than light speed), Bob performs one of the following operations (9 to 21):
Anyway, that's the argument against true instant messaging. The uncertainty of quantum entanglement defeats its purpose. To know which state is the right state, Bob must wait for Alice's signal which could take light years to reach him. The Bohr will surely destroy Einstein IV! On the other hand, what if Alice is an airhead? Or, what if the communication system is full of tribbles? Let's try thinking outside the box and see what can be done. To simplify matters let's convert those messy entangled states to simple variables:
Now let's add a second entangled system. That gives two entangled systems. We label each with i or j subscripts:
Alice measures both Ai and Aj and gets A2 and A3 respectively. Bob sees B2 and B3 respectively. (See row 1 at table below.) When Bob was a cadet he was trained to ignore this first initial state. Alice continues to measure Ai until she gets the desired state A1 (see row 3). How does she let Bob know that B1 is the right state? Does she give him a call? No! She measures Aj until its state changes while leaving Ai alone. At row 4, Bob sees that B3 has changed to B2. Any change is a signal that Bi is now the correct state. So Bob knows B1 is the state Alice intended.
At close range, this form of communication is highly impractical; however, since Alice and Bob are light years apart, the up to 16 measurements needed to make getting the proper state a 99% certainty is well worth it! That being said, perhaps we can improve this new system. How about making the probabilities work for us instead of against us?
Below are the new rules Alice needs to get the state she wants 75% of the time instead of 25%:
The best way to understand the above rules is a straightforward example. Alice measures both Ai an Aj. No matter what state she gets, both she and Bob understand that this random state shall be labeled 1. (See row one at the table below.) This 1 bit is not part of the message--it simply sets things up. Alice wants to send Bob a 0 bit, so she measures Ai until it changes from A1. She does nothing to Aj which is A2. At row 3, Bob notices that B1 has changed to B4 and B2 is unchanged. He (or a CPU) interprets this as 0. At row 4 he sees that B4 has not changed and B2 has changed, so he records another 0 bit. At row 5 Alice does not measure Ai, only Aj and gets A3. Bob notices the change from B1 to B3. Another 0 bit. At row 6, Alice measures Ai, not Aj. Bob sees the change in Bi but not Bj, so he records a 1 bit ... and so on.
When all is said and done, Bob receives the following message from Alice: 100011. The first bit is ignored, so the message is really 00011.
Alice can improve the odds of getting the desired bit by using superpositions with more than four possible states. Imagine two pairs of dice (36 states). Roll both pairs. Label whatever you get 1. To make the same bit, leave the first pair alone and roll the second pair until you get a different state. To make the opposite bit, leave the second pair alone and roll the first pair until its state changes. The chance of getting the bit you want on the first try is 35/36 or 97%--practically a sure thing!