The above state is a "Bell Pair" or "The Einstein-Podolsky-Rosen (EPR) Pair", or "Maximally Entangled Pair"
There are many maximally entangled pair of qubits, but the above is the most famous one. Although isn't a way to quantify how "entangled" a pair is, but the |MEP\rangle is definitely as most entangled as a pair can be.
Property of MEP:
doing an operation M on one bit is the same as doing the inverse operation M^\dagger on the other.
doing an operation M twice on different qubits will cancel out the effect.
any rotation of individual bits of |MEP\rangle is still a |MEP\rangle (only joint operations can get the state out of |MEP\rangle)
measuring any bits in |MEP\rangle will give \frac{1}{2} chance of being 1 and \frac{1}{2} chance of being 0.
Super-dense Coding
We saw, in one homework, that you can hide one extra bit of information in the amplitude. Sending one of the qubit in |MEP\rangle can convey 2 classical bits of information.
Remote State Preparation
We use the fact that Alice doing operation M on a joined state is the same as Bob doing operation M^\dagger on the joined state. So we can transmit information by "helping" Bob to do certain operation.
Sending 1 classical bit after preparing |MEP\rangle can convey 1 qubit of information. Here is how we do it:
We encode message in \theta = 67.79661016....
Our goal is to send a state Rot_{\theta}|0\rangle where \theta is our sample space.
Prepare |MEP\rangle and distribute to message sender and receiver.
If we now let the receiver measure the qubit, the receiver would not be able to know which angle we have sent \theta or \theta + 90\degree, which loses all meanings. // QUESTION: is it meaning less? how do we extract information in theta?
So we have to measure first and give instruction to receiver on how to interpret the result.
With Pr\{A = 0\} = \frac{1}{2} state collapse to |0\rangle \otimes Rot_\theta|0\rangle. With Pr\{A = 1\} = \frac{1}{2} state collapse to |1\rangle \otimes Rot_\theta|1\rangle
So we send 1 bit information to receiver let him know which one is intended.