Editing Bell state (section)

=== Superdense coding ===
[[Superdense coding]] allows two individuals to communicate two bits of classical information by only sending a single qubit. The basis of this phenomenon is the entangled states or Bell states of a two qubit system. In this example, Alice and Bob are very far from each other, and have each been given one qubit of the entangled state.

<math>|\psi \rangle = \frac{|00\rangle + |11\rangle}{\sqrt{2}}</math>.

In this example, Alice is trying to communicate two bits of classical information, one of four two bit strings: <math>'00', '01', '10',</math>or <math>'11'</math>. If Alice chooses to send the two bit message <math>'01'</math>, she would perform the <math>X</math> gate to her qubit. Similarly, if Alice wants to send <math>'10'</math>, she would apply the phase flip <math>Z</math>; if she wanted to send <math>'11'</math>, she would apply the <math>iY</math>gate to her qubit; and finally, if Alice wanted to send the two bit message <math>'00'</math>, she would do nothing to her qubit. Alice performs these [[quantum gate]] transformations locally, transforming the initial entangled state <math>|\psi\rangle</math> into one of the four Bell states.

The steps below show the necessary quantum gate transformations, and resulting Bell states, that Alice needs to apply to her qubit for each possible two bit message she desires to send to Bob.

<math>00: I = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \longrightarrow |\psi \rangle = \frac{|00\rangle + |11\rangle}{\sqrt2}\equiv |{\Phi^+}\rangle</math>

<math>01: X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix}\longrightarrow |\psi \rangle = \frac{|01\rangle + |10\rangle}{\sqrt2}\equiv |{\Psi^+}\rangle</math>

<math>10: Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix}\longrightarrow |\psi \rangle = \frac{|00\rangle - |11\rangle}{\sqrt2}\equiv |{\Phi^-}\rangle</math>                                                     

<math>11: iY = ZX = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix}\longrightarrow |\psi \rangle = \frac{|01\rangle -  |10\rangle}{\sqrt2}\equiv |{\Psi^-}\rangle</math>.

After Alice applies the desired transformations to her qubit, she sends it to Bob. Bob then performs a measurement on the Bell state, which projects the entangled state onto one of the four two-qubit basis vectors, one of which will coincide with the original two bit message Alice was trying to send.