Superdense Coding

I think we are now ready to show Cardy a quantum information application that demonstrates an advantage over classical computing. Some QIS people call this “superdense coding,” which sounds fantastic. It’s great rhetoric but the reality, though impressive, is more modest. Perhaps dense coding would be a better name, which is what we will call it.

The basic idea behind dense coding is that you can send two classical bits with just one single qubit if you are clever about preparing the qubit in an entangled state and then having appropriate gates ready at the receiving end to process that qubit. It turns out that this method is a nice example of secure quantum communication. But it works only if the message sender has already sent you a properly prepared qubit.

I want to send Bob two classical bits. The four possibilities are $00$, $01$, $10$, or $11$. I will show that I can do this by sending Bob just one qubit if we do some preliminary work. First, I prepare two qubits $C$ and $D$ in the entangled Bell state:

$|\psi\rangle=\mid$Bell$\left._{00}\right\rangle=\frac{1}{\sqrt{2}}\left|0_C 0_D\right\rangle+\frac{1}{\sqrt{2}}\left|1_C 1_D\right\rangle.$

I keep one of the qubits and send the other to Bob.