Categories: Quantum information.

# Superdense coding

In quantum information, (super)dense coding is a protocol to enhance classical communication. It uses a quantum communication channel and entanglement to send two bits of classical data with just one qubit. It is conceptually similar to quantum teleportation.

Suppose that Alice wants to send two bits of classical data to Bob, but she can only communicate with him over a quantum channel. She could send a qubit, which has a larger state space than a classical bit, but it can only be measured once, thereby yielding only one bit of data.

However, they are already sharing an entangled pair of qubits in the Bell state $$\ket*{\Phi^{+}}_{AB}$$, where $$A$$ and $$B$$ are qubits belonging to Alice and Bob, respectively.

Based on the values of the two classical bits $$(a_1, a_2)$$, Alice performs the following operations on her side $$A$$ of the Bell state:

$$(a_1, a_2)$$ Operator $$\qquad$$ Result
$$00$$ $$\hat{I}$$ $$\ket*{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\ket{0}_A \ket{0}_B + \ket{1}_A \ket{1}_B \Big)$$
$$01$$ $$\hat{\sigma}_z$$ $$\ket*{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\ket{0}_A \ket{0}_B - \ket{1}_A \ket{1}_B \Big)$$
$$10$$ $$\hat{\sigma}_x$$ $$\ket*{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\ket{0}_A \ket{1}_B + \ket{1}_A \ket{0}_B \Big)$$
$$11$$ $$\hat{\sigma}_x \hat{\sigma}_z$$ $$\ket*{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\ket{0}_A \ket{1}_B - \ket{1}_A \ket{0}_B \Big)$$

Her actions affect the state on Bob’s side $$B$$ due to entanglement. Alice then sends her qubit $$A$$ to Bob over the quantum channel, so he has both sides of the entangled pair.

Finally, Bob performs a measurement of his pair in the Bell basis, which will yield a Bell state that he can then look up in the table above to recover the values of the bits $$(a_1, a_2)$$. In the end, Alice only sent a single qubit, and the rest of the information transfer was via entanglement.

## References

1. J.B. Brask, Quantum information: lecture notes, 2021, unpublished.