# Super-dense coding

Superdense coding (aka quantum dense coding, or dense coding) is a method of utilizing shared quantum entanglement to increase the rate at which information may be sent through a noiseless quantum channel. Sending a single qubit noiselessly between two parties gives a maximum rate of communication of one bit per qubit (by the HSW Theorem). If the sender's qubit is maximally entangled with a qubit in the receiver's possession, then dense coding increases the maximum rate to two bits per qubit.

### Dense Coding for Qubits

If the sender (Alice) and receiver (Bob) share a maximally entangled state

$$|\Psi^+_{AB}\rangle = \frac{1}{\sqrt{2}}\big( |0_A\rangle |0_B \rangle + |1_A\rangle |1_B \rangle$$

then Alice may perform encode two bits of information into the shared state by using one of four unitary operations corresponding to the different two bit strings. The operations consist of the identity (doing nothing), a bit flip σX (where ∣0⟩ → ∣1⟩ and ∣1⟩ → ∣0⟩), a phase flip σZ (where ∣0⟩ → ∣0⟩ and ∣1⟩ →  − ∣1⟩), or a combination of both σY. After encoding, Alice and Bob share one of the states

00 → (IA ⊗ IB)∣ΨAB + ⟩ = ∣ΨAB + ⟩

$$01 \rightarrow (\sigma^X_A \otimes \mathbb{I}_B)|\Psi^+_{AB}\rangle = \frac{1}{\sqrt{2}}\big( |1_A\rangle |0_B \rangle + |0_A\rangle |1_B \rangle = |\Phi^+_{AB}\rangle$$

$$10 \rightarrow (\sigma^Y_A \otimes \mathbb{I}_B)|\Psi^+_{AB}\rangle = \frac{-i}{\sqrt{2}}\big( |1_A\rangle |0_B \rangle - |0_A\rangle |1_B \rangle = |\Phi^-_{AB}\rangle$$

$$11 \rightarrow (\sigma^Z_A \otimes \mathbb{I}_B)|\Psi^+_{AB}\rangle = \frac{1}{\sqrt{2}}\big( |0_A\rangle |0_B \rangle - |1_A\rangle |1_B \rangle = |\Psi^-_{AB}\rangle$$

These resultant shared states are orthogonal, and if Alice sends her state to Bob he can undertake an orthogonal measurement to determine which of the four operators Alice used, and hence determine what the original two bits of Alice's message are.