Notes - Quantum Information HT24, Dense coding protocol
Flashcards
Can you briefly state what the dense coding protocol allows you to achieve?
When both Alice and Bob have access to an entangled state, Alice can communicate two bits to Bob, while only sending him one qubit.
Given the following definitions,
\[U_{00} = I, \quad U_{01} = X, \quad U_{10} = Z, \quad U_{11} = ZX\]
and
\[|\Psi_{00}\rangle = |\Phi^+\rangle, \quad |\Psi_{01}\rangle = |\Psi^+\rangle, \quad |\Psi_{10}\rangle = |\Phi^-\rangle, \quad |\Psi_{11}\rangle = |\Psi^-\rangle\]
where $ \vert \Phi^+\rangle, \vert \Psi^+\rangle, \vert \Phi^-\rangle, \vert \Phi^+\rangle$ are the Bell basis and the relations
\[\begin{align*}
(U_{00} \otimes I_B) |\Phi^+\rangle &= (I_A \otimes I_B) |\Phi^+\rangle &= |\Phi^+\rangle &= |\Psi_{00}\rangle \\\\
(U_{01} \otimes I_B) |\Phi^+\rangle &= (X_A \otimes I_B) |\Phi^+\rangle &= |\Psi^+\rangle &= |\Psi_{01}\rangle \\\\
(U_{10} \otimes I_B) |\Phi^+\rangle &= (Z_A \otimes I_B) |\Phi^+\rangle &= |\Phi^-\rangle &= |\Psi_{10}\rangle \\\\
(U_{11} \otimes I_B) |\Phi^+\rangle &= (Z_A X_A \otimes I_B) |\Phi^+\rangle &= |\Psi^-\rangle &= |\Psi_{11}\rangle.
\end{align*}\]
and further suppose:
- Alice has access to qubit $A$
- Bob has access to qubit $B$
- The composite system is initially in the Bell state $ \vert \Phi^+\rangle$.
Explain and quickly validate the idea behind dense coding, which lets Alice communicate two bits to Bob using only one qubit.
- Alice encodes the value of two bits $\{00, 01, 10, 11\}$ using the corresponding unitary gate
- Alice sends qubit $A$ to Bob
- Bob measures $AB$ in the ONB $\{ \vert \Psi _ {00}\rangle, \vert \Psi _ {01}\rangle, \vert \Psi _ {10}\rangle, \vert \Psi _ {11}\rangle\}$
- Then, by the relations above, Bob will see $\Psi _ {ij}$ corresponding to the unitary matrix Alice used