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+---
+title: "Superdense coding"
+date: 2021-03-07
+categories:
+- Quantum information
+layout: "concept"
+---
+
+In quantum information, **(super)dense coding**
+is a protocol to enhance classical communication.
+It uses a quantum communication channel and
+[entanglement](/know/concept/quantum-entanglement/)
+to send two bits of classical data with just one qubit.
+It is conceptually similar to [quantum teleportation](/know/concept/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](/know/concept/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:
+
+<table style="width:70%;margin:auto;text-align:center;">
+    <tr>
+        <th>$(a_1, a_2)$</th>
+        <th>Operator</th>
+        <th>Result</th>
+    </tr>
+    <tr>
+        <td>$00$</td>
+        <td>$\hat{I}$</td>
+        <td>$\ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B + \Ket{1}_A \Ket{1}_B \Big)$</td>
+    </tr>
+    <tr>
+        <td>$01$</td>
+        <td>$\hat{\sigma}_z$</td>
+        <td>$\ket{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B - \Ket{1}_A \Ket{1}_B \Big)$</td>
+    </tr>
+    <tr>
+        <td>$10$</td>
+        <td>$\hat{\sigma}_x$</td>
+        <td>$\ket{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B + \Ket{1}_A \Ket{0}_B \Big)$</td>
+    </tr>
+    <tr>
+        <td>$11$</td>
+        <td>$\hat{\sigma}_x \hat{\sigma}_z$</td>
+        <td>$\ket{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B - \Ket{1}_A \Ket{0}_B \Big)$</td>
+    </tr>
+</table>
+
+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.
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