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+---
+title: "Superdense coding"
+firstLetter: "S"
+publishDate: 2021-03-07
+categories:
+- Quantum information
+
+date: 2021-03-07T20:30:41+01:00
+draft: false
+markup: pandoc
+---
+
+# Superdense coding
+
+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:
+
+| $(a_1, a_2)$ | 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+    2021, unpublished.