---
title: "Superdense coding"
sort_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 markdown="1">
  $$(a_1, a_2)$$
 </th>
 <th>
  Operator
 </th>
 <th>
  Result
 </th>
</tr>
<tr>
 <td markdown="1">
  $$00$$
 </td>
 <td markdown="1">
  $$\hat{I}$$
 </td>
 <td markdown="1">
  $$\displaystyle \ket{\Phi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B + \Ket{1}_A \Ket{1}_B \Big)$$
 </td>
</tr>
<tr>
 <td markdown="1">
  $$01$$
 </td>
 <td markdown="1">
  $$\hat{\sigma}_z$$
 </td>
 <td markdown="1">
  $$\displaystyle \ket{\Phi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{0}_B - \Ket{1}_A \Ket{1}_B \Big)$$
 </td>
</tr>
<tr>
 <td markdown="1">
  $$10$$
 </td>
 <td markdown="1">
  $$\hat{\sigma}_x$$
 </td>
 <td markdown="1">
  $$\displaystyle \ket{\Psi^{+}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B + \Ket{1}_A \Ket{0}_B \Big)$$
 </td>
</tr>
<tr>
 <td markdown="1">
  $$11$$
 </td>
 <td markdown="1">
  $$\hat{\sigma}_x \hat{\sigma}_z$$
 </td>
 <td markdown="1">
  $$\displaystyle \ket{\Psi^{-}} = \frac{1}{\sqrt{2}} \Big(\Ket{0}_A \Ket{1}_B - \Ket{1}_A \Ket{0}_B \Big)$$
 </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.