---
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:

<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.