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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2021, unpublished.
|
