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---
title: "No-cloning theorem"
sort_title: "No-cloning theorem"
date: 2021-03-06
categories:
- Physics
- Quantum mechanics
- Quantum information
layout: "concept"
---
In quantum mechanics, the **no-cloning theorem** states
there is no general way to make copies of an arbitrary quantum state $\ket{\psi}$.
This has profound implications for quantum information.
To prove this theorem, let us pretend that a machine exists
that can do just that: copy arbitrary quantum states.
Given an input $\ket{\psi}$ and a blank $\ket{?}$,
this machines turns $\ket{?}$ into $\ket{\psi}$:
$$\begin{aligned}
\ket{\psi} \ket{?}
\:\:\longrightarrow\:\:
\ket{\psi} \ket{\psi}
\end{aligned}$$
We can use this device to make copies of the basis vectors $\ket{0}$ and $\ket{1}$:
$$\begin{aligned}
\ket{0} \ket{?}
\:\:\longrightarrow\:\:
\ket{0} \ket{0}
\qquad \quad
\ket{1} \ket{?}
\:\:\longrightarrow\:\:
\ket{1} \ket{1}
\end{aligned}$$
If we feed this machine a superposition $\ket{\psi} = \alpha \ket{0} + \beta \ket{1}$,
we *want* the following behaviour:
$$\begin{aligned}
\Big( \alpha \ket{0} + \beta \ket{1} \Big) \ket{?}
\:\:\longrightarrow\:\:
&\Big( \alpha \ket{0} + \beta \ket{1} \Big) \Big( \alpha \ket{0} + \beta \ket{1} \Big)
\\
&= \Big( \alpha^2 \ket{0} \ket{0} + \alpha \beta \ket{0} \ket{1} + \alpha \beta \ket{1} \ket{0} + \beta^2 \ket{1} \ket{1} \Big)
\end{aligned}$$
Note the appearance of the cross terms with a factor of $\alpha \beta$.
The problem is that the fundamental linearity of quantum mechanics
dictates different behaviour:
$$\begin{aligned}
\Big( \alpha \ket{0} + \beta \ket{1} \Big) \ket{?}
= \alpha \ket{0} \ket{?} + \beta \ket{1} \ket{?}
\:\:\longrightarrow\:\:
\alpha \ket{0} \ket{0} + \beta \ket{1} \ket{1}
\end{aligned}$$
This is clearly not the same as before: we have a contradiction,
which implies that such a general cloning machine cannot ever exist.
## References
1. N. Brunner,
*Quantum information theory: lecture notes*,
2019, unpublished.
2. J.B. Brask,
*Quantum information: lecture notes*,
2021, unpublished.
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