---
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    2019, unpublished.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    2021, unpublished.