Categories: Physics, Quantum information, Quantum mechanics.

# No-cloning theorem

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 N. Brunner, Quantum information theory: lecture notes, 2019, unpublished.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 J.B. Brask, Quantum information: lecture notes, 2021, unpublished.