Categories: Physics, Quantum information, Quantum mechanics.
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.