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diff --git a/source/know/concept/no-cloning-theorem/index.md b/source/know/concept/no-cloning-theorem/index.md new file mode 100644 index 0000000..e5e5be4 --- /dev/null +++ b/source/know/concept/no-cloning-theorem/index.md @@ -0,0 +1,70 @@ +--- +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 N. Brunner, + *Quantum information theory: lecture notes*, + 2019, unpublished.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 J.B. Brask, + *Quantum information: lecture notes*, + 2021, unpublished. |