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+---
+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+    2019, unpublished.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+    2021, unpublished.