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 ∣ψ⟩\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 ∣ψ⟩\ket{\psi} and a blank ∣?⟩\ket{?}, this machines turns ∣?⟩\ket{?} into ∣ψ⟩\ket{\psi}:

∣ψ⟩∣?⟩  ⟶  ∣ψ⟩∣ψ⟩\begin{aligned} \ket{\psi} \ket{?} \:\:\longrightarrow\:\: \ket{\psi} \ket{\psi} \end{aligned}

We can use this device to make copies of the basis vectors ∣0⟩\ket{0} and ∣1⟩\ket{1}:

∣0⟩∣?⟩  ⟶  ∣0⟩∣0⟩∣1⟩∣?⟩  ⟶  ∣1⟩∣1⟩\begin{aligned} \ket{0} \ket{?} \:\:\longrightarrow\:\: \ket{0} \ket{0} \qquad \qquad \ket{1} \ket{?} \:\:\longrightarrow\:\: \ket{1} \ket{1} \end{aligned}

If we feed this machine a superposition ∣ψ⟩=α∣0⟩+β∣1⟩\ket{\psi} = \alpha \ket{0} + \beta \ket{1}, we want the following behavior:

(α∣0⟩+β∣1⟩)∣?⟩  ⟶  (α∣0⟩+β∣1⟩)(α∣0⟩+β∣1⟩)=(α2∣0⟩∣0⟩+αβ∣0⟩∣1⟩+αβ∣1⟩∣0⟩+β2∣1⟩∣1⟩)\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 αβ\alpha \beta. The problem is that the fundamental linearity of quantum mechanics dictates different behaviour:

(α∣0⟩+β∣1⟩)∣?⟩=α∣0⟩∣?⟩+β∣1⟩∣?⟩  ⟶  α∣0⟩∣0⟩+β∣1⟩∣1⟩\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 exist.

References

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 N. Brunner, Quantum information theory: lecture notes, 2019, unpublished.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 J.B. Brask, Quantum information: lecture notes, 2021, unpublished.