1.2.5 No-cloning theorem

Course subject(s) Module 1: Working with Single Qubits

Before moving on to the next module, we should introduce a simple yet profound result known as the no-cloning theorem. A very common theme is science fiction is cloning, or producing an exact copy of something that already exists. As you will learn below, such a feat can never be possible.

What you have learned so far is enough to understand the proof of the “no-cloning” theorem, which goes as follows. We want to know if it is possible to devise some circuit U which can take an unknown state |ψ⟩ and copy it into another register. Therefore, we desire the following behavior from U

U(|ψ⟩|e⟩)=|ψ⟩|ψ⟩

which must hold for any state to-be-copied |ψ⟩ and |e⟩, which is some arbitrary state of the copy register before the copy occurs. It turns out that such a U is not possible. Imagine that we have cloned both |ψ⟩ and some other state |φ⟩, and then compute the following projection:

⟨ψ|φ⟩=⟨ψ|⟨e||φ|e⟩=⟨ψ|⟨e|U^†U|φ|e⟩=⟨ψ|⟨ψ||φ|φ⟩=(⟨ψ|φ⟩)²

The equation x=x² only holds if x= 0 or x=1, which implies that |ψ⟩ and |φ⟩ must either be collinear (so the same up to an overall phase) or orthogonal. In any other case, the proof fails, so we cannot copy any arbitrary input state. Interestingly, you could also run this proof in reverse, which uses the same logic to show that no-cloning implies no-deleting, which means we cannot in general devise U some that can always take |ψ⟩|ψ⟩→|ψ⟩|e⟩ for any |ψ⟩.

Copying information is something very normal and intuitive to us classical creatures. Every day, millions of people back up their phones and photos to cloud services. Unfortunately, such a world is not achievable with quantum hardware, which raises an important point. How can we protect our quantum information if we cannot copy it? The earliest classical error-correcting codes were based on the simple repetition of data; if we cannot clone, how can we ever hope to preserve our fragile qubits for long periods of time? Fortunately, the field of quantum error correction is not so bleak, and entanglement can be harnessed to protect quantum information without copying it. You will learn more about entanglement in the next module!

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