You can’t copy a qubit

The BB84 algorithm works because no eavesdropper can make a copy of a qubit’s state. Imagine that Eve intercepts one of Alice’s qubits, makes a measurement, and gets a value of 1. Eve has no way of knowing whether the qubit she measured was in the |1⟩ state, the state, the state, or some other exotic in-between state. So, Eve doesn’t know exactly what to forward to Bob.

But wait! Can we be sure that Eve has no way to make a copy of Alice’s qubit? Yes, we can. The No-Cloning theorem shows that assuming that qubits can be copied leads to a nasty contradiction.

Let’s start by agreeing on three properties of tensor products:

• For any three matrices, x, y, and z, a left distributive law holds – that is, .

You can write out a formal proof of this fact, but I always like to test with a simple example:

An example is never as good as proof, but an example helps us...