The No-Cloning Theorem in Quantum Information
During my undergraduate days, I was notorious for not attending lectures. So, when exams came, I had to inevitably seek “charity” for the lecture notes. A major portion of my bachelor’s degree is owed to many of my good friends who were kind enough to share their notes with me.
Now, I want you to imagine yourself as one of those friends (and if you are reading this, you most probably are!). You have attended every lecture and compiled all your lecture notes into a chef’s-kiss-worthy PDF file. And I have come to you, as usual, hoping you’d share that file with me. Being the good friend that you are, you decide to make a copy of the file and send it to me so that I can pass the exam.
BUT, what if I told you that in the process of making a copy, you were bound to irreversibly damage — partly or completely — your original set of notes? Would you still want to make a copy of your notes for me? That’s a peculiar thought, but it captures the essence of the topic that we are going to discuss in this post, the no-cloning theorem in quantum information.
In the classical world, you can make as many copies of your lecture notes and distribute it to all your shiftless friends (like me!), and end up not failing the exam yourself. But if the thing you wish to copy isn’t a PDF, but a quantum particle (a photon or an electron, for example) then I am afraid it wouldn’t matter how good friends we are. If somehow you manage to get hold of a photon in an unknown state — and again, being the nice friend that you are — you wish to share it with me while keeping the original, you wouldn’t be allowed by the rules of quantum mechanics to do that. Nature forbids that. She does not allow you to make a copy of your photon without necessarily disturbing its state. This is essentially the spirit of the no-cloning theorem!
Formally put, it states that you cannot make a perfect copy of an unknown quantum state.
A modern mathematical proof of the no-cloning theorem requires no more than five-six lines. It makes use of one of the fundamental postulates of quantum mechanics — the unitarity of quantum operations. I will describe that proof in the later half, but before that, let’s try to understand it by building some circuits.
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The following sections are inspired from the classic textbook, the certified bible of the field, “Quantum Computation and Quantum Information” by Michael Nielsen and Isaac Chuang (Mike and Ike!).
Can we build a qubit copying circuit?
Before moving on to the quantum case, let’s ask how does a classical copying circuit work? How can we build one? Well, it’s really simple, you just need to know how an XOR gate works. Figure 1 depicts the implementation of a classical copying circuit. On the left side of the circuit, you have an unknown bit x (0 or 1) that you wish to copy, and a ‘scratchpad’ bit (usually 0, its sole purpose is to store the copy of the first bit). The right side shows the output bits, which are both clearly in the same state x. That was not too hard, was it?
Now, let’s try to construct a circuit which can achieve this “copying” task for quantum bits (or qubits, in short). The gate depicting the entire process in Figure 1 is an example of a classical CNOT gate. Well, we also have a quantum version, called the quantum CNOT gate, arguably the most important two-qubit gate. Figure 2 depicts the operation of a quantum CNOT gate. The top qubit is sometimes also referred to as the control qubit as depending on its state, and bottom one — or the target — gets negated (negate here means converts 0 to 1, and vice versa). As can be seen from the truth table, if the control qubit is 0, the target qubit remains unchanged. However, when the control qubit is 1, the operation is to negate state of the target qubit.
Let’s examine if the quantum CNOT gate can help us make a copy of an unknown quantum state as the classical CNOT most comfortably did! We have the first qubit in an unknown quantum state |ψ⟩ = α|0⟩ + β|1⟩, and the second one is the ‘scratchpad’ qubit, initialised to state |0⟩. Figure 3 depicts the circuit for the procedure, and we can simply write the input state of the two qubits as:
The function of the CNOT gate is to negate the second qubit when the first qubit is 1, otherwise do nothing. Thus, the output is simply α|00⟩ + β|11⟩ as shown in Figure 3.
But, have we achieved what we came for, have we successfully copied the unknown state |ψ⟩? That is, have we created the state |ψ⟩|ψ⟩? I don’t think we have as for a general state |ψ⟩, we notice that,
Hence, unless αβ = 0, which is obviously not true in general, we note that the ‘copying circuit’ in Figure 3 does not clone the input quantum state. In fact, it is impossible to build such a circuit, all because of the no-cloning theorem.
The modern proof
The simplest way to prove the non-cloning theorem is through a proof by contradiction. We start by assuming that there exists a machine which can produce a copy of an unknown pure quantum state |ψ⟩. The machine has two slots labeled 𝐴 and 𝐵. Slot 𝐴 starts out in the state |ψ⟩, which is to be copied, while the second one, slot 𝐵, contains some standard pure state (can be the state |0⟩, for example). Thus, the initial state of the copying machine is
Some unitary evolution U now effects the copying procedure, ideally,
Since this procedure works for an arbitrary pure state |ψ⟩, it should also be able to clone an another unknown state |φ⟩. From this, we have
Taking the inner product of these two equations, and using the fact that
U is unitary (so it preserves inner products), we have
On the other hand, taking the inner product of the outputs gives
Since unitarity implies that the inner product must be the same before and after the transformation U, we obtain
But this equation which is clearly of the form x² = x can have only two solutions, either x = 0 or x = 1. Hence, either |ψ⟩ = |φ⟩ or |ψ⟩ and |φ⟩ are orthogonal. This implies that a cloning device can only clone states which are orthogonal to one another, and therefore a general quantum cloning device is impossible. This concludes the proof of the no-cloning theorem!
The no-cloning theorem is a surprising result, both for its content and even more so for its simplicity. One would think it would have been discovered in the late 1920s or the early 30s, right along side the preliminary developments in the new quantum theory. However, this now-so-obvious theorem was not discovered until 1970. Buried inside a paper published in 1970 lies the first mathematical proof of the no-cloning theorem, but that proof was unfortunately neglected and forgotten. Until 2018, every quantum information text book (including Mike and Ike’s) traced the (re)discovery of the theorem to 1982. But the story of the no-cloning theorem had actually begun way earlier. The events around its discovery and the rediscovery are equally fascinating as the theorem itself. I have also written about that rather unknown history, and if you wish to read about that, you can find it here.
That’s all for now!
Cheers, SA!
