The No-Cloning Theorem

On any classical computer, copying is free and unremarkable: select, Ctrl-C, Ctrl-V, and you have a perfect duplicate of a file, a bit, a whole disk. It is so ordinary you never think of it as an operation at all. Quantum information is different in a way that sounds almost unbelievable the first time you meet it: there is no Ctrl-C for an unknown qubit. No circuit, no machine, no clever sequence of gates can take an arbitrary unknown state and produce two copies of it. This is the no-cloning theorem, and it is not an engineering shortfall — it is forbidden by the linearity of quantum mechanics itself.

A cloning machine would be a single fixed U — one device that works on every input — taking the qubit you want to copy together with a blank scratch qubit |0\rangle, and returning two copies:

U\big(|\psi\rangle \otimes |0\rangle\big) = |\psi\rangle \otimes |\psi\rangle \quad \text{for every } |\psi\rangle.

The claim of the theorem is stark: no such U exists. The machine you are being asked to build cannot be built.

The machine we are told to build

Picture the device as a box on two wires: the unknown |\psi\rangle on the top wire, a blank |0\rangle on the bottom, and out the far side two wires each carrying |\psi\rangle. It looks perfectly reasonable — until you try to make one gate do it for all inputs at once.

Worked example: linearity kills the cloner

Suppose, for contradiction, that a cloner U exists. Then in particular it must copy the two basis states — that is the least we can ask of it:

Now feed it the balanced superposition |{+}\rangle = \tfrac{1}{\sqrt2}\big(|0\rangle + |1\rangle\big). A gate is a linear map, so it must act on the sum term by term — it cannot peek at the whole state and decide what to do:

U|{+}\rangle|0\rangle = \tfrac{1}{\sqrt2}\Big(U|0\rangle|0\rangle + U|1\rangle|0\rangle\Big) = \tfrac{1}{\sqrt2}\big(|00\rangle + |11\rangle\big).

That output is a Bell state — the two qubits are entangled, not two independent copies. But genuine cloning demanded |{+}\rangle|{+}\rangle, which multiplies out to

|{+}\rangle|{+}\rangle = \tfrac12\big(|00\rangle + |01\rangle + |10\rangle + |11\rangle\big).

Compare the two results: \tfrac{1}{\sqrt2}(|00\rangle + |11\rangle) has no |01\rangle or |10\rangle terms, while the true copy is drenched in them. They are simply different states — so a machine that copies |0\rangle and |1\rangle cannot also copy |{+}\rangle. The contradiction is complete: no universal cloner exists.

Worked example: the inner-product proof

Here is a second, slicker argument that says the same thing. Suppose one gate U cloned two arbitrary states |\psi\rangle and |\varphi\rangle:

Gates are unitary, and unitaries preserve inner products. Taking the inner product of the two inputs, and separately of the two outputs, must give the same number. The inputs give \langle\psi|\varphi\rangle\langle 0|0\rangle = \langle\psi|\varphi\rangle; the outputs give \langle\psi|\varphi\rangle^2. Writing x = \langle\psi|\varphi\rangle, both must be equal:

x = x^2 \;\Longrightarrow\; x = 0 \ \text{ or } \ x = 1.

So cloning can only work when the two states are orthogonal (x = 0) or identical (x = 1). For any pair of distinct, non-orthogonal states — such as |0\rangle and |{+}\rangle, where x = \tfrac{1}{\sqrt2} — no cloner can exist.

there is no unitary U with U(|\psi\rangle \otimes |0\rangle) = |\psi\rangle \otimes |\psi\rangle for every |\psi\rangle;
the obstruction is linearity/unitarity, not any technological limit;
cloning is only possible for orthogonal states (e.g. the basis |0\rangle, |1\rangle), never for arbitrary superpositions;
a device that copies |0\rangle and |1\rangle entangles |{+}\rangle instead of copying it.

What you can do: copy known states, and move states

No-cloning forbids copying an unknown state — it does not outlaw copying altogether. Two things are perfectly legal.

First, you can copy classical (orthogonal) information. The CNOT gate, applied to a basis state and a blank, duplicates the bit:

\text{CNOT}\,|0\rangle|0\rangle = |0\rangle|0\rangle, \qquad \text{CNOT}\,|1\rangle|0\rangle = |1\rangle|1\rangle.

This is exactly the everyday copying of a classical bit — allowed because |0\rangle and |1\rangle are orthogonal. But watch what the very same gate does to a superposition control |{+}\rangle:

\text{CNOT}\,|{+}\rangle|0\rangle = \text{CNOT}\,\tfrac{1}{\sqrt2}\big(|00\rangle + |10\rangle\big) = \tfrac{1}{\sqrt2}\big(|00\rangle + |11\rangle\big) \neq |{+}\rangle|{+}\rangle.

Instead of two copies of |{+}\rangle you get an entangled Bell state — the very same contradiction as before. CNOT copies basis states but is not a cloner.

Second, you can move a state. The SWAP gate exchanges the contents of two qubits, and quantum teleportation transfers a state across a distance. Both are allowed precisely because they relocate the state and leave nothing behind: the original is destroyed. Cloning is not the same as moving — copying makes a second original while keeping the first; that is what is impossible.

Why it matters

No-cloning quietly shapes the whole field. You cannot back up a qubit to measure it repeatedly: since measurement collapses the state and destroys the superposition, the obvious fix — clone it a million times first, then measure the copies to recover the amplitudes — is flatly impossible. There is no fan-out of an unknown state to feed many parts of a circuit, so quantum algorithms must be designed around a single travelling copy. And, most famously, no-cloning is the bedrock of quantum cryptography: an eavesdropper cannot silently copy qubits in transit to read them at leisure, which is exactly what makes quantum key distribution provably secure. The impossibility of copying is not just a curiosity — it is a resource.

Every secure-messaging scheme lives in dread of a silent eavesdropper who copies the traffic and cracks it later at leisure. Classical bits offer no defence: light can be split, wires can be tapped, and a copy leaves the original untouched. Quantum key distribution turns no-cloning into a wall. If two parties send secret keys encoded in unknown qubit states, an eavesdropper who wants a copy simply cannot make one — the best she can do is measure, which collapses the state and disturbs it in a way the legitimate parties can detect by comparing notes. Copying without detection is not hard; it is forbidden by physics. So the same law that stops you backing up a qubit is the one that lets two strangers share a key no computer, however powerful, can steal in transit. The absence of a copy button is, remarkably, a security feature.

Two easy over-corrections. First, no-cloning forbids copying an unknown state — it does not say "you can never copy a qubit". You absolutely can copy a known or orthogonal basis state: CNOT duplicates |0\rangle and |1\rangle all day long, because those are just classical bits. What is impossible is one fixed machine that copies every state, including superpositions it was never told about. Second, cloning is not the same as moving: you are free to SWAP a state to another qubit or teleport it across the lab — those relocate the state and destroy the original. The forbidden act is making a second original while the first survives. Copy vs move, unknown vs known: keep those two distinctions straight and the theorem stops sounding paradoxical.