1  ·  What “cloning” would actually mean

A cloning machine would be a single fixed unitary U together with a standard blank register |e⟩, such that for every possible input state |ψ⟩ it produces:

Three pieces of that statement deserve unpacking, because the whole theorem turns on them.

What |e⟩ is, and what “blank” means. The register |e⟩ is simply the second slot, the one that will end up holding the copy. We call it “blank” by analogy with an empty sheet of paper or an uninitialised memory cell: before the operation it carries no information whatsoever about the input |ψ⟩. The conventional choice is |0⟩, or |0…0⟩ when the copy needs several qubits, but |e⟩ does not have to be the zero vector specifically. Any fixed pure state works equally well for the theorem. The property that actually matters is not its literal value but that it is independent of the unknown input. A register that already secretly equalled |ψ⟩ would not be blank, and you would not have copied anything, you would have been handed the answer.

What “fixed” means, for both U and |e⟩. Fixed means chosen in advance and identical on every run, never allowed to depend on which |ψ⟩ happens to come in. This is the condition that keeps the problem honest. If you were permitted to pick |e⟩ or U based on the input, you would first have to know what |ψ⟩ is, and at that point there is nothing left to clone: you could just prepare a second copy yourself. So the cloner gets one unitary and one starting register, and they must work for an arbitrary, unknown input. The claim of the theorem is that no such pair (U, |e⟩) exists.

Why cloning a known state is allowed, and how you do it. The theorem forbids copying an unknown arbitrary state. It says nothing about states you already understand, and there are two everyday ways such a state can be “known”:

As a concrete example of the first route, if you are told the state is |ψ⟩ = R_y(θ)|0⟩ for a known angle θ, you simply apply R_y(θ) to two separate qubits. No universal device is involved, and no rule is broken. The contradiction in the next two sections appears only when no such side information exists, that is, when the same machine must handle a continuum of unknown inputs at once.

2  ·  Proof I: linearity forbids it

Assume, for contradiction, that a universal cloner U exists. Pick two arbitrary states |ψ⟩ and |φ⟩. By assumption U clones both of them:

Now feed the machine a superposition of the two inputs, |χ⟩ = (1/√2)(|ψ⟩ + |φ⟩). Quantum evolution is linear, so U acts term by term on the input:

But if U is genuinely a cloner, then by definition it must also clone |χ⟩ itself, producing |χ⟩|χ⟩. Expanding that out:

The two results disagree. Linearity gave us only the “diagonal” terms |ψ⟩|ψ⟩ and |φ⟩|φ⟩, whereas true cloning of |χ⟩ demands the cross terms |ψ⟩|φ⟩ and |φ⟩|ψ⟩ as well. They can only match if the cross terms vanish, which happens precisely when |ψ⟩ and |φ⟩ are identical or orthogonal. For general, non-orthogonal states the equality is impossible. The cloner cannot exist.

3  ·  Proof II: unitarity forbids it

The linearity argument shows cloning fails for superpositions. A second proof, often considered the cleaner one, uses the fact that unitary operations preserve inner products. Suppose again that U clones two states:

The inner product is taken over the full two-register states that U acts on, not over |ψ⟩ and |φ⟩ on their own. Label the two inputs and their images under U:

That identity, ⟨A|B⟩ = ⟨UA|UB⟩, is the definition of a unitary: it preserves inner products. U never acts on the inner product itself, which is only a number; it acts on the states, and unitarity says doing so cannot change their overlap. Now evaluate both sides, using the tensor-product factorisation ⟨a|⟨b| · |c⟩|d⟩ = ⟨a|c⟩ ⟨b|d⟩ and ⟨e|e⟩ = 1:

Write x = ⟨ψ|φ⟩. The condition x = x² rearranges to x(x − 1) = 0, whose only solutions are x = 0 or x = 1. In other words, cloning is consistent only when the two states are orthogonal (x = 0) or identical (x = 1). Any pair of distinct, non-orthogonal states, the generic case, breaks the equality. No universal cloner can exist.

Both proofs converge on the same boundary. Orthogonal states carry distinguishable classical labels and can be copied; overlapping quantum states cannot.

4  ·  “But doesn’t CNOT copy a qubit?”

This is the single most common point of confusion, and it is worth resolving carefully. A CNOT with the input qubit as control and a fresh |0⟩ as target looks exactly like a copy operation. On computational basis states it behaves like one:

So far it really is copying: the second qubit ends up matching the first. The trouble appears the moment the control is a superposition. Take |ψ⟩ = a|0⟩ + b|1⟩ as control and |0⟩ as target:

Compare that with what a true clone would have produced, namely |ψ⟩|ψ⟩:

These are different states. The CNOT output a|00⟩ + b|11⟩ is an entangled pair, not two independent copies. If you discard the original and look at the “copy” on its own, you do not find |ψ⟩; you find a mixed state that has lost the phase relationship between a and b entirely. CNOT copies the value in the computational basis but destroys the superposition structure that made the state quantum in the first place. That is exactly the gap between copying classical information, which is always allowed, and cloning a quantum state, which is not.

5  ·  Why this single theorem matters so much

No-cloning is not a footnote. Three of the most important results in quantum information rest directly on it.

Quantum key distribution. In BB84, Alice sends qubits encoded in randomly chosen bases. An eavesdropper, Eve, would love to intercept each qubit, keep a copy, and forward the original untouched so she can measure her copies later once the bases are announced. No-cloning makes that impossible. Eve cannot duplicate an unknown qubit, so she is forced to measure in transit, and measuring in the wrong basis disturbs the state and injects a detectable error rate. The security of the protocol is the no-cloning theorem wearing a different hat.

No faster-than-light signalling. Entanglement correlates distant measurement outcomes, which has always tempted people into superluminal-communication schemes. Many of those schemes secretly assume Bob can clone his half of an entangled pair, make many copies, and read off the statistics to learn which basis Alice measured in. No-cloning closes that loophole and is one of the reasons quantum mechanics and relativity coexist without paradox.

Quantum error correction. Classical error correction leans on the simplest trick imaginable: copy the bit several times and take a majority vote. That is illegal in the quantum setting, because you cannot copy an unknown logical state. Quantum codes had to be built on a different principle, spreading logical information across many physical qubits using entanglement and measuring error syndromes without ever reading or duplicating the encoded state. The whole architecture of fault tolerance is shaped by the fact that the obvious classical fix is off the table.

6  ·  Watching a clone fail in Qiskit

The abstract proof is convincing, but it is satisfying to watch the failure happen numerically. We attempt the naive CNOT “clone” of the state |+⟩ = H|0⟩, then measure how badly it falls short by two yardsticks: fidelity against the intended |+⟩|+⟩, and the purity of the copy qubit on its own.

Running it gives:

Every number tells the same story. The output is a maximally entangled Bell state, not the product |+⟩|+⟩ we were hoping for, and the fidelity to the intended clone is only 0.5. Trace out the original and the supposed “copy” is the maximally mixed state I/2, with purity 0.5 rather than 1. The copy retains no memory of the |+⟩ superposition at all; it is as good as a fair coin. The CNOT did precisely what the theorem says it must: it produced correlation, not duplication.

Malcolm Low is an Associate Professor at the Singapore Institute of Technology, writing on quantum computing, programming, and applied computing from Singapore.

Website: malcolmlow.com  ·  Singapore