The no-cloning theorem: why quantum information cannot be copied

It is impossible to copy an unknown quantum state. This ban is both the difficulty of quantum communication and the source of its security. See why an attempt to copy with a CNOT fails.

Classical information copies freely

Copying a file, forwarding an email or amplifying a signal are the most ordinary operations on classical information. The copy is identical to the original and leaves the original untouched.

Quantum information is different. In 1982 Wootters and Zurek (and, independently, Dieks) proved that a machine that copies an unknown quantum state cannot exist. This is called the no-cloning theorem.

An attempt to copy

At first sight the CNOT gate looks like a copying machine: if q0 is 0, q1 stays 0; if q0 is 1, q1 becomes 1 too. So it copies the states 0 and 1 perfectly.

But what happens if q0 is in superposition? With two independent copies of |+⟩, each of the four outcomes would appear 25% of the time. The CNOT produces a Bell state instead: only 00 and 11. We got entanglement instead of a copy. The reason is that quantum gates are linear; a gate that copies 0 and 1 cannot copy their superposition.

from qiskit import QuantumCircuit
from qiskit.quantum_info import Statevector

# A "copying machine" attempt: copy q0 onto q1 with a CNOT
for name, superposition in [('|0>', False), ('|+>', True)]:
    qc = QuantumCircuit(2)
    if superposition:
        qc.h(0)
    qc.cx(0, 1)
    probs = Statevector(qc).probabilities_dict(decimals=3)
    print(name, {str(k): float(p) for k, p in probs.items()})
# |0> {'00': 1.0}
# |+> {'00': 0.5, '11': 0.5}   two independent copies would give all four outcomes 25%

Why does it matter?

No-cloning makes it hard to carry quantum information over long distances: you cannot amplify a photon in an optical fibre the way you amplify a classical signal. That is why quantum networks need new methods such as teleportation and “quantum repeaters”.

The same ban is a security guarantee: an eavesdropper cannot take a copy of a quantum state and send the original on undisturbed. The BB84 protocol you will see in this course rests on exactly this idea.

Try it yourself

This circuit is in fact a copying attempt: H makes q0 |+⟩ and the CNOT tries to “copy” it onto q1. The result is not two independent coin flips but only 00 and 11. Now remove the H: 0 is copied perfectly and the result is 00.

Open the circuit in your browser

The circuit playground is in Turkish. Button guide: “1000 kez ölç” = measure 1000 times, “Durum vektörünü göster” = show the state vector, “Bu devrenin Qiskit kodu” = this circuit’s Qiskit code, “Geri al” = undo, “Temizle” = clear. Click a gate in the circuit to remove it.

Key points

The no-cloning theorem: an unknown quantum state cannot be copied.
A CNOT copies 0 and 1 but not a superposition; it produces entanglement instead of a copy.
That is why quantum information cannot be amplified like classical signals.
The same ban is the basis of the security of quantum cryptography.

Test yourself

What do you get when you apply a CNOT to q0 in the state |+⟩ to “copy” it onto q1?

Two independent copies of |+⟩
|+⟩|0⟩ with q0 left untouched
An entangled Bell state that only gives 00 and 11

Show the answer

An entangled Bell state that only gives 00 and 11. Quantum gates are linear: the CNOT produces the superposition of the copies of |0⟩ and |1⟩, that is a Bell state.

Quantum teleportation: moving a state with entanglement →