No-Cloning Theorem

Statement

There is no universal physical operation that takes an arbitrary unknown pure state and produces two exact copies while using the same blank state for every input. More generally, one exact deterministic device can clone a pure-state family only when each pair is orthogonal or represents the same ray.

In channel notation, no completely positive trace-preserving map can satisfy

\mathcal C \left( |\psi\rangle\langle\psi| \otimes |0\rangle\langle0| \right) = |\psi\rangle\langle\psi| \otimes |\psi\rangle\langle\psi|

for every pure state $|\psi\rangle$ of a nontrivial quantum system. The canonical inner-product proof, channel dilation, allowed relaxations, and no-signaling consequences are developed in No-Cloning and No-Signaling.

Assumptions

The promised input family contains distinct nonorthogonal states.
The cloning process is exact and deterministic.
The same physical operation and the same blank system are used for all inputs.
The process obeys ordinary linear quantum evolution, either unitary evolution on a larger closed system or a quantum channel.
The claim concerns copying the unknown state itself, not learning a classical label for a known orthogonal preparation.

Quantum-Mechanical Meaning

No-cloning is a direct consequence of the linear structure of state space. Orthogonal alternatives can be copied because they can be measured without ambiguity and then re-prepared. Nonorthogonal alternatives cannot be perfectly distinguished, and the canonical operational proof shows that exact copying would change their overlap.

This is why an unknown qubit cannot be backed up like a classical bit string. It is also why eavesdropping in quantum communication cannot be treated as harmless copying: any exact record of an unknown nonorthogonal signal would violate the theorem.

No-cloning is only one ingredient in Quantum Key Distribution. A QKD proof must additionally quantify Eve’s partial information, authenticate the classical transcript, account for finite statistics and implementation behavior, and extract a composably secret key.

The theorem does not forbid Quantum Teleportation. Teleportation transfers an unknown state using shared entanglement, a Bell-basis measurement, classical communication, and a correction operation; the original input is consumed by the measurement. Nor does no-cloning forbid entanglement distribution, quantum error correction, or approximate cloning machines with fidelity below one.

Allowed and Forbidden Cases

Allowed:

copying known classical data stored in orthogonal states;
copying a known quantum state by preparing another system in the same state;
copying all states in a fixed orthonormal basis;
approximate cloning with less-than-perfect fidelity;
probabilistic exact cloning for linearly independent sets when failure outcomes are allowed.

Forbidden:

exact deterministic cloning of every pure state in a Hilbert space;
exact deterministic cloning of an arbitrary set containing nonorthogonal states;
treating teleportation as a second copy of the original state;
using measurement to identify an unknown nonorthogonal state and then reprepare it exactly.

Canonical Links

Common Mistakes

Saying quantum information cannot be copied at all. Orthogonal states and known states can be copied.
Forgetting the word “unknown.” If the state is known, one may prepare another system in the same state.
Treating approximate cloning as a counterexample to the theorem.
Treating teleportation as cloning.
Proving only the unitary case and then assuming without comment that open-system processes evade the conclusion. They do not.

Quick Check

Why can the states $\lvert0\rangle$ and $\lvert1\rangle$ be copied, while $\lvert0\rangle$ and $(\lvert0\rangle+\lvert1\rangle)/\sqrt2$ cannot both be copied by the same exact cloner?

Solution

The first pair is orthogonal, so one device may copy its classical basis label. The second pair is distinct and has nonzero overlap $1/\sqrt2$ , so it violates the pairwise criterion. Therefore one exact deterministic device cannot clone both states. See the inner-product proof for the derivation.

References

W. K. Wootters and W. H. Zurek, “A single quantum cannot be cloned,” Nature 299, 802–803 (1982), doi:10.1038/299802a0.
D. Dieks, “Communication by EPR devices,” Physics Letters A 92, 271–272 (1982), doi:10.1016/0375-9601(82)90084-6.
V. Bužek and M. Hillery, “Quantum copying: Beyond the no-cloning theorem,” Physical Review A 54, 1844–1852 (1996), doi:10.1103/PhysRevA.54.1844.
M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, Cambridge University Press, 2010.
J. Watrous, The Theory of Quantum Information, Cambridge University Press, 2018.