No-Cloning Theorem

A fundamental theorem proving it’s impossible to create an exact copy of an arbitrary unknown quantum state.

The no-cloning theorem states that there is no quantum operation that can copy an arbitrary unknown quantum state. This is one of the most important results in quantum information theory.

The Theorem

Statement: There is no unitary operation $U$ such that: $$U|\psi ⟩|0⟩=|\psi ⟩|\psi ⟩$$ for all quantum states $|\psi ⟩$.

The Proof (Simplified)

Suppose such a cloning machine existed. Consider two states $|\psi ⟩$ and $|\phi ⟩$:

$$U|\psi ⟩|0⟩=|\psi ⟩|\psi ⟩$$ $$U|\phi ⟩|0⟩=|\phi ⟩|\phi ⟩$$

Taking the inner product: $$⟨\psi |\phi ⟩⟨0|0⟩=⟨\psi |\phi {⟩}^2$$

So $⟨\psi |\phi ⟩=⟨\psi |\phi {⟩}^2$, which means $⟨\psi |\phi ⟩$ equals 0 or 1.

This means you can only clone states that are either identical or orthogonal, not arbitrary states. Contradiction.

What You CAN Do

Clone Known States

If you know what state you have, you can prepare copies: $$|0⟩\to |0⟩|0⟩✓$$

Clone Classical Information

Measure first (destroying quantum information), then copy: $$|\psi ⟩\stackrel{\text{measure}}{\to }\text{classical bit}\stackrel{\text{copy}}{\to }\text{many classical bits}$$

Approximate Cloning

Create imperfect copies with bounded fidelity. The best universal cloner achieves fidelity 5/6 for qubits.

Clone Orthogonal States

A set of mutually orthogonal states can be cloned (they’re distinguishable, hence effectively classical).

Implications

Quantum Cryptography Security

QKD security relies on no-cloning. An eavesdropper can’t copy quantum states to analyze later without disturbing them.

Quantum Information is Fragile

You can’t backup quantum states. Loss or error destroys information permanently (without error correction).

Teleportation Doesn’t Clone

Quantum teleportation destroys the original when creating the copy, satisfying no-cloning.

Distinguishing Quantum from Classical

Classical information (bits) can be freely copied. Quantum information (qubits) cannot. This is a fundamental difference.

Related Theorems

Historical Note

Proved independently by Wootters-Zurek and Dieks in 1982. It was initially seen as a limitation, but later recognized as the foundation of quantum cryptography’s security.

See also: Quantum Key Distribution, Quantum Teleportation, Quantum State, Quantum Money