The No-Cloning Theorem

No quantum operation can make an identical copy of an arbitrary unknown quantum state, a direct consequence of linearity.

Statement

The no-cloning theorem says there is no unitary operation U that maps |psi>|blank> to |psi>|psi> for every unknown state |psi>. You cannot build a universal copier for quantum information.

The one-line proof

Suppose a cloner existed: U|psi>|b> = |psi>|psi> and U|phi>|b> = |phi>|phi> for two states. Take the inner product of the two equations. The left side gives (times the blank overlap, which is 1). The right side gives ^2. So = ^2, which forces the overlap to be 0 or 1. Cloning therefore works only for states that are identical or orthogonal — not for arbitrary unknown states. Linearity of quantum mechanics forbids the general case.

What is and is not allowed

• Allowed: copying known basis states, e.g. CNOT copies |0> and |1> in the computational basis
• Allowed: moving a state with a swap or teleportation (the original is destroyed)
• Forbidden: duplicating an arbitrary superposition into two independent copies

CNOT applied to |x>|0> gives |x>|x> only for x in {0,1}. Applied to a superposition it produces an entangled state, not two copies — a common point of confusion.

Consequences

No-cloning is the reason a quantum state cannot be backed up or amplified the way a classical signal can. It is the security foundation of quantum key distribution: an eavesdropper cannot copy qubits in transit without disturbing them. It also shapes quantum error correction, which must protect information without ever copying the unknown logical state, forcing the use of redundant entangled encodings rather than simple duplication.

It is closely related to the no-deleting and no-broadcasting theorems, which together fence off the classical intuition that information can be freely copied and erased.