No-Cloning and QEC
The no-cloning theorem forbids copying an unknown quantum state, which rules out classical triple-modular redundancy but not error correction itself.
The no-cloning theorem
There is no unitary U and fixed blank state |b> such that U(|psi>|b>) = |psi>|psi> for every unknown |psi>. The proof is a two-line consequence of linearity. Suppose cloning worked for |0> and |1>. Then by linearity it acts on (|0>+|1>)/sqrt(2) to give (|00>+|11>)/sqrt(2), but genuine cloning would demand (|0>+|1>)(|0>+|1>)/2. These differ, so no such U exists.
This immediately blocks the classical error-correction strategy of storing three identical copies and majority voting. You cannot produce |psi>|psi>|psi> from a single unknown |psi>.
Encoding is not cloning
QEC sidesteps the theorem by encoding rather than copying. The three-qubit repetition code maps a|0>+b|1> to a|000>+b|111>, which is an entangled state, not three copies a|0>+b|1> each. No information about a and b is duplicated into separable copies; instead it is delocalized across correlations. The map is a valid unitary because it acts on the specific input |psi>|0>|0>, not on an arbitrary joint state.
- Cloning: produce independent identical copies of an unknown state (impossible).
- Encoding: spread one logical state across many physical qubits as an entangled codeword (allowed).
- Recovery: measure joint parities to detect errors without learning the encoded amplitudes.
A related result, the no-deleting theorem, forbids the reverse operation. Together these theorems shape the design of every quantum code: information must be protected by nonlocal encoding and syndrome measurement rather than by redundancy of readable copies.
The deep reason QEC still works is that errors, though continuous, act locally and can be diagnosed by measurements that commute with the encoded information. The stabilizer formalism makes this precise.