No-cloning and no-broadcasting theorems

Suppose a fixed device receives one unknown pure input and an input-independent blank and environment. Any deterministic quantum channel has an isometric description when its discarded environment is retained. If the two desired output copies are exactly pure, the final global state factors into those copies and a possibly input-dependent environment state. Preservation of the inner product between two alternative inputs then relates their initial overlap to its square times the final environment overlap.

The absolute environment overlap cannot exceed one. For distinct nonorthogonal inputs, their overlap magnitude lies strictly between zero and one, so its square is smaller than itself. The required equality is impossible. Orthogonal states avoid the contradiction because their overlap is zero; identical rays avoid it because the magnitude is one. A fixed basis copier is therefore consistent with quantum mechanics, whereas a deterministic exact copier for a family containing two distinct nonorthogonal pure states is not.

Controlled-NOT copies the zero–one label into a blank zero: zero-zero stays zero-zero and one-zero becomes one-one. By linearity, a superposition input produces a superposition of these two correlated terms. It does not produce the tensor square of the original superposition, which would also contain cross-label terms with products of amplitudes. Each local output can consequently be mixed even though the total output is pure.

The theorem does not prevent repeatedly preparing a state from a known classical recipe. It also does not forbid approximate copies with a defined fidelity, or certain heralded probabilistic copying tasks on restricted families. A heralded success branch changes the deterministic assumptions and must include its success probability. Quantum error correction encodes information into correlations and reverses a specified class of errors; it does not leave several independent perfect copies of an arbitrary logical state available for separate inspection.

Cloning a mixed state requires the product output rho tensor rho. Broadcasting asks only that both output marginals equal rho, allowing correlations between them. A single fixed channel can broadcast a finite-dimensional family of density operators exactly if and only if the operators commute pairwise. This is a theorem, not a conclusion obtained merely by applying the pure-state inner-product proof to arbitrary mixed matrices.

For the commuting direction, choose a common eigenbasis and copy that classical basis label. The output is a mixture of matched labels, with both marginals equal to the input. Its correlations generally prevent it from being a tensor product of independent copies. Noncommuting families cannot be broadcast by one input-independent deterministic channel, even under this weaker requirement. A state-specific preparation device is outside that unknown-family task, so saying that a known fixed mixed state can be prepared twice does not challenge the theorem.