No deterministic quantum operation can make two perfect copies of every unknown pure state. For a proposed unitary operator acting with a fixed blank and environment, preservation of inner products gives , where is the input overlap and is the overlap of final environment states. Nonorthogonal distinct states have , contradicting this equality. The no-cloning theorem for two pure states is its elementary two-state form.
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The no-cloning theorem is a fundamental principle in quantum mechanics that states it is impossible to create an identical copy (or "clone") of an arbitrary unknown quantum state. This theorem is significant because it highlights a key difference between classical information and quantum information. In classical physics, if you have a piece of information, you can make copies of it easily.