The no-cloning theorem states the impossibility to create an identical copy of an arbitrary unknown quantum state, which was discovered by Wootters and Zurek and Dieks 1 in 1982. The no-cloning theorem forms the basis of quantum cryptography, an important part of quantum information science: it is not possible eavesdrop a quantum communication, since a copy a the qubit would be needed for that.
In order to prove the no-cloning theorem we assume that such an operation does exist and proceed to show that this leads to a contradiction. We have two arbitrary quantum states, denoted as and that shall be copied onto a state . Since the scalar product has to bee conserved our copy operation can only be a unitary operation having the following properties: and . For the scalar product we get two equations: and , from which directly follows , which can be rewritten as . Since we finally get . This equation has only two solutions namly and . While the latter means ) the former states that and are orthogonal (). However, this cannot be the case for two arbitrary states. Therefore, a single universal operation cannot clone a general quantum state. This proves the no-cloning theorem.
There is also another even simpler proof: Agein we start with the mode mode of operation of the unitary operator, using a slightly differnt way, that is (so the copy is written on a basis state instead of an arbitrary state as we had bafore). An arbitrary quantum state is given by a linear combination of the basis states as with (more precisely and ). According to the unitarity of the operator the operator must obey the following equation: . On the oter hand from liniarity of the operator we also have . This two equation are in contradiction ! They are only valid for (or ) but not for an arbitray quantum state with .
1. W. K. Wootters and W. H. Zurek, Nature (London) 299 802 – 803 (1982). ↩