Question

What exactly does No cloning mean, in the context of Quantum Computing? Can you explain and proof the no-cloning theorem ?

Answer 1

--> No-cloning theorem states that no unitary cloning machine exists that would clone arbitrary initial states or we can say Quantum information cannot be copied exactly.

Theorem : There is no unitary operator U on H⊗H such that for all normalised states |ϕ〉A and |e〉B in H.

   for some real number α depending on ϕ and e.

The extra phase factor expresses the fact that a quantum mechanical state defines a normalised vector in Hilbert space only up to a phase factor

prove : we select an arbitrary pair of states |ϕ〉A and |ψ〉A in the Hilbert space H. Because U is unitary,

Since the quantum state |e〉 is assumed to be normalized we thus get

This implies that either |〈ϕ|ψ〉|=1 or |〈ϕ|ψ〉|=0. Hence by the Cauchy–Schwarz inequality either ϕ=eiβψ or ϕ is orthogonal to ψ. However, this cannot be the case for two arbitrary states. Therefore, a single universal U cannot clone a general quantum state. This proves the no-cloning theorem.