A register of four qubits is always prepared in one of the sixteen computational…
PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · The quantum circuit model and quantum algorithms
Quantum Information and Technology, Ch Qubits, registers and the Bloch sphere
Problem
A register of four qubits is always prepared in one of the sixteen computational basis strings, for instance ∣0110⟩; it is never a superposition. To back up the data, a blank register of four qubits, initialised in ∣0000⟩, is available, and CNOT gates are used with qubit i of the data register as control and qubit i of the blank register as target (i=1,2,3,4). A colleague objects: "The no-cloning theorem says an unknown quantum state cannot be copied, so this backup scheme is impossible."
((a)) Assess the objection: does the CNOT scheme succeed in duplicating ∣0110⟩ onto the blank register? Justify your answer.
((b)) Suppose instead qubit 1 of the data register were prepared in the superposition a∣0⟩+b∣1⟩ with a,b=0 (qubits 2 to 4 still in definite basis states). Apply the same CNOT (qubit 1 controlling the corresponding blank qubit) and show explicitly that the result is not the state one would want from cloning, (a∣0⟩+b∣1⟩)⊗(a∣0⟩+b∣1⟩).