A proposed universal cloning device is a unitary acting on a system qubit and a blank…

PHYSICS · Multi-part · Fermi University Physics · Quantum Information & Technology · Quantum states, measurement and entanglement · Quantum states, measurement and entanglement

Quantum Information and Technology, Ch Qubits, registers and the Bloch sphere

Problem

A proposed universal cloning device is a unitary UU acting on a system qubit and a blank ancilla qubit prepared in ∣0⟩|0\rangle, such that for every normalised qubit state ∣ψ⟩|\psi\rangle, U(∣ψ⟩⊗∣0⟩)=∣ψ⟩⊗∣ψ⟩.U\big(|\psi\rangle\otimes|0\rangle\big)=|\psi\rangle\otimes|\psi\rangle. Consider the two states ∣0⟩|0\rangle and ∣ϕ⟩=cos⁡(π/6)∣0⟩+sin⁡(π/6)∣1⟩|\phi\rangle=\cos(\pi/6)|0\rangle+\sin(\pi/6)|1\rangle (Bloch polar angle θ=π/3\theta=\pi/3 from the north pole, using the half-angle convention θ/2=π/6\theta/2=\pi/6). ((a)) Compute the overlap ⟨0∣ϕ⟩\langle0|\phi\rangle. ((b)) A more general copier is also allowed to change the state of its own machinery, so that U(∣ψ⟩⊗∣0⟩⊗∣s⟩)=∣ψ⟩⊗∣ψ⟩⊗∣mψ⟩U\big(|\psi\rangle\otimes|0\rangle\otimes|s\rangle\big)=|\psi\rangle\otimes|\psi\rangle\otimes|m_\psi\rangle for every ∣ψ⟩|\psi\rangle in a chosen set of states, where ∣s⟩|s\rangle is the machinery's fixed starting state and each ∣mψ⟩|m_\psi\rangle is some normalised state of the machinery, possibly different for different ψ\psi. Show that such a UU can clone two distinct states ∣ψ⟩|\psi\rangle and ∣ϕ⟩|\phi\rangle from this set only if ∣⟨ψ∣ϕ⟩∣|\langle\psi|\phi\rangle| is 00 or 11, and say what this implies for a copier that is allowed to keep, in its machinery, a record of which state passed through it.

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