A qubit is prepared in the pure state with

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

Quantum Information and Technology, Ch Quantum metrology and sensing

Problem

A qubit is prepared in the pure state ∣ψ⟩=cos⁡(θ/2)∣0⟩+sin⁡(θ/2)∣1⟩|\psi\rangle = \cos(\theta/2)|0\rangle + \sin(\theta/2)|1\rangle with θ=60∘\theta = 60^\circ, and then evolves under the phase e−iφZ/2e^{-i\varphi Z/2} so that ∣ψφ⟩=e−iφZ/2∣ψ⟩|\psi_\varphi\rangle = e^{-i\varphi Z/2}|\psi\rangle. The root fidelity between two pure states, as defined for density operators, is F(ψ,ϕ)=∣⟨ψ∣ϕ⟩∣F(\psi,\phi) = |\langle\psi|\phi\rangle|. ((a)) Compute ⟨ψφ∣ψφ+δ⟩\langle\psi_\varphi|\psi_{\varphi+\delta}\rangle and hence F(ψφ,ψφ+δ)F(\psi_\varphi,\psi_{\varphi+\delta}) as a function of δ\delta for this state (it will not depend on φ\varphi). ((b)) Expand F(ψφ,ψφ+δ)F(\psi_\varphi,\psi_{\varphi+\delta}) to second order in δ\delta and read off the quantum Fisher information FQF_Q. ((c)) Using the Cramer-Rao bound, find the smallest achievable Δφ\Delta\varphi with ν=5000\nu = 5000 shots of the best possible measurement.

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