Consider qubits

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

Quantum Information and Technology, Ch Quantum metrology and sensing

Problem

Consider NN qubits, each undergoing the phase evolution e−iφZj/2e^{-i\varphi Z_j/2} generated by G=∑j=1NZj/2G = \sum_{j=1}^N Z_j/2. ((a)) For NN independent qubits, each prepared in ∣+⟩|+\rangle and evolved and measured independently, show that the total quantum Fisher information is FQ=NF_Q = N, and hence derive the standard quantum limit ΔφSQL=1/νN\Delta\varphi_{\rm SQL} = 1/\sqrt{\nu N} for ν\nu repetitions of the whole NN-qubit shot. ((b)) For the NN-qubit GHZ state (∣0⟩⊗N+∣1⟩⊗N)/2(|0\rangle^{\otimes N} + |1\rangle^{\otimes N})/\sqrt{2} evolved by the same GG, show that FQ=N2F_Q = N^2 and hence derive the Heisenberg limit ΔφHL=1/(Nν)\Delta\varphi_{\rm HL} = 1/(N\sqrt{\nu}). ((c)) State in one sentence why part (b)'s advantage over part (a) is not free, referring to the range of φ\varphi each state reads unambiguously.

Browse the problem library