Quantum metrology
How to sense fluctuations of a phase-like parameter φ?
Prepare N systems in entangled state ρN.
Prepare N systems in entangled state ρN. A chain of N spins.
Prepare N systems in entangled state ρN. A chain of N spins. A cluster state with linear optics.
The phase φ is encoded by unitary Uφ on each particle
The final state ρNφ is measured.
Repeat to increase precision.
"An observable generates a unitary transformation of the system with a parameter θ: ρ(θ) = exp(−iAθ) ρ(0) exp(iAθ).
Learn from noisy quantum experiments
import Quantum_state /-- Quantum Fischer information -/ def quantum_Fisher_info [quantum_state ρ] [op A] : ℝ := 2 * ∑ i j, (e i - e j)²/(e i + e j) ∣⟨i|A|j⟩∣² -- e i are the eigenvalues of ρ. -- |i⟩ are the eigenvectors.
/-- The quantum Fisher information constrains the achievable precision in statistical estimation of the parameter θ. Let m be the number of independent repetitions. -/ lemma more_precision_allowed_of_larger_quantum_Fischer_info (m : ℕ) : (Δθ)² ≥ 1/(m * (quantum_Fisher_info ρ A)) := sorry
/-- Sensing with classical states has resolution limited to 1/N -/ theorem precision_of_classical_sensing : ... := sorry /-- No quantum-enhanced sensitivity with separable states -/ theorem no_enhanced_precision_of_separable : ... := sorry
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Applications:
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Enhanced sensing of phase-like parameters in optical interferometry.
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In atomic-ensemble experiments of spectroscopy and magnetometry,
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atomic clocks.
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i.i.d noise channels on the N particles limit the quantum enhancement to a constant factor.
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Group the constituent particles into separate entangled blocks of finite size.
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How much entanglement is needed for quantum metrology;
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There exist states where all the systems are entangled but nonetheless are useless for metrology.
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By employing large-scale but yet undistillable entanglement one may still attain the HL resolution in phase sensing.
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noise-robust metrology schemes that attain super-classical resolutions by employing quantum states with just the necessary entanglement properties.
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“Open question: What if one considers metrology protocols without repetitions? The estimated parameter may not be assumed to be fluctuating around a known value. And the Bayesian approach to estimation must be pursued.
- We expect the requirements on the entanglement to be then much more stringent due to the lack of sufficient statistics.”
Quantum metrology and Quantum error correction?
notation `FQ` := quantum_Fischer_info
notation `F` := Uhlmann_fidelity
theorem theorem1 {N : ℕ} :
∀ [quantum_state ρN σN],
|(FQ ρN) - (FQ σN)| ≤ 8 * N² * √(1 - (F ρN σN)) :=
sorry
corollary cor1 {N : ℕ} :
∀ [quantum_state ρN σN],
FQ ρN ≤ FQ σN + 8 * N² * √(2 * (trace_dist ρN σN)) :=
sorry
critical quantum metrology
tune sensor to near phase transition. more sensitivity. like BJT amplifier. amplifies small signal.
1D Ising spin chain for a probe
prior prob dist of the parameter to be estimated