Quantum Thermometry
Cramér–Rao bound and Fisher information for temperature estimation
System: Two-level probe
True temperature T
1.00
Energy gap Δ
1.00
Measurement shots n
200
Fisher Information
F(T) = …
σ_min = 1/√F = …
σ_emp = …
Theory
Cramér–Rao bound
: Var(T̂) ≥ 1/[n·F(T)]
For a two-level probe with gap Δ in thermal equilibrium at temperature T (k_B=1):
p↑ = e^{-Δ/T}/(1+e^{-Δ/T})
Quantum Fisher Information
:
F(T) = (Δ/T²)² · p↑(1-p↑)
Optimal sensitivity near Δ≈2.4T. Below that, the probe is too easily saturated; above, excited state probability → 0.