Physics Quantum Fisher Info3

Quantum Fisher Information

Quantum Fisher information (QFI) is a quantity from quantum metrology that bounds the best possible precision of any measurement of a parameter. For a pure quantum state, the QFI equals four times the variance of the generator of the parameter shift, written QFI = 4 Var(H). A larger QFI means a smaller minimum variance in the estimated parameter, so it directly sets the shot-noise limit for sensors and interferometers.

The concept emerged from classical statistics, where the Fisher information measures how much a probability distribution changes with a parameter. Quantum Fisher information extends this to quantum states and measurements, and the Cramér-Rao bound then states that the variance of any unbiased estimator is at least the inverse of the QFI. This bound is fundamental: no clever measurement strategy can beat it.

In Recognition Science, the module ledger (a discrete record of events) models the cost of recognition events, and the framework's central cost function J(x) = (x + 1/x)/2 - 1 is forced by five plain conditions. The module applies this cost to the ratio of two masses, m/e, and proves three general facts about the resulting cost: it equals zero when the masses are equal, it is never negative for positive masses, and the golden-ratio threshold phi - 3/2 is positive. These are formal theorems checked by the framework's machine-checked library of formal theorems.

The framework then connects this cost to QFI through a research note: the minimum QFI per recognition tick is 4 times (J(phi) * hbar)^2, which at the golden-ratio cost gives about 4 J(phi)^2 hbar^2. This would describe the smallest detectable parameter change in a recognition-based measurement. However, the module itself proves nothing specific to quantum metrology, because the ratio m/e is defined without reference to any particular physical system.

The practical consequence is that the framework offers a candidate floor for quantum measurement precision, tied to the golden ratio and the reduced Planck constant, but the step from a general cost function to a specific QFI bound remains a research goal, not a proved result.

def domainCost (m e : ℝ) : ℝ := Jcost (m / e)

theorem domainCost_at_eq (r : ℝ) (h : r ≠ 0) : domainCost r r = 0 := by
  unfold domainCost; rw [div_self h]; exact Jcost_unit0

theorem domainCost_nonneg (m e : ℝ) (hm : 0 < m) (he : 0 < e) : 0 ≤ domainCost m e := by
  unfold domainCost; exact Jcost_nonneg (div_pos hm he)

theorem canonicalThreshold_pos : 0 < canonicalThreshold := by
  unfold canonicalThreshold; linarith [phi_gt_onePointFive]

What this page does not claim

This module proves a specific quantum metrology bound, only a general cost function property. The minimum QFI formula is derived in this module; it is a research note, not a theorem.

Verify this page

Every tagged claim above names its theorem. To check one yourself rather than trust this page, elaborate the source module with Lean 4 and audit its axiom basis:

$ lake env lean IndisputableMonolith/Physics/Quantum_Fisher_Info3.lean
expected axiom basis: [propext, Classical.choice, Quot.sound] (the Lean kernel's standard three; no RS-specific axioms)

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