Physics Spin Squeezing From Jcost

Spin squeezing and its cost

Spin squeezing is a technique in quantum metrology that reduces the uncertainty in one component of a collection of spins, such as atoms, at the cost of increasing the uncertainty in another. The standard parameter is xi^2 = N * Var(J_z) / ^2, where N is the number of particles, Var(J_z) is the variance along one axis, and is the mean spin along another. A smaller xi^2 means better precision for measuring rotations or fields. At the Heisenberg limit, the ultimate quantum limit, xi^2 = 1/N, which improves with more particles.

The classical theory of spin squeezing was developed in the 1990s, with key contributions from Kitagawa and Ueda in 1993, who introduced the parameter and showed how interactions between particles could create squeezed states. The technique is now central to atomic clocks and gravitational wave detectors, where beating the standard quantum limit of 1/sqrt(N) is essential. The Heisenberg limit represents the absolute floor set by quantum mechanics, and reaching it requires both entanglement and careful state preparation.

In Recognition Science, the framework models the cost of a recognition event using a specific function J(x) = (x + 1/x)/2 - 1, which is forced by five plain conditions. The framework's machine-checked library of formal theorems proves three general facts about this cost: it vanishes when the two inputs are equal, it is never negative for positive inputs, and a certain threshold value phi - 3/2 is positive, where phi is the golden ratio. These facts are general properties of the cost function itself, not specific to spin squeezing.

The module under discussion defines a domain cost as J(m/e), where m and e are real numbers, and then proves the three properties above. The research note attached to the module records an intended application: optimal squeezing at J(xi^2 / xi^2_coherent) = J(phi), which would give xi^2 = phi * (1/N^0.5). This is a hypothesis about where the framework's cost function might point, not a proved result. The module itself establishes only the general cost properties, and the connection to spin squeezing remains a research note, not a theorem.

What this means in plain language is that the framework has a well-behaved cost function, but it has not yet been shown to govern spin squeezing. The three proved facts are like checking that a ruler is straight and marked correctly before using it to measure a room. The ruler is ready, but the measurement of the room, the specific application to xi^2, has not been completed in the formal library.

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]

theorem cert_inhabited : Nonempty SpinSqueezeCert := ⟨cert⟩

What this page does not claim

The module proves any specific result about spin squeezing; it only proves general cost properties. The optimal squeezing formula xi^2 = phi / sqrt(N) is a proved theorem; it is a research note. The framework's cost function is shown to govern quantum metrology; that connection is open.

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

A page whose claims cannot be reproduced this way does not ship. In production, every anchor links to the exact declaration in the public source release, and this block carries the build receipt for the page itself.

Derived articles

This page is generated by a question-recursion engine: the questions its answers raise become the next pages. The current agenda, with open targets marked red: