Finite Calibration Sets for Rational Scalar Controls

Abstract

Every finite set above the fixed threshold is exactly the calibration jet set of a strictly feasible rational scalar control with degree at most twice its cardinality.

The parameters a and delta are real, with 0 < a < 1, 0 < delta, delta < (1-a)/4, and delta < (1-a squared)/16. The last inequality is retained among the hypotheses although the scalar proof does not use it. The finite set S consists of arbitrary real nodes i satisfying a/(1-delta) < i < 1; its cardinality has no fixed upper bound.

The scalar parameters are L(a) = 1-a, lam(a,delta) = L(a)-delta, gamma(a,delta) = lam(a,delta)/(2 delta), and center(a,delta) = lam(a,delta)/L(a). The radius is the product of the square roots of a/lam(a,delta) and (1-p)/p. The energy at s is s squared plus gamma squared times the square of s + 1/s - 2. The required derivative is beta(p) = 1/(2 p (1-p)), taken with respect to p.

Definition 1.1 (All scalar control requirements).

$$\mathrm{FullControl}\left(a,delta,S,N,D\right)⟺\mathrm{natDegree}\left(N\right)\le 2\mathrm{card}\left(S\right)\wedge \mathrm{natDegree}\left(D\right)\le 2\mathrm{card}\left(S\right)\wedge (\forall p:\mathbb{R},0<D\left(p\right))\wedge (\forall p\in \mathrm{Icc}\left(a,1\right),(0<s\left(p\right)\wedge {\mathrm{radius}\left(a,delta,p\right)}^2\mathrm{energy}\left(a,delta,s\left(p\right)\right)<1))\wedge \mathrm{AnalyticOnNhd}\left(\mathbb{R},s,\mathrm{Ioo}\left(a,1\right)\right)\wedge (\forall p\in \mathrm{Ioo}\left(a,1\right),(s\left(p\right)=1\wedge \mathrm{deriv}\left(s,p\right)=\mathrm{beta}\left(p\right))⟺p\in S)\wedge (S=\varnothing \Rightarrow \forall p:\mathbb{R},s\left(p\right)=\mathrm{center}\left(a,delta\right))$$

Formalization. D5/S3/Quantum/Information/FiniteCalibrationControl.FullControl (✓ std3).

Source. Repository-derived.

Commentary.

Here N and D are real polynomials and s(p) denotes N(p)/D(p). Both natural degrees are at most twice card S. D is strictly positive on the entire real line. For every p in the closed interval [a,1], s(p) is positive and radius squared times energy is strictly less than one, including both endpoints.

The quotient is real analytic in a neighborhood of every point of (a,1). At each point of that open interval, value one together with derivative beta(p) holds if and only if p belongs to S. This specifies the complete set of first jets. When S is empty, the quotient equals center(a,delta) at every real p.

Theorem 1.2 (Every prescribed finite set is realized exactly).

$$\forall a,delta:\mathbb{R},S:\mathrm{Finset}\left(\mathbb{R}\right),(0<a\wedge a<1\wedge 0<delta\wedge delta<\frac{1-a}{4}\wedge delta<\frac{1-a^2}{16}\wedge (\forall i\in S,\frac{a}{1-delta}<i\wedge i<1))\Rightarrow \exists N,D:\mathbb{R}[X],\mathrm{FullControl}\left(a,delta,S,N,D\right)$$

Proof. Machine-checked in Lean as D5/S3/Quantum/Information/FiniteCalibrationControl.result (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let P be the nodal polynomial of S and let B_i be its Lagrange basis polynomials. The polynomial hermite(S) is one plus the sum of beta(i) times (X-i) times B_i squared. Its value is one and its derivative is beta(i) at every prescribed node.

For nonempty S the construction uses N = H + k center P squared and D = 1 + k P squared, with one positive real k. The center is strictly feasible throughout [a,1]. Convexity of the energy preserves feasibility as k increases. Pointwise eventual feasibility gives a directed open cover of the compact interval; compactness supplies a single k valid at every point. The polynomial degrees satisfy the stated bound.

Away from S, (H-1)/P squared is a sum of positive weights divided by p-i. Its derivative is strictly negative. At every additional root where s(p)=1, the derivative of s is therefore strictly negative, whereas beta(p) is positive on (a,1). Such roots cannot supply additional calibration jets. For empty S take N constant equal to the center and D=1; the center is strictly below one.

The conclusion concerns the rational scalar control and its complete calibration jet set. Constructing a completely positive, trace preserving processor, an actual pure-state program, its quantum Fisher information, and a minimum program dimension requires separate operator and state results. Those conclusions are not asserted here.

References

• Truth anchor: D5/S3/Quantum/Information/FiniteCalibrationControl.FullControl
• Truth anchor: D5/S3/Quantum/Information/FiniteCalibrationControl.result