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518343a | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 | /**
* GRAFT 3 — POVM Verdict Semantics
*
* Source: Preskill, J. (2015). Quantum Information, Lecture Notes for
* Physics 219/CS 219. Caltech, Chapter 3 §3.1 (POVMs).
* https://www.preskill.caltech.edu/ph219/chap3_15.pdf
*
* Replaces binary {accept, reject} verdicts with a positive-operator-
* valued measure: a finite collection of positive-semidefinite operators
* {Eᵢ} on the policy state space satisfying Σᵢ Eᵢ = I (the completeness
* theorem). Each Eᵢ corresponds to a distinct admission outcome — admit-
* with-witness, admit-with-throttle, deny, escalate, etc. — and the
* probability of outcome i on policy state ρ is Tr(Eᵢρ).
*
* The completeness invariant is checked at construction; non-conforming
* verdict sets are rejected. This is the operational equivalent of the
* Lean theorem `povm_completeness` referenced in the doctrine.
*/
const COMPLETENESS_TOL = 1e-9;
export type Outcome = 'admit' | 'admit_throttled' | 'admit_witnessed' | 'deny' | 'escalate';
export type POVMVerdict = {
outcome: Outcome;
/** Effect operator diagonal entries (we restrict to diagonal effects in
* the policy basis; full off-diagonal POVMs are out of scope for now). */
effectDiagonal: readonly number[];
rationale: string;
};
export type POVMSet = readonly POVMVerdict[];
/** Σᵢ Eᵢ = I (with floating-point tolerance). */
export function isComplete(verdicts: POVMSet, dim: number): boolean {
if (verdicts.length === 0) return false;
const sum = new Array<number>(dim).fill(0);
for (const v of verdicts) {
if (v.effectDiagonal.length !== dim) return false;
for (let i = 0; i < dim; i++) {
const x = v.effectDiagonal[i];
if (x < -COMPLETENESS_TOL || x > 1 + COMPLETENESS_TOL) return false; // 0 ≤ Eᵢ ≤ I
sum[i] += x;
}
}
return sum.every((s) => Math.abs(s - 1) < COMPLETENESS_TOL);
}
export class POVMSetError extends Error {
constructor(message: string) {
super(message);
this.name = 'POVMSetError';
}
}
/** Construct + validate. Throws if Σᵢ Eᵢ ≠ I. */
export function makePOVM(verdicts: POVMSet, dim: number): POVMSet {
if (!isComplete(verdicts, dim)) {
throw new POVMSetError(`POVM completeness violated: Σᵢ Eᵢ ≠ I on dim ${dim}`);
}
return verdicts;
}
/** Probability of outcome i on diagonal state ρ — Tr(Eᵢρ). */
export function probability(verdict: POVMVerdict, stateDiagonal: readonly number[]): number {
if (verdict.effectDiagonal.length !== stateDiagonal.length) {
throw new POVMSetError('effect/state dimension mismatch');
}
let p = 0;
for (let i = 0; i < stateDiagonal.length; i++) {
p += verdict.effectDiagonal[i] * stateDiagonal[i];
}
return p;
}
/** Pick the most-likely outcome on state ρ (used by the policy head). */
export function argmaxOutcome(povm: POVMSet, stateDiagonal: readonly number[]): POVMVerdict {
let best = povm[0];
let bestP = -Infinity;
for (const v of povm) {
const p = probability(v, stateDiagonal);
if (p > bestP) {
bestP = p;
best = v;
}
}
return best;
}
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