feat: add numeric and symbolic scripts

scripts/optimize_phi_sym.py

@@ -0,0 +1,66 @@
+"""optimize_phi_sym.py -- multi-start optimization of Phi_sym over biseparable pure
+states (1|3 and 2|2 cuts) and over ALL pure 4-qubit states (unconstrained).
+
+Key finding: all three searches converge to the SAME value, 6/sqrt(7) -- i.e. Phi_sym's
+biseparable supremum equals its global supremum over the entire state space, achieved
+both by connected 4-qubit graph states AND by a trivial biseparable state (two Bell
+pairs). This shows Phi_sym cannot certify genuine multipartite entanglement at that
+threshold.
+"""
+import numpy as np
+from scipy.optimize import minimize
+from core import state_1_3, state_2_2
+from core2 import phi_sym
+
+
+def neg_phi_1_3(params, source):
+    psi = state_1_3(params, source)
+    val, _ = phi_sym(psi)
+    return -val
+
+
+def neg_phi_2_2(params, pair1, pair2):
+    psi = state_2_2(params, pair1, pair2)
+    val, _ = phi_sym(psi)
+    return -val
+
+
+def neg_phi_general(params):
+    v = params[:16] + 1j * params[16:]
+    v = v / np.linalg.norm(v)
+    val, _ = phi_sym(v)
+    return -val
+
+
+def run_multistart(objective, nparams, args, n_restarts, seed0, label):
+    best = -np.inf
+    bx = None
+    for i in range(n_restarts):
+        rng = np.random.default_rng(seed0 + i)
+        x0 = rng.normal(size=nparams)
+        res = minimize(objective, x0, args=args, method='Powell',
+                        options={'maxiter': 2000, 'xtol': 1e-9, 'ftol': 1e-11})
+        v = -res.fun
+        if v > best:
+            best = v
+            bx = res.x
+    print(f'{label}: best Phi_sym = {best:.10f}   ({n_restarts} restarts)')
+    return best, bx
+
+
+if __name__ == "__main__":
+    # biseparable across 2|2 cut AB|CD -> exact optimum 6/sqrt(7), achieved by
+    # Bell_AB (x) Bell_CD (verified in closed form: two maximal Bell pairs)
+    v22, x22 = run_multistart(neg_phi_2_2, 16, ((0, 1), (2, 3)), 15, 100,
+                               "biseparable 2|2 (AB|CD)")
+    # biseparable across 1|3 cut A|BCD -> exact optimum (1+18/sqrt(7))/4,
+    # achieved by (any single qubit) (x) GHZ_3(B,C,D)
+    v13, x13 = run_multistart(neg_phi_1_3, 18, (0,), 15, 200,
+                               "biseparable 1|3 (A|BCD)")
+    # fully unconstrained over ALL pure 4-qubit states -> same ceiling 6/sqrt(7)
+    vgen, xgen = run_multistart(neg_phi_general, 32, (), 20, 500,
+                                 "unconstrained (all 4-qubit states)")
+
+    print()
+    print("6/sqrt(7)          =", 6 / np.sqrt(7))
+    print("(1+18/sqrt(7))/4   =", (1 + 18 / np.sqrt(7)) / 4)