Add scripts for exact Casimir projectors and recoupling proof

- Implemented `2_exact_casimir_projectors.py` to construct exact SO(3) isotypic projectors using the Casimir operator, replacing Monte-Carlo methods.
- Created `3_apply_exact_projectors.py` to apply the exact projectors to example states, calculating ||A_j||_* estimates with improved precision.
- Developed `4_spherical_basis.py` to build a Condon-Shortley-consistent spherical basis for a single spin-1 leg using ladder operators.
- Introduced `6_six_j_recoupling_proof.py` to provide a complete proof of the cut-recoupling formula for full collective SU(2) symmetry, verifying the relationship between reduced blocks A_j^(1) and A_p^(2).
- Added a README file to document the execution order and purpose of each script in the symmetric states project.

scripts/symmetric_states/1_recoupling_check.py

@@ -0,0 +1,134 @@
+"""
+Numerical check of the 'one invariant tensor, many cuts' claim
+for the 6-qubit singlet-network example states.
+
+Qubits ordered A,B,C,D,E,F -> tensor axes 0..5.
+S1 = ABC | DEF   (3|3 cut)
+S2 = AB  | CDEF  (2|4 cut)
+
+We build the two 'basis' states
+  |psi1> = singlets (A,D)(B,E)(C,F)
+  |psi2> = singlets (A,E)(B,F)(C,D)
+and the family
+  |Xi(alpha)> = (cos(a) psi1 + sin(a) psi2) / norm
+
+For rho_Xi = |Xi><Xi|, linearity in rho gives EXACTLY
+  T(Xi) = [ cos^2(a) T1 + sin^2(a) T2 + cos(a)sin(a) C12 ] / N2
+where T1 = <psi1|O|psi1>, T2 = <psi2|O|psi2>, C12 = <psi1|O|psi2> + <psi2|O|psi1>,
+N2 = <Xi_raw|Xi_raw>.
+
+Key point: T1, T2, C12 do NOT depend on alpha or on the cut.
+Once computed ONCE (full 6-leg tensors), every cut's shadow-map block
+for every alpha is obtained by (i) taking this fixed linear combination
+of THREE fixed numbers times three fixed tensors, and (ii) a plain
+numpy .reshape() -- no further contraction over the 64-dim Hilbert space.
+
+This script verifies that against two independent brute-force
+quantum simulations (cut S1 and cut S2, both done from scratch).
+"""
+
+import numpy as np
+
+# ---------- Pauli matrices ----------
+X = np.array([[0,1],[1,0]], dtype=complex)
+Y = np.array([[0,-1j],[1j,0]], dtype=complex)
+Z = np.array([[1,0],[0,-1]], dtype=complex)
+paulis = [X, Y, Z]
+
+# ---------- build the two basis states (6-qubit amplitude tensors) ----------
+def s(a,b):
+    if (a,b) == (0,1): return 1/np.sqrt(2)
+    if (a,b) == (1,0): return -1/np.sqrt(2)
+    return 0.0
+
+def build_pairing(pairs):
+    psi = np.zeros((2,)*6, dtype=complex)
+    for idx in np.ndindex(2,2,2,2,2,2):
+        val = 1.0
+        for (p,q) in pairs:
+            val *= s(idx[p], idx[q])
+            if val == 0: break
+        psi[idx] = val
+    return psi
+
+psi1 = build_pairing([(0,3),(1,4),(2,5)])   # (A,D)(B,E)(C,F)
+psi2 = build_pairing([(0,4),(1,5),(2,3)])   # (A,E)(B,F)(C,D)
+overlap = np.vdot(psi1, psi2)
+print("overlap <psi1|psi2> =", overlap)
+
+def apply_pauli_leg(psi, axis, P):
+    psi2 = np.moveaxis(psi, axis, 0)
+    out = np.tensordot(P, psi2, axes=([1],[0]))
+    return np.moveaxis(out, 0, axis)
+
+def corr_tensor(bra, ket):
+    """<bra| sigma_i1 x ... x sigma_i6 |ket>, all six legs active (i in {0,1,2}=x,y,z)."""
+    c = np.zeros((3,3,3,3,3,3), dtype=complex)
+    for iA in range(3):
+     for iB in range(3):
+      for iC in range(3):
+       for iD in range(3):
+        for iE in range(3):
+         for iF in range(3):
+            ket_ = ket
+            for axis,ii in zip(range(6),(iA,iB,iC,iD,iE,iF)):
+                ket_ = apply_pauli_leg(ket_, axis, paulis[ii])
+            c[iA,iB,iC,iD,iE,iF] = np.vdot(bra, ket_)
+    return c
+
+print("computing T1 = <psi1|O|psi1> ...")
+T1 = corr_tensor(psi1, psi1)
+print("computing T2 = <psi2|O|psi2> ...")
+T2 = corr_tensor(psi2, psi2)
+print("computing cross term <psi1|O|psi2> ...")
+X12 = corr_tensor(psi1, psi2)
+X21 = corr_tensor(psi2, psi1)
+C12 = X12 + X21
+
+print("max imag part T1,T2,C12:",
+      np.abs(T1.imag).max(), np.abs(T2.imag).max(), np.abs(C12.imag).max())
+T1, T2, C12 = T1.real, T2.real, C12.real
+
+np.save("T1.npy", T1); np.save("T2.npy", T2); np.save("C12.npy", C12)
+
+# ---------- ground truth: brute-force Xi(alpha) for a couple of alphas, both cuts ----------
+alpha_list = [np.pi/5, 0.9, -0.3]
+
+def build_Xi(alpha):
+    raw = np.cos(alpha)*psi1 + np.sin(alpha)*psi2
+    n = np.linalg.norm(raw)
+    return raw/n, n**2
+
+def predict_from_basis(alpha):
+    N2 = 1 + np.sin(2*alpha)*overlap.real
+    return (np.cos(alpha)**2*T1 + np.sin(alpha)**2*T2 + np.cos(alpha)*np.sin(alpha)*C12) / N2
+
+max_err_cut1 = 0.0
+max_err_cut2 = 0.0
+for a in alpha_list:
+    Xi, N2_check = build_Xi(a)
+    Tgt = corr_tensor(Xi, Xi).real          # brute-force ground truth, full simulation
+    Tpred = predict_from_basis(a)           # from the 3 fixed tensors, no new simulation
+
+    err = np.abs(Tgt - Tpred).max()
+    print(f"alpha={a:+.4f}: max|T_bruteforce - T_predicted| = {err:.3e}   (N2 check: {N2_check:.6f})")
+
+    # cut 1: ABC|DEF  (27x27)
+    M1_true = Tgt.reshape(27,27)
+    M1_pred = Tpred.reshape(27,27)
+    e1 = np.abs(M1_true - M1_pred).max()
+    max_err_cut1 = max(max_err_cut1, e1)
+
+    # cut 2: AB|CDEF  (9x81)
+    M2_true = Tgt.reshape(9,81)
+    M2_pred = Tpred.reshape(9,81)
+    e2 = np.abs(M2_true - M2_pred).max()
+    max_err_cut2 = max(max_err_cut2, e2)
+
+    print(f"   cut ABC|DEF : max matrix error = {e1:.3e},  ||M||_* true={np.linalg.svd(M1_true,compute_uv=False).sum():.4f} "
+          f"pred={np.linalg.svd(M1_pred,compute_uv=False).sum():.4f}")
+    print(f"   cut AB|CDEF : max matrix error = {e2:.3e},  ||M||_* true={np.linalg.svd(M2_true,compute_uv=False).sum():.4f} "
+          f"pred={np.linalg.svd(M2_pred,compute_uv=False).sum():.4f}")
+
+print()
+print(f"WORST CASE over all tested alpha, both cuts: {max(max_err_cut1, max_err_cut2):.3e}")