initial commit

scripts/tiles_upb.py

@@ -0,0 +1,105 @@
+import numpy as np
+
+np.set_printoptions(precision=5, suppress=True)
+
+# ---------- Gell-Mann generators for d=3, normalized so Tr(sigma_i sigma_j) = 3*delta_ij ----------
+i_ = 1j
+lam = [None]*9
+lam[1] = np.array([[0,1,0],[1,0,0],[0,0,0]], dtype=complex)
+lam[2] = np.array([[0,-i_,0],[i_,0,0],[0,0,0]], dtype=complex)
+lam[3] = np.array([[1,0,0],[0,-1,0],[0,0,0]], dtype=complex)
+lam[4] = np.array([[0,0,1],[0,0,0],[1,0,0]], dtype=complex)
+lam[5] = np.array([[0,0,-i_],[0,0,0],[i_,0,0]], dtype=complex)
+lam[6] = np.array([[0,0,0],[0,0,1],[0,1,0]], dtype=complex)
+lam[7] = np.array([[0,0,0],[0,0,-i_],[0,i_,0]], dtype=complex)
+lam[8] = (1/np.sqrt(3))*np.array([[1,0,0],[0,1,0],[0,0,-2]], dtype=complex)
+
+# check standard normalization Tr(lam_a lam_b) = 2 delta_ab
+for a in range(1,9):
+    for b in range(1,9):
+        val = np.trace(lam[a]@lam[b])
+        if a==b and not np.isclose(val,2):
+            print("WARN std norm", a,b,val)
+        if a!=b and not np.isclose(val,0):
+            print("WARN std orth", a,b,val)
+
+sigma = [None] + [np.sqrt(3/2)*lam[k] for k in range(1,9)]  # d=3 -> Tr(sigma_i sigma_j)=3 delta_ij
+
+# sanity check
+for a in range(1,9):
+    for b in range(1,9):
+        val = np.trace(sigma[a]@sigma[b]).real
+        expected = 3.0 if a==b else 0.0
+        assert abs(val-expected) < 1e-9, (a,b,val)
+print("Generator normalization OK: Tr(sigma_i sigma_j) = 3 delta_ij")
+
+# ---------- Tiles UPB (Bennett, DiVincenzo, Mor, Shor, Smolin, Terhal 1999) ----------
+e0 = np.array([1,0,0], dtype=complex)
+e1 = np.array([0,1,0], dtype=complex)
+e2 = np.array([0,0,1], dtype=complex)
+
+def nrm(v):
+    return v/np.linalg.norm(v)
+
+psi = []
+psi.append(np.kron(e0, nrm(e0-e1)))
+psi.append(np.kron(e2, nrm(e1-e2)))
+psi.append(np.kron(nrm(e0-e1), e2))
+psi.append(np.kron(nrm(e1-e2), e0))
+psi.append(np.kron(nrm(e0+e1+e2), nrm(e0+e1+e2)))
+
+# check orthonormality
+G = np.array([[np.vdot(p,q) for q in psi] for p in psi])
+print("\nGram matrix of the 5 UPB vectors (should be identity):")
+print(np.round(G,6))
+
+P = sum(np.outer(p, p.conj()) for p in psi)
+I9 = np.eye(9, dtype=complex)
+rho = (I9 - P)/4.0
+
+print("\nTr(rho) =", np.trace(rho).real, " (should be 1)")
+print("rho is Hermitian:", np.allclose(rho, rho.conj().T))
+eigvals_rho = np.linalg.eigvalsh(rho)
+print("eigenvalues of rho (should be >=0, rank 4 nonzero):", np.round(eigvals_rho,5))
+
+# ---------- PPT check ----------
+def partial_transpose_B(rho, dA=3, dB=3):
+    r = rho.reshape(dA,dB,dA,dB)
+    rpt = r.transpose(0,3,2,1)
+    return rpt.reshape(dA*dB, dA*dB)
+
+rho_pt = partial_transpose_B(rho)
+eig_pt = np.linalg.eigvalsh(rho_pt)
+print("\nEigenvalues of partial transpose (PPT check):")
+print(np.round(eig_pt,6))
+print("min eigenvalue of PT:", eig_pt.min(), " -> PPT" if eig_pt.min() > -1e-9 else " -> NPT (entangled via ordinary PPT already)")
+
+# ---------- correlation tensor / shadow map ----------
+T = np.zeros((8,8))
+for a in range(1,9):
+    for b in range(1,9):
+        op = np.kron(sigma[a], sigma[b])
+        T[a-1,b-1] = np.trace(rho @ op).real
+
+s = np.linalg.svd(T, compute_uv=False)
+nuclear_T = s.sum()
+dA, dB = 3, 3
+norm_const = np.sqrt((dA-1)*(dB-1))
+M_norm = nuclear_T / norm_const
+
+print("\nSingular values of correlation tensor T:", np.round(s,5))
+print("Nuclear norm ||T||_* =", nuclear_T)
+print("Normalization constant sqrt((dA-1)(dB-1)) =", norm_const)
+print("Shadow-map value ||M_A(rho)||_* =", M_norm, " (separable bound: <= 1)")
+
+# ---------- CCNR / realignment criterion for comparison ----------
+def realign(rho, dA=3, dB=3):
+    r = rho.reshape(dA,dB,dA,dB)
+    # standard realignment: R_{(i mu),(j nu)} = rho_{ij,mu nu}
+    R = r.transpose(0,2,1,3).reshape(dA*dA, dB*dB)
+    return R
+
+R = realign(rho)
+s_R = np.linalg.svd(R, compute_uv=False)
+ccnr = s_R.sum()
+print("\nCCNR (realignment) trace norm:", ccnr, " (separable bound: <= 1)")