feat: add numeric and symbolic scripts

scripts/cluster.py

@@ -0,0 +1,50 @@
+"""cluster.py -- the full 15x15 bigraduated cluster shadow map M_S for a 2-qubit source
+cluster S (within 4 qubits), and helpers for building the Pauli tensor of a general
+(possibly mixed) density matrix."""
+import numpy as np
+
+I2 = np.eye(2, dtype=complex)
+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)
+_PAULI = [I2, X, Y, Z]
+
+
+def cluster_map(C, s0, s1):
+    """C: (4,4,4,4) Pauli correlation tensor. S = {s0,s1} (source cluster), complement =
+    the other two qubits. Returns the normalized 15x15 matrix M_S(rho), normalization
+    1/sqrt((d_S-1)(d_Sc-1)) = 1/sqrt(3*3) = 1/3 for two-qubit clusters in 4 qubits."""
+    others = [k for k in range(4) if k not in (s0, s1)]
+    c0, c1 = others
+    rows = [(i, j) for i in range(4) for j in range(4) if (i, j) != (0, 0)]
+    cols = [(i, j) for i in range(4) for j in range(4) if (i, j) != (0, 0)]
+    M = np.zeros((15, 15))
+    for ri, (ia, ib) in enumerate(rows):
+        for ci, (ic, idd) in enumerate(cols):
+            idx = [0, 0, 0, 0]
+            idx[s0] = ia
+            idx[s1] = ib
+            idx[c0] = ic
+            idx[c1] = idd
+            M[ri, ci] = C[tuple(idx)]
+    return M / 3.0
+
+
+def nuc(M):
+    return np.linalg.svd(M, compute_uv=False).sum()
+
+
+def full_tensor_mixed(rho):
+    """Pauli-tensor extraction for a general (possibly mixed) 16x16 density matrix rho,
+    via direct trace. Slower than core2.full_tensor (which needs a pure-state vector)
+    but works for explicit mixtures (e.g. the Smolin state)."""
+    def kron4(a, b, c, d):
+        return np.kron(np.kron(a, b), np.kron(c, d))
+    C = np.zeros((4, 4, 4, 4))
+    for i0 in range(4):
+        for i1 in range(4):
+            for i2 in range(4):
+                for i3 in range(4):
+                    op = kron4(_PAULI[i0], _PAULI[i1], _PAULI[i2], _PAULI[i3])
+                    C[i0, i1, i2, i3] = np.real(np.trace(rho @ op))
+    return C