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