46 lines
2 KiB
Python
46 lines
2 KiB
Python
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"""universal_ceiling.py -- closed-form universal ceiling for shadow-map nuclear norms,
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derived via Cauchy-Schwarz (rank bound) + a trace identity for the Pauli correlation
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tensor of a pure n-qubit state.
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Single-party source (m=1) in n qubits:
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max_rho ||M_a(rho)||_* = 3 * sqrt(2^(n-2) / (2^(n-1)-1))
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reproduces sqrt(6) (n=3), 6/sqrt(7) (n=4), 2.19089... (n=5) -- exactly the "common
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values" the paper reports numerically for GHZ_n / line_n / ring_n / connected graph
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states.
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General m-qubit cluster source S (m <= n/2):
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max_rho ||M_S(rho)||_* = sqrt( (2^(2m)-1)(2^n - 2^(n-2m)) / ((2^m-1)(2^(n-m)-1)) )
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which reduces to the m=1 formula above, and gives exactly 5 for (n=4, m=2) -- matching
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the numerically found ceiling for the 2-qubit cluster maps M_AB, M_AC, M_AD.
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Equality holds iff (i) the m-qubit source marginal rho_S is maximally mixed
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(tr(rho_S^2) = 1/2^m), and (ii) the resulting shadow map has all singular values equal
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("isotropic"). This ceiling is saturated not only by highly symmetric stabilizer states
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(GHZ_n, connected graph states) but also by simple biseparable states across an
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UNRELATED cut (e.g. two Bell pairs) -- which is why Phi_sym / a single ||M_S||_* cannot
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serve as a genuine multipartite entanglement witness on their own.
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"""
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import numpy as np
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def universal_ceiling(n, m):
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num = (2 ** (2 * m) - 1) * (2 ** n - 2 ** (n - 2 * m))
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den = (2 ** m - 1) * (2 ** (n - m) - 1)
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return np.sqrt(num / den)
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def universal_ceiling_singleparty(n):
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return 3 * np.sqrt(2 ** (n - 2) / (2 ** (n - 1) - 1))
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if __name__ == "__main__":
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print("Single-party (m=1) ceiling for n=3,4,5:")
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for n in [3, 4, 5]:
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print(f" n={n}: {universal_ceiling_singleparty(n):.6f} "
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f"(general formula gives: {universal_ceiling(n, 1):.6f})")
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print(" compare: sqrt(6) =", np.sqrt(6), " 6/sqrt(7) =", 6 / np.sqrt(7))
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print()
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print("2-qubit cluster (m=2) ceiling for n=4 qubits:")
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print(f" {universal_ceiling(4, 2):.6f} (matches the numerically found value 5.0)")
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