Quantum-Computation-course-Basel/extra_resources/Entanglement_extra/7_Separability.ipynb

In [1]:

"""
For 2×2 and 2×3 systems, the Positive Partial Transpose (PPT) criterion is necessary and sufficient for separability.

Exercises:
- Extend to a 2×3 example by padding qubits or adding a qutrit toy model with dims=(2,3).
- Wrap into a helper function `is_entangled_ppt(rho, dims)`.
"""

Out [1]:

'\nFor 2×2 and 2×3 systems, the Positive Partial Transpose (PPT) criterion is necessary and sufficient for separability.\n\nExercises:\n- Extend to a 2×3 example by padding qubits or adding a qutrit toy model with dims=(2,3).\n- Wrap into a helper function `is_entangled_ppt(rho, dims)`.\n'

In [2]:

import numpy as np
from qiskit.quantum_info import Statevector, DensityMatrix

phi = Statevector([1,0,0,1])/np.sqrt(2)
rho_pure = DensityMatrix(phi).data
I4 = np.eye(4)/4

def isotropic(p):
    # |Φ+> density matrix with simple depolarizing noise when p<1
    return p*rho_pure + (1-p)*I4

for p in [1.0, 0.8, 0.6, 0.4]:
    rho = isotropic(p)
    rho_pt = DensityMatrix.partial_transpose(DensityMatrix(rho), [1]).data
    eig = np.linalg.eigvalsh(rho_pt)
    print(f"p={p}  min eig(ρ^T_B)={eig.min(): .6e}  {'ENTANGLED' if eig.min()< -1e-10 else 'PPT'}")

p=1.0  min eig(ρ^T_B)=-5.000000e-01  ENTANGLED
p=0.8  min eig(ρ^T_B)=-3.500000e-01  ENTANGLED
p=0.6  min eig(ρ^T_B)=-2.000000e-01  ENTANGLED
p=0.4  min eig(ρ^T_B)=-5.000000e-02  ENTANGLED

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