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      "text/plain": [
       "'\\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'"
      ]
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    }
   ],
   "source": [
    "\"\"\"\n",
    "For 2×2 and 2×3 systems, the Positive Partial Transpose (PPT) criterion is necessary and sufficient for separability.\n",
    "\n",
    "Exercises:\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",
    "\"\"\""
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "p=1.0  min eig(ρ^T_B)=-5.000000e-01  ENTANGLED\n",
      "p=0.8  min eig(ρ^T_B)=-3.500000e-01  ENTANGLED\n",
      "p=0.6  min eig(ρ^T_B)=-2.000000e-01  ENTANGLED\n",
      "p=0.4  min eig(ρ^T_B)=-5.000000e-02  ENTANGLED\n"
     ]
    }
   ],
   "source": [
    "import numpy as np\n",
    "from qiskit.quantum_info import Statevector, DensityMatrix\n",
    "\n",
    "phi = Statevector([1,0,0,1])/np.sqrt(2)\n",
    "rho_pure = DensityMatrix(phi).data\n",
    "I4 = np.eye(4)/4\n",
    "\n",
    "def isotropic(p):\n",
    "    # |Φ+> density matrix with simple depolarizing noise when p<1\n",
    "    return p*rho_pure + (1-p)*I4\n",
    "\n",
    "for p in [1.0, 0.8, 0.6, 0.4]:\n",
    "    rho = isotropic(p)\n",
    "    rho_pt = DensityMatrix.partial_transpose(DensityMatrix(rho), [1]).data\n",
    "    eig = np.linalg.eigvalsh(rho_pt)\n",
    "    print(f\"p={p}  min eig(ρ^T_B)={eig.min(): .6e}  {'ENTANGLED' if eig.min()< -1e-10 else 'PPT'}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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