{ "cells": [ { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "'\\n### Overview\\nSimulate a simple entanglement purification step (BBPSSW-like) on isotropic states and track fidelity gains.\\n\\nExercises:\\n- Chain multiple rounds and observe convergence threshold behavior.\\n'" ] }, "execution_count": 1, "metadata": {}, "output_type": "execute_result" } ], "source": [ "\"\"\"\n", "### Overview\n", "Simulate a simple entanglement purification step (BBPSSW-like) on isotropic states and track fidelity gains.\n", "\n", "Exercises:\n", "- Chain multiple rounds and observe convergence threshold behavior.\n", "\"\"\"" ] }, { "cell_type": "code", "execution_count": 1, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "F_in=0.50 F_out≈0.500 p_succ≈0.556\n", "F_in=0.60 F_out≈0.620 p_succ≈0.609\n", "F_in=0.70 F_out≈0.735 p_succ≈0.680\n", "F_in=0.80 F_out≈0.838 p_succ≈0.769\n", "F_in=0.90 F_out≈0.926 p_succ≈0.876\n" ] } ], "source": [ "import numpy as np\n", "def bbpssw_round(F):\n", " # Toy mapping for isotropic states under BBPSSW (not exact; illustrative)\n", " # Successful output fidelity (heuristic):\n", " F_out = (F**2 + ((1-F)/3)**2) / (F**2 + 2*F*(1-F)/3 + 5*((1-F)/3)**2)\n", " p_succ = F**2 + 2*F*(1-F)/3 + 5*((1-F)/3)**2\n", " return F_out, p_succ\n", "for F in [0.5,0.6,0.7,0.8,0.9]:\n", " Fo, ps = bbpssw_round(F)\n", " print(f\"F_in={F:.2f} F_out≈{Fo:.3f} p_succ≈{ps:.3f}\")" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "Bell States Density Matrices:\n", "Phi+:\n", " DensityMatrix([[0.5+0.j, 0. +0.j, 0. +0.j, 0.5+0.j],\n", " [0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n", " [0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n", " [0.5+0.j, 0. +0.j, 0. +0.j, 0.5+0.j]],\n", " dims=(2, 2))\n", "Phi-:\n", " DensityMatrix([[ 0.5+0.j, 0. +0.j, 0. +0.j, -0.5+0.j],\n", " [ 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n", " [ 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n", " [-0.5+0.j, 0. +0.j, 0. +0.j, 0.5+0.j]],\n", " dims=(2, 2))\n", "Psi+:\n", " DensityMatrix([[0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n", " [0. +0.j, 0.5+0.j, 0.5+0.j, 0. +0.j],\n", " [0. +0.j, 0.5+0.j, 0.5+0.j, 0. +0.j],\n", " [0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j]],\n", " dims=(2, 2))\n", "Psi-:\n", " DensityMatrix([[ 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n", " [ 0. +0.j, 0.5+0.j, -0.5+0.j, 0. +0.j],\n", " [ 0. +0.j, -0.5+0.j, 0.5+0.j, 0. +0.j],\n", " [ 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j]],\n", " dims=(2, 2))\n" ] } ], "source": [ "import numpy as np\n", "from qiskit import QuantumCircuit\n", "from qiskit.quantum_info import Statevector, DensityMatrix, partial_trace\n", "\n", "qc_phip = QuantumCircuit(2)\n", "qc_phip.h(0)\n", "qc_phip.cx(0, 1)\n", "rho_phip = DensityMatrix.from_instruction(qc_phip)\n", "\n", "qc_phim = QuantumCircuit(2)\n", "qc_phim.x(0)\n", "qc_phim.h(0)\n", "qc_phim.cx(0, 1)\n", "rho_phim = DensityMatrix.from_instruction(qc_phim)\n", "\n", "\n", "qc_psip = QuantumCircuit(2)\n", "qc_psip.x(1)\n", "qc_psip.h(0)\n", "qc_psip.cx(0, 1)\n", "rho_psip = DensityMatrix.from_instruction(qc_psip)\n", "\n", "\n", "qc_psim = QuantumCircuit(2)\n", "qc_psim.x(0)\n", "qc_psim.x(1)\n", "qc_psim.h(0)\n", "qc_psim.cx(0, 1)\n", "rho_psim = DensityMatrix.from_instruction(qc_psim)\n", "\n", "print(\"Bell States Density Matrices:\")\n", "print(\"Phi+:\\n\", rho_phip)\n", "print(\"Phi-:\\n\", rho_phim)\n", "print(\"Psi+:\\n\", rho_psip)\n", "print(\"Psi-:\\n\", rho_psim)\n" ] }, { "cell_type": "code", "execution_count": 5, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "F_in=0.50 F_out=0.900\n", "F_in=0.60 F_out=0.953\n", "F_in=0.70 F_out=0.980\n", "F_in=0.80 F_out=0.993\n", "F_in=0.90 F_out=0.999\n" ] } ], "source": [ "rho_id = DensityMatrix(np.eye(4) / 4)\n", "\n", "def isotropic_state(F):\n", "\t\"\"\"Create an isotropic state with fidelity F to |Φ+>.\"\"\"\n", "\treturn F * rho_phip + (1 - F) / 3 * (rho_phim + rho_psip + rho_psim)\n", "\n", "def bbpssw_circuit():\n", "\t\"\"\"Construct the BBPSSW purification circuit.\"\"\"\n", "\tcirc = QuantumCircuit(4, 2)\n", "\t# CNOTs\n", "\tcirc.cx(0, 2)\n", "\tcirc.cx(1, 3)\n", "\treturn circ\n", "\n", "def simulate_bbpssw(F):\n", "\t\"\"\"Simulate one round of BBPSSW purification on isotropic states.\"\"\"\n", "\t# Prepare initial state\n", "\trho1 = isotropic_state(F)\n", "\trho2 = isotropic_state(F)\n", "\trho_in = rho1.tensor(rho2)\n", "\tproj = rho_id.tensor(rho_phip+rho_phim)\n", "\n", "\t# Create BBPSSW circuit\n", "\tcirc = bbpssw_circuit()\n", "\n", "\t# Apply circuit to the state\n", "\trho_out = DensityMatrix(circ).evolve(rho_in)\n", "\n", "\t# Trace out measured qubits (2 and 3) over only successful outcomes and renormalize\n", "\trho_reduced = partial_trace(DensityMatrix(proj.data @ rho_out.data), [2, 3])\n", "\tnorm = np.trace(rho_reduced.data)\n", "\trho_reduced = DensityMatrix(rho_reduced.data / norm)\n", "\n", "\t# Calculate fidelity with |Φ+>\n", "\tfid = np.real(np.trace(rho_reduced.data @ rho_phip.data))\n", "\n", "\treturn fid, rho_reduced\n", "\n", "for F in [0.5, 0.6, 0.7, 0.8, 0.9]:\n", "\tFo, rho_final = simulate_bbpssw(F)\n", "\tprint(f\"F_in={F:.2f} F_out={Fo:.3f}\")" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": ".venv", "language": "python", "name": "python3" }, "language_info": { "codemirror_mode": { "name": "ipython", "version": 3 }, "file_extension": ".py", "mimetype": "text/x-python", "name": "python", "nbconvert_exporter": "python", "pygments_lexer": "ipython3", "version": "3.12.8" } }, "nbformat": 4, "nbformat_minor": 2 }