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https://github.com/quantumjim/Quantum-Computation-course-Basel.git
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Added entanglement lecture applications notebooks
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PFmix
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extra_resources/Entanglement_extra/12_Purification.ipynb
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extra_resources/Entanglement_extra/12_Purification.ipynb
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{
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"cells": [
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"'\\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'"
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]
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},
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"execution_count": 1,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"\"\"\"\n",
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"### Overview\n",
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"Simulate a simple entanglement purification step (BBPSSW-like) on isotropic states and track fidelity gains.\n",
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"\n",
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"Exercises:\n",
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"- Chain multiple rounds and observe convergence threshold behavior.\n",
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"\"\"\""
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]
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"F_in=0.50 F_out≈0.500 p_succ≈0.556\n",
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"F_in=0.60 F_out≈0.620 p_succ≈0.609\n",
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"F_in=0.70 F_out≈0.735 p_succ≈0.680\n",
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"F_in=0.80 F_out≈0.838 p_succ≈0.769\n",
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"F_in=0.90 F_out≈0.926 p_succ≈0.876\n"
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]
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}
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],
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"source": [
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"import numpy as np\n",
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"def bbpssw_round(F):\n",
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" # Toy mapping for isotropic states under BBPSSW (not exact; illustrative)\n",
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" # Successful output fidelity (heuristic):\n",
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" F_out = (F**2 + ((1-F)/3)**2) / (F**2 + 2*F*(1-F)/3 + 5*((1-F)/3)**2)\n",
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" p_succ = F**2 + 2*F*(1-F)/3 + 5*((1-F)/3)**2\n",
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" return F_out, p_succ\n",
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"for F in [0.5,0.6,0.7,0.8,0.9]:\n",
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" Fo, ps = bbpssw_round(F)\n",
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" print(f\"F_in={F:.2f} F_out≈{Fo:.3f} p_succ≈{ps:.3f}\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"Bell States Density Matrices:\n",
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"Phi+:\n",
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" DensityMatrix([[0.5+0.j, 0. +0.j, 0. +0.j, 0.5+0.j],\n",
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" [0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n",
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" [0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n",
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" [0.5+0.j, 0. +0.j, 0. +0.j, 0.5+0.j]],\n",
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" dims=(2, 2))\n",
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"Phi-:\n",
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" DensityMatrix([[ 0.5+0.j, 0. +0.j, 0. +0.j, -0.5+0.j],\n",
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" [ 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n",
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" [ 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n",
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" [-0.5+0.j, 0. +0.j, 0. +0.j, 0.5+0.j]],\n",
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" dims=(2, 2))\n",
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"Psi+:\n",
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" DensityMatrix([[0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n",
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" [0. +0.j, 0.5+0.j, 0.5+0.j, 0. +0.j],\n",
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" [0. +0.j, 0.5+0.j, 0.5+0.j, 0. +0.j],\n",
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" [0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j]],\n",
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" dims=(2, 2))\n",
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"Psi-:\n",
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" DensityMatrix([[ 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j],\n",
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" [ 0. +0.j, 0.5+0.j, -0.5+0.j, 0. +0.j],\n",
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" [ 0. +0.j, -0.5+0.j, 0.5+0.j, 0. +0.j],\n",
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" [ 0. +0.j, 0. +0.j, 0. +0.j, 0. +0.j]],\n",
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" dims=(2, 2))\n"
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]
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}
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],
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"source": [
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"import numpy as np\n",
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"from qiskit import QuantumCircuit\n",
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"from qiskit.quantum_info import Statevector, DensityMatrix, partial_trace\n",
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"\n",
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"qc_phip = QuantumCircuit(2)\n",
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"qc_phip.h(0)\n",
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"qc_phip.cx(0, 1)\n",
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"rho_phip = DensityMatrix.from_instruction(qc_phip)\n",
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"\n",
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"qc_phim = QuantumCircuit(2)\n",
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"qc_phim.x(0)\n",
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"qc_phim.h(0)\n",
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"qc_phim.cx(0, 1)\n",
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"rho_phim = DensityMatrix.from_instruction(qc_phim)\n",
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"\n",
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"\n",
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"qc_psip = QuantumCircuit(2)\n",
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"qc_psip.x(1)\n",
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"qc_psip.h(0)\n",
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"qc_psip.cx(0, 1)\n",
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"rho_psip = DensityMatrix.from_instruction(qc_psip)\n",
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"\n",
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"\n",
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"qc_psim = QuantumCircuit(2)\n",
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"qc_psim.x(0)\n",
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"qc_psim.x(1)\n",
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"qc_psim.h(0)\n",
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"qc_psim.cx(0, 1)\n",
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"rho_psim = DensityMatrix.from_instruction(qc_psim)\n",
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"\n",
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"print(\"Bell States Density Matrices:\")\n",
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"print(\"Phi+:\\n\", rho_phip)\n",
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"print(\"Phi-:\\n\", rho_phim)\n",
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"print(\"Psi+:\\n\", rho_psip)\n",
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"print(\"Psi-:\\n\", rho_psim)\n"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"metadata": {},
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"outputs": [
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{
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"name": "stdout",
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"output_type": "stream",
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"text": [
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"F_in=0.50 F_out=0.900\n",
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"F_in=0.60 F_out=0.953\n",
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"F_in=0.70 F_out=0.980\n",
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"F_in=0.80 F_out=0.993\n",
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"F_in=0.90 F_out=0.999\n"
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]
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}
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],
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"source": [
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"rho_id = DensityMatrix(np.eye(4) / 4)\n",
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"\n",
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"def isotropic_state(F):\n",
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"\t\"\"\"Create an isotropic state with fidelity F to |Φ+>.\"\"\"\n",
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"\treturn F * rho_phip + (1 - F) / 3 * (rho_phim + rho_psip + rho_psim)\n",
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"\n",
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"def bbpssw_circuit():\n",
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"\t\"\"\"Construct the BBPSSW purification circuit.\"\"\"\n",
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"\tcirc = QuantumCircuit(4, 2)\n",
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"\t# CNOTs\n",
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"\tcirc.cx(0, 2)\n",
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"\tcirc.cx(1, 3)\n",
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"\treturn circ\n",
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"\n",
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"def simulate_bbpssw(F):\n",
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"\t\"\"\"Simulate one round of BBPSSW purification on isotropic states.\"\"\"\n",
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"\t# Prepare initial state\n",
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"\trho1 = isotropic_state(F)\n",
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"\trho2 = isotropic_state(F)\n",
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"\trho_in = rho1.tensor(rho2)\n",
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"\tproj = rho_id.tensor(rho_phip+rho_phim)\n",
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"\n",
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"\t# Create BBPSSW circuit\n",
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"\tcirc = bbpssw_circuit()\n",
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"\n",
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"\t# Apply circuit to the state\n",
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"\trho_out = DensityMatrix(circ).evolve(rho_in)\n",
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"\n",
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"\t# Trace out measured qubits (2 and 3) over only successful outcomes and renormalize\n",
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"\trho_reduced = partial_trace(DensityMatrix(proj.data @ rho_out.data), [2, 3])\n",
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"\tnorm = np.trace(rho_reduced.data)\n",
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"\trho_reduced = DensityMatrix(rho_reduced.data / norm)\n",
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"\n",
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"\t# Calculate fidelity with |Φ+>\n",
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"\tfid = np.real(np.trace(rho_reduced.data @ rho_phip.data))\n",
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"\n",
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"\treturn fid, rho_reduced\n",
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"\n",
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"for F in [0.5, 0.6, 0.7, 0.8, 0.9]:\n",
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"\tFo, rho_final = simulate_bbpssw(F)\n",
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"\tprint(f\"F_in={F:.2f} F_out={Fo:.3f}\")"
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]
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},
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{
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"cell_type": "code",
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"execution_count": null,
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"metadata": {},
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"outputs": [],
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"source": []
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}
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],
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"metadata": {
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"kernelspec": {
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"display_name": ".venv",
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"language": "python",
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"name": "python3"
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},
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"language_info": {
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"codemirror_mode": {
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"name": "ipython",
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"version": 3
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},
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"file_extension": ".py",
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"mimetype": "text/x-python",
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"name": "python",
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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"version": "3.12.8"
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}
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},
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"nbformat": 4,
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"nbformat_minor": 2
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}
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