main > main: presentation for paper-dilation
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docs/examples_dilations.ipynb
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docs/examples_dilations.ipynb
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{
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"cells": [
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"# Examples - non-dilatable $d$-parameter $C_{0}$-semigroups #\n",
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"\n",
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"This notebook provides supplementary material to the research paper <https://doi.org/10.48550/arXiv.2210.02353>.\n",
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"\n",
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"In §5.3 a construction is given which yields for any $d \\geq 2$\n",
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"a $d$-parameter $C_{0}$-semigroup, $T$ satisfying:\n",
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"\n",
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"- $T$ is contractive (equivalently its marginals $T_{i}$ are for each $i$);\n",
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"- the generator $A_{i}$ of $T_{i}$ has strictly negative spectral bound for each $i$;\n",
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"\n",
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"and such that $T$ is __not__ **completely dissipative**.\n",
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"Thus by the classification Theorem (Thm 1.1), $T$ does not have a regular unitary dilation.\n",
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"\n",
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"This Notebook demonstrates this general result empirically."
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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": 12,
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"metadata": {},
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"outputs": [],
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"source": [
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"import os;\n",
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"import sys;\n",
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"\n",
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"# NOTE: need this to force jupyter to reload imports:\n",
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"for key in list(sys.modules.keys()):\n",
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" if key.startswith('src.'):\n",
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" del sys.modules[key];\n",
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"\n",
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"os.chdir(os.path.dirname(_dh[0]));\n",
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"sys.path.insert(0, os.getcwd());\n",
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"\n",
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"from src.examples_dilations import *;"
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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": 13,
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"metadata": {},
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"outputs": [],
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"source": [
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"# User input:\n",
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"N = 4; # dimension of the Hilbert space.\n",
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"d = 4;\n",
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"\n",
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"# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n",
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"# + you want S_TK > 0 (strictly) for all K ≠ {1,2,...,d}:\n",
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"alpha = 1/math.sqrt(d - 0.5);\n",
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"\n",
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"# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n",
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"# alpha = 1/math.sqrt(d - 1);\n",
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"\n",
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"# Otherwise:\n",
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"# alpha = 1;"
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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": 14,
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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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"\n",
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"The marginal semigroups T_i and their generators A_i:\n",
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"\n",
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" i spec bound of A_i A_i dissipative (<==> T_i contractive)?\n",
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"--- ------------------- -----------------------------------------\n",
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" 1 -0.465478 True\n",
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" 2 -0.465478 True\n",
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" 3 -0.465478 True\n",
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" 4 -0.465478 True\n"
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]
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}
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],
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"source": [
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"# create the generators `A_i` of the marginal semigroups `T_i`:`\n",
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"A = [\n",
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" generate_semigroup_generator(\n",
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" shape = [N, N],\n",
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" rational = True,\n",
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" base = 100,\n",
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" alpha = alpha,\n",
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" )\n",
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" for _ in range(d)\n",
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"];\n",
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"\n",
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"data = [];\n",
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"for i, A_i in enumerate(A):\n",
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" omega_Re = spec_bounds((1/2)*(A_i + A_i.T.conj()));\n",
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" omega = spec_bounds(A_i);\n",
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" data.append((i+1, omega_Re, True if (omega_Re <= 0) else False));\n",
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"\n",
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"repr = tabulate(\n",
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" tabular_data = data,\n",
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" headers = ['i', 'spec bound of A_i', 'A_i dissipative (<==> T_i contractive)?'],\n",
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" showindex = False,\n",
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" floatfmt = '.6f',\n",
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" colalign = ['center', 'center', 'center'],\n",
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" tablefmt = 'simple',\n",
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");\n",
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"print(f'\\nThe marginal semigroups T_i and their generators A_i:\\n\\n{repr}');"
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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": 15,
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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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"\n",
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"Dissipation operators:\n",
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"\n",
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" K min σ(S_{T,K}) S_{T,K} >= 0?\n",
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"------------ ---------------- ---------------\n",
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" [] 1.000000 True\n",
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" [3] 0.465478 True\n",
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" [2] 0.465478 True\n",
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" [1] 0.465478 True\n",
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" [0] 0.465478 True\n",
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" [2, 3] 0.212675 True\n",
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" [1, 3] 0.183738 True\n",
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" [1, 2] 0.217453 True\n",
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" [0, 3] 0.200206 True\n",
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" [0, 2] 0.301332 True\n",
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" [0, 1] 0.215681 True\n",
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" [1, 2, 3] 0.058427 True\n",
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" [0, 2, 3] 0.075037 True\n",
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" [0, 1, 3] 0.056030 True\n",
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" [0, 1, 2] 0.077350 True\n",
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"[0, 1, 2, 3] -0.082403 False\n"
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]
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}
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],
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"source": [
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"# compute the dissipation operators `S_TK` for each `K ⊆ {1,2,...,d}``:\n",
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"S, beta_T = dissipation_operators(shape=[N, N], A=A);\n",
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"\n",
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"data = [];\n",
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"for K, S_TK, b in sorted(S, key=lambda x: len(x[0])):\n",
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" data.append((K, b, True if b >= -MACHINE_EPS else False))\n",
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"\n",
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"repr = tabulate(\n",
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" tabular_data = data,\n",
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" headers = ['K', 'min σ(S_{T,K})', 'S_{T,K} >= 0?'],\n",
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" showindex = False,\n",
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" floatfmt = '.6f',\n",
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" colalign = ['center', 'center', 'center'],\n",
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" tablefmt = 'simple',\n",
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");\n",
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"print(f'\\nDissipation operators:\\n\\n{repr}');"
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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": 16,
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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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"\n",
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"β_T = min_K min σ(S_{T,K}) = -0.082403\n",
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"⟹ T is not compeletely dissipative.\n",
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"⟹ (by Thm 1.1) T does not have a regular unitary dilation.\n"
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]
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}
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],
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"source": [
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"# Display summary:\n",
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"print('')\n",
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"print(f'β_T = min_K min σ(S_{{T,K}}) = {beta_T:.6f}');\n",
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"if beta_T == 0:\n",
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" print('⟹ T is compeletely dissipative.');\n",
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" print('⟹ (by Thm 1.1) T has a regular unitary dilation.');\n",
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"elif beta_T > 0:\n",
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" print('⟹ T is compeletely super dissipative.');\n",
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" print('⟹ (by Thm 1.1) T has a regular unitary dilation.');\n",
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"else:\n",
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" print('⟹ T is not compeletely dissipative.');\n",
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" print('⟹ (by Thm 1.1) T does not have a regular unitary dilation.');"
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]
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}
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],
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"metadata": {
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"jupytext": {
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"cell_metadata_filter": "-all"
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},
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"kernelspec": {
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"display_name": "Python 3.10.6 64-bit",
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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": "3.10.6"
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},
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"widgets": {
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"application/vnd.jupyter.widget-state+json": {
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"state": {},
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"version_major": 2,
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"version_minor": 0
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},
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"nbformat": 4,
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"nbformat_minor": 4
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}
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0
src/__init__.py
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0
src/__init__.py
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288
src/examples_dilations.py
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src/examples_dilations.py
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#!/usr/bin/env python3
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# -*- coding: utf-8 -*-
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# AUTHOR: Raj Dahya
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# CREATED: 05.09.2022
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# EDITED: 08.10.2022
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# TYPE: Source code
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# DEPARTMENT: Fakult\"at for Mathematik und Informatik
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# INSTITUTE: Universit\"at Leipzig
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# NOTE: Collapse this block.
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# IMPORTS
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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import re;
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import math
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from tkinter import E;
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import numpy as np;
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from typing import Generator;
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import itertools;
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from tabulate import tabulate;
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# CONSTANTS + SETTINGS
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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np.set_printoptions(precision=3);
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MACHINE_EPS = 0.5e-12;
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# SECONDARY METHODS - Generate C0-Semigroup Generator
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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def generate_semigroup_generator(
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shape: list[int, int],
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rational: bool = False,
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base: int = 2,
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alpha: float = 1,
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) -> np.ndarray:
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'''
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Creates bounded generators for d commuting C0-semigroups as in the paper (cf. §5.3).
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@inputs
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- `shape` - <int, int> Desired shape of output.
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- `rational` - <bool> Whether or not entries are to be rational.
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- `base` - <int> If `rational = True`, fixes the denominator of the rational numbers.
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- `alpha` - <float> Additional parameter to scale the D_i operators.
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NOTE: in the paper `α = 1` was chosen. However one can choose any value in `(1/√d, \infty)`.
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By choosing any value `α ≥ 1/√(d-1)`, by the computations in Proposition 5.3
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one can force that the S_TK operators only fail to be positive when |K| > d-1.
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@returns
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Randomly created bounded generators for C0-semigroups as in the paper (cf. §5.3).
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'''
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sh = tuple(shape)
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m, n = sh;
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assert m == n, 'must be square';
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assert m % 2 == 0, 'dimensions must be divisible by 2';
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N = int(m/2);
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I = generate_op_identity(shape=[2*N, 2*N]);
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O = generate_op_zero(shape=[N, N]);
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U = generate_op_unitary(
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shape = [N, N],
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rational = rational,
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base = base,
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);
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D = alpha * U;
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A = -I + np.vstack([
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np.hstack([ O, -2*D ]),
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np.hstack([ O, O ]),
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]);
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return A;
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# SECONDARY METHODS - Compute spectral bounds
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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def spec_bounds(A: np.ndarray) -> float:
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'''
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Computes the spectral bounds of an operator:
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```tex
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\max_{i} \max \{ Re \lambda : \lambda \in \sigma(A_{i}) \}
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```
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'''
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sigma_A = np.linalg.eigvals(A);
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return max(np.real(sigma_A));
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# SECONDARY METHODS - Generate Dissipation Operator
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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def dissipation_operators(
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shape: list[int, int],
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A: list[np.ndarray],
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) -> tuple[
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list[tuple[
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list[int],
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np.ndarray,
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float,
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]],
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float,
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]:
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'''
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Computes the dissipation operators and their minimum spectral values.
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'''
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d = len(A);
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indexes = list(range(d));
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data = [];
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beta = np.inf;
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for K in power_set(indexes):
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S_TK, eig_K = dissipation_operator(shape=shape, A=A, K=K);
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data.append((K, S_TK, eig_K));
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beta = min(eig_K, beta);
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return data, beta;
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def dissipation_operator(
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shape: list[int, int],
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A: list[np.ndarray],
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K: list[int],
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) -> tuple[np.ndarray, float]:
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'''
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Computes the dissipation operator `S_{T,K}` and its minimum spectral values.
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'''
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sh = tuple(shape)
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m, n = sh;
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assert m == n, 'must be square';
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N = m;
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O = generate_op_zero(shape=[N, N]);
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I = generate_op_identity(shape=[N, N]);
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S = O;
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for C, C_ in partitions_two(K):
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A_C = I;
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A_C_ = I;
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for i in C:
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A_C = A_C @ -A[i];
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for i in C_:
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A_C_ = A_C_ @ -A[i];
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S += A_C.conj().T @ A_C_;
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S = (1/2**len(K)) * S;
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eig = min(np.real(np.linalg.eigvals(S)));
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return S, eig;
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# MISCELLANEOUS METHODS - SETS
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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def power_set(K: list[int]) -> Generator[list[int], None, None]:
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'''
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Iterates through all subsets `C` of a set `K`.
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'''
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n = len(K);
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for ch in itertools.product(*([[0,1]]*n)):
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yield [ K[i] for i in range(n) if ch[i] == 1 ];
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def partitions_two(K: list[int]) -> Generator[tuple[list[int],list[int]], None, None]:
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'''
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Iterates through all partitions `[C1, C2]` of a set `K` of size 2.
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'''
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n = len(K);
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for ch in itertools.product(*([[0,1]]*n)):
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yield (
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[ K[i] for i in range(n) if ch[i] == 1 ],
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[ K[i] for i in range(n) if ch[i] == 0 ],
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);
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# MISCELLANEOUS METHODS - BASIC OPERATORS
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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def generate_op_zero(shape: list[int, int]) -> np.ndarray:
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'''
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Returns the zero operator with the right shape and type.
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'''
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sh = tuple(shape)
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O = np.zeros(sh).astype(complex);
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return O;
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def generate_op_identity(shape: list[int, int]) -> np.ndarray:
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'''
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Returns the identity operator with the right shape and type.
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'''
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sh = tuple(shape)
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m, n = sh;
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assert m == n, 'must be square';
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I = np.eye(m).astype(complex);
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return I;
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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# MISCELLANEOUS METHODS - GENERATE RANDOM MATRICES
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# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
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def generate_op_selfadjoint(
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shape: list[int, int],
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bound: float = 1.,
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) -> np.ndarray:
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'''
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Generates a self-adjoint operator.
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@inputs
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- `shape` - <int, int> Desired shape of the output.
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- `bound` - <float> The desired norm of output operator.
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@returns
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A randomly generated self-adjoint operator as a matrix.
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'''
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sh = tuple(shape)
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m, n = sh;
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assert m == n, 'must be square';
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A = np.random.random(sh) + 1j * np.random.random(sh);
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S = (A + A.conj().T)/2.;
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M = np.linalg.norm(S);
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if M > 0:
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S = (bound/M) * S;
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return S;
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def generate_onb(shape: list[int, int]) -> np.ndarray:
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'''
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Generates an ONB.
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@inputs
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- `shape` - <int, int> Desired shape of matrices.
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@returns
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An ONB for an n-dim complex Hilbert space as a matrix.
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'''
|
||||
sh = tuple(shape)
|
||||
m, n = sh;
|
||||
assert m == n, 'must be square';
|
||||
N = m;
|
||||
O = generate_op_zero(shape=[N, N]);
|
||||
P = O;
|
||||
for k in range(N):
|
||||
while True:
|
||||
v = np.random.random((N,)) + 1j * np.random.random((N,));
|
||||
for i in range(k):
|
||||
# NOTE: np.inner does not work.
|
||||
coeff = np.sum([ v[j] * P[j, i].conj() for j in range(N)]);
|
||||
v -= coeff * P[:, i];
|
||||
norm_v = np.linalg.norm(v);
|
||||
if norm_v > 0.:
|
||||
P[:, k] = v / norm_v;
|
||||
break;
|
||||
return P;
|
||||
|
||||
def generate_op_unitary(
|
||||
shape: list[int, int],
|
||||
rational: bool = False,
|
||||
base: int = 2,
|
||||
) -> np.ndarray:
|
||||
'''
|
||||
Generates a unitary operator.
|
||||
|
||||
@inputs
|
||||
- `shape` - <int, int> Desired shape of output.
|
||||
- `rational` - <bool> Whether or not entries are to be rational.
|
||||
- `base` - <int> If `rational = True`, fixes the denominator of the rational numbers.
|
||||
|
||||
@returns
|
||||
A randomly generated unitary operator as a matrix.
|
||||
'''
|
||||
sh = tuple(shape)
|
||||
m, n = sh;
|
||||
assert m == n, 'must be square';
|
||||
N = m;
|
||||
P = generate_onb(shape=shape);
|
||||
if rational:
|
||||
theta = 2 * np.pi * np.random.randint(low=0, high=base, size=N) / base;
|
||||
else:
|
||||
theta = 2 * np.pi * np.random.random(size=N);
|
||||
D = np.diag(np.exp(1j * theta));
|
||||
U = P @ D @ P.conj().T;
|
||||
return U;
|
||||
|
||||
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
|
||||
# MISCELLANEOUS METHODS - Get user input
|
||||
# ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
|
||||
|
||||
def ask_confirmation(message: str):
|
||||
while True:
|
||||
answer = input(f'{message} (y/n): ');
|
||||
if re.match(pattern='^y(es)?|1$', string=answer, flags=re.IGNORECASE):
|
||||
return True;
|
||||
elif re.match(pattern='^n(o)?|0$', string=answer, flags=re.IGNORECASE):
|
||||
return False;
|
||||
Loading…
x
Reference in New Issue
Block a user