230 lines
7.0 KiB
Plaintext
230 lines
7.0 KiB
Plaintext
{
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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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"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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"nbconvert_exporter": "python",
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"pygments_lexer": "ipython3",
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"widgets": {
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"nbformat": 4,
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}
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