{ "cells": [ { "cell_type": "markdown", "metadata": {}, "source": [ "# Examples - non-dilatable $d$-parameter $C_{0}$-semigroups #\n", "\n", "This notebook provides supplementary material to the research paper .\n", "\n", "In §5.3 a construction is given which yields for any $d \\geq 2$\n", "a $d$-parameter $C_{0}$-semigroup, $T$ satisfying:\n", "\n", "- $T$ is contractive (equivalently its marginals $T_{i}$ are for each $i$);\n", "- the generator $A_{i}$ of $T_{i}$ has strictly negative spectral bound for each $i$;\n", "\n", "and such that $T$ is __not__ **completely dissipative**.\n", "Thus by the classification Theorem (Thm 1.1), $T$ does not have a regular unitary dilation.\n", "\n", "This Notebook demonstrates this general result empirically." ] }, { "cell_type": "code", "execution_count": 12, "metadata": {}, "outputs": [], "source": [ "import os;\n", "import sys;\n", "\n", "# NOTE: need this to force jupyter to reload imports:\n", "for key in list(sys.modules.keys()):\n", " if key.startswith('src.'):\n", " del sys.modules[key];\n", "\n", "os.chdir(os.path.dirname(_dh[0]));\n", "sys.path.insert(0, os.getcwd());\n", "\n", "from src.examples_dilations import *;" ] }, { "cell_type": "code", "execution_count": 13, "metadata": {}, "outputs": [], "source": [ "# User input:\n", "N = 4; # dimension of the Hilbert space.\n", "d = 4;\n", "\n", "# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n", "# + you want S_TK > 0 (strictly) for all K ≠ {1,2,...,d}:\n", "alpha = 1/math.sqrt(d - 0.5);\n", "\n", "# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n", "# alpha = 1/math.sqrt(d - 1);\n", "\n", "# Otherwise:\n", "# alpha = 1;" ] }, { "cell_type": "code", "execution_count": 14, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "The marginal semigroups T_i and their generators A_i:\n", "\n", " i spec bound of A_i A_i dissipative (<==> T_i contractive)?\n", "--- ------------------- -----------------------------------------\n", " 1 -0.465478 True\n", " 2 -0.465478 True\n", " 3 -0.465478 True\n", " 4 -0.465478 True\n" ] } ], "source": [ "# create the generators `A_i` of the marginal semigroups `T_i`:`\n", "A = [\n", " generate_semigroup_generator(\n", " shape = [N, N],\n", " rational = True,\n", " base = 100,\n", " alpha = alpha,\n", " )\n", " for _ in range(d)\n", "];\n", "\n", "data = [];\n", "for i, A_i in enumerate(A):\n", " omega_Re = spec_bounds((1/2)*(A_i + A_i.T.conj()));\n", " omega = spec_bounds(A_i);\n", " data.append((i+1, omega_Re, True if (omega_Re <= 0) else False));\n", "\n", "repr = tabulate(\n", " tabular_data = data,\n", " headers = ['i', 'spec bound of A_i', 'A_i dissipative (<==> T_i contractive)?'],\n", " showindex = False,\n", " floatfmt = '.6f',\n", " colalign = ['center', 'center', 'center'],\n", " tablefmt = 'simple',\n", ");\n", "print(f'\\nThe marginal semigroups T_i and their generators A_i:\\n\\n{repr}');" ] }, { "cell_type": "code", "execution_count": 15, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "Dissipation operators:\n", "\n", " K min σ(S_{T,K}) S_{T,K} >= 0?\n", "------------ ---------------- ---------------\n", " [] 1.000000 True\n", " [3] 0.465478 True\n", " [2] 0.465478 True\n", " [1] 0.465478 True\n", " [0] 0.465478 True\n", " [2, 3] 0.212675 True\n", " [1, 3] 0.183738 True\n", " [1, 2] 0.217453 True\n", " [0, 3] 0.200206 True\n", " [0, 2] 0.301332 True\n", " [0, 1] 0.215681 True\n", " [1, 2, 3] 0.058427 True\n", " [0, 2, 3] 0.075037 True\n", " [0, 1, 3] 0.056030 True\n", " [0, 1, 2] 0.077350 True\n", "[0, 1, 2, 3] -0.082403 False\n" ] } ], "source": [ "# compute the dissipation operators `S_TK` for each `K ⊆ {1,2,...,d}``:\n", "S, beta_T = dissipation_operators(shape=[N, N], A=A);\n", "\n", "data = [];\n", "for K, S_TK, b in sorted(S, key=lambda x: len(x[0])):\n", " data.append((K, b, True if b >= -MACHINE_EPS else False))\n", "\n", "repr = tabulate(\n", " tabular_data = data,\n", " headers = ['K', 'min σ(S_{T,K})', 'S_{T,K} >= 0?'],\n", " showindex = False,\n", " floatfmt = '.6f',\n", " colalign = ['center', 'center', 'center'],\n", " tablefmt = 'simple',\n", ");\n", "print(f'\\nDissipation operators:\\n\\n{repr}');" ] }, { "cell_type": "code", "execution_count": 16, "metadata": {}, "outputs": [ { "name": "stdout", "output_type": "stream", "text": [ "\n", "β_T = min_K min σ(S_{T,K}) = -0.082403\n", "⟹ T is not compeletely dissipative.\n", "⟹ (by Thm 1.1) T does not have a regular unitary dilation.\n" ] } ], "source": [ "# Display summary:\n", "print('')\n", "print(f'β_T = min_K min σ(S_{{T,K}}) = {beta_T:.6f}');\n", "if beta_T == 0:\n", " print('⟹ T is compeletely dissipative.');\n", " print('⟹ (by Thm 1.1) T has a regular unitary dilation.');\n", "elif beta_T > 0:\n", " print('⟹ T is compeletely super dissipative.');\n", " print('⟹ (by Thm 1.1) T has a regular unitary dilation.');\n", "else:\n", " print('⟹ T is not compeletely dissipative.');\n", " print('⟹ (by Thm 1.1) T does not have a regular unitary dilation.');" ] } ], "metadata": { "jupytext": { "cell_metadata_filter": "-all" }, "kernelspec": { "display_name": "Python 3.10.6 64-bit", "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": "3.10.6" }, "widgets": { "application/vnd.jupyter.widget-state+json": { "state": {}, "version_major": 2, "version_minor": 0 } } }, "nbformat": 4, "nbformat_minor": 4 }