notebooks/docs/examples_dilations.ipynb

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    "# Examples - non-dilatable $d$-parameter $C_{0}$-semigroups #\n",
    "\n",
    "This notebook provides supplementary material to the research paper <https://doi.org/10.48550/arXiv.2210.02353>.\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."
   ]
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   "source": [
    "import os;\n",
    "import sys;\n",
    "\n",
    "# NOTE: need this to force jupyter to reload imports:\n",
    "for key in filter(lambda key: key.startswith('src.'), list(sys.modules.keys())):\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 *;\n",
    "np.random.seed(7098123); # for repeatability"
   ]
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   "execution_count": null,
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   "source": [
    "# User input:\n",
    "N = 6; # dimension of the Hilbert space; must be divisible by 2 for this construction.\n",
    "d = 4;\n",
    "# force k-th order dissipation operators to be strictly postive for k < k0\n",
    "# force k-th order dissipation operators to be non-positive for k >= k0\n",
    "# one can choose 2 <= k0 <= d\n",
    "k0 = d;"
   ]
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "# create the generators `A_i` of the marginal semigroups `T_i`:`\n",
    "alpha = 1/math.sqrt(k0 - 0.5);\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, 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}');"
   ]
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  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "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",
    "repr = tabulate(\n",
    "    tabular_data = [\n",
    "        (K, b, True if b >= -MACHINE_EPS else False)\n",
    "        for K, S_TK, b in sorted(S, key=lambda x: (len(x[0]), x[0]) )\n",
    "    ],\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": null,
   "metadata": {},
   "outputs": [],
   "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.');"
   ]
  }
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