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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   "execution_count": 16,
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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 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 *;\n",
    "np.random.seed(7098123); # for repeatability"
   ]
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   "execution_count": 17,
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   "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;"
   ]
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   "execution_count": 18,
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      "\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        -1.000000                         True\n",
      " 2        -1.000000                         True\n",
      " 3        -1.000000                         True\n",
      " 4        -1.000000                         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, 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": 19,
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     "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]         0.207789           True\n",
      "   [1, 3]         0.179513           True\n",
      "   [1, 2]         0.179239           True\n",
      "   [0, 3]         0.315586           True\n",
      "   [0, 2]         0.205222           True\n",
      "   [0, 1]         0.327619           True\n",
      " [1, 2, 3]        0.049226           True\n",
      " [0, 2, 3]        0.088339           True\n",
      " [0, 1, 3]        0.098730           True\n",
      " [0, 1, 2]        0.081731           True\n",
      "[0, 1, 2, 3]     -0.133914           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",
    "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]))\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": 20,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
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     "text": [
      "\n",
      "β_T = min_K min σ(S_{T,K}) = -0.133914\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.');"
   ]
  }
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