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master > master: updated comments and user input

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RD 2022-10-12 12:20:58 +02:00
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commit 2d0adf1038
2 changed files with 9 additions and 15 deletions
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20
docs/examples_dilations.ipynb
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@ -30,8 +30,7 @@
"import sys;\n", "import sys;\n",
"\n", "\n",
"# NOTE: need this to force jupyter to reload imports:\n", "# NOTE: need this to force jupyter to reload imports:\n",
"for key in list(sys.modules.keys()):\n", "for key in filter(lambda key: key.startswith('src.'), list(sys.modules.keys())):\n",
" if key.startswith('src.'):\n",
" del sys.modules[key];\n", " del sys.modules[key];\n",
"\n", "\n",
"os.chdir(os.path.dirname(_dh[0]));\n", "os.chdir(os.path.dirname(_dh[0]));\n",
@ -48,18 +47,12 @@
"outputs": [], "outputs": [],
"source": [ "source": [
"# User input:\n", "# User input:\n",
"N = 4; # dimension of the Hilbert space.\n", "N = 6; # dimension of the Hilbert space; must be divisible by 2 for this construction.\n",
"d = 4;\n", "d = 4;\n",
"\n", "# force k-th order dissipation operators to be strictly postive for k < k0\n",
"# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}\n", "# force k-th order dissipation operators to be non-positive for k >= k0\n",
"# + you want S_TK > 0 (strictly) for all K ≠ {1,2,...,d}:\n", "# one can choose 2 <= k0 <= d\n",
"alpha = 1/math.sqrt(d - 0.5);\n", "k0 = d;"
"\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;"
] ]
}, },
{ {
@ -69,6 +62,7 @@
"outputs": [], "outputs": [],
"source": [ "source": [
"# create the generators `A_i` of the marginal semigroups `T_i`:`\n", "# create the generators `A_i` of the marginal semigroups `T_i`:`\n",
"alpha = 1/math.sqrt(k0 - 0.5);\n",
"A = [\n", "A = [\n",
" generate_semigroup_generator(\n", " generate_semigroup_generator(\n",
" shape = [N, N],\n", " shape = [N, N],\n",

2
src/examples_dilations.py
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@ -46,7 +46,7 @@ def generate_semigroup_generator(
- `base` - <int> If `rational = True`, fixes the denominator of the rational numbers. - `base` - <int> If `rational = True`, fixes the denominator of the rational numbers.
- `alpha` - <float> Additional parameter to scale the D_i operators. - `alpha` - <float> Additional parameter to scale the D_i operators.
NOTE: in the paper `α = 1` was chosen. However one can choose any value in `(1/√d, \infty)`. NOTE: One can choose any value of `α ∈ (1/√d, \infty)`.
By choosing any value `α ≥ 1/√(d-1)`, by the computations in Proposition 5.3 By choosing any value `α ≥ 1/√(d-1)`, by the computations in Proposition 5.3
one can force that the S_TK operators only fail to be positive when |K| > d-1. one can force that the S_TK operators only fail to be positive when |K| > d-1.