master > master: updated comments and user input
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@ -30,8 +30,7 @@
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"import sys;\n",
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"import sys;\n",
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"\n",
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"\n",
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"# NOTE: need this to force jupyter to reload imports:\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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"for key in filter(lambda key: key.startswith('src.'), 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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" del sys.modules[key];\n",
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"\n",
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"\n",
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"os.chdir(os.path.dirname(_dh[0]));\n",
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"os.chdir(os.path.dirname(_dh[0]));\n",
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@ -48,18 +47,12 @@
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"outputs": [],
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"outputs": [],
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"source": [
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"source": [
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"# User input:\n",
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"# User input:\n",
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"N = 4; # dimension of the Hilbert space.\n",
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"N = 6; # dimension of the Hilbert space; must be divisible by 2 for this construction.\n",
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"d = 4;\n",
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"d = 4;\n",
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"\n",
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"# force k-th order dissipation operators to be strictly postive for k < k0\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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"# force k-th order dissipation operators to be non-positive for k >= k0\n",
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"# + you want S_TK > 0 (strictly) for all K ≠ {1,2,...,d}:\n",
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"# one can choose 2 <= k0 <= d\n",
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"alpha = 1/math.sqrt(d - 0.5);\n",
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"k0 = d;"
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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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},
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{
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{
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@ -69,6 +62,7 @@
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"outputs": [],
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"outputs": [],
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"source": [
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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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"# create the generators `A_i` of the marginal semigroups `T_i`:`\n",
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"alpha = 1/math.sqrt(k0 - 0.5);\n",
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"A = [\n",
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"A = [\n",
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" generate_semigroup_generator(\n",
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" generate_semigroup_generator(\n",
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" shape = [N, N],\n",
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" shape = [N, N],\n",
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@ -46,7 +46,7 @@ def generate_semigroup_generator(
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- `base` - <int> If `rational = True`, fixes the denominator of the rational numbers.
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- `base` - <int> If `rational = True`, fixes the denominator of the rational numbers.
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- `alpha` - <float> Additional parameter to scale the D_i operators.
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- `alpha` - <float> Additional parameter to scale the D_i operators.
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NOTE: in the paper `α = 1` was chosen. However one can choose any value in `(1/√d, \infty)`.
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NOTE: One can choose any value of `α ∈ (1/√d, \infty)`.
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By choosing any value `α ≥ 1/√(d-1)`, by the computations in Proposition 5.3
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By choosing any value `α ≥ 1/√(d-1)`, by the computations in Proposition 5.3
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one can force that the S_TK operators only fail to be positive when |K| > d-1.
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one can force that the S_TK operators only fail to be positive when |K| > d-1.
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