notebooks/docs/examples_dilations.ipynb

Examples - non-dilatable $d$-parameter $C_{0}$-semigroups

This notebook provides supplementary material to the research paper https://doi.org/10.48550/arXiv.2210.02353.

In §5.3 a construction is given which yields for any d \geq 2 a $d$-parameter $C_{0}$-semigroup, T satisfying:

• T is contractive (equivalently its marginals T_{i} are for each i);
• the generator A_{i} of T_{i} has strictly negative spectral bound for each i;

and such that T is not completely dissipative. Thus by the classification Theorem (Thm 1.1), T does not have a regular unitary dilation.

This Notebook demonstrates this general result empirically.

In [16]:

import os;
import sys;

# NOTE: need this to force jupyter to reload imports:
for key in list(sys.modules.keys()):
    if key.startswith('src.'):
        del sys.modules[key];

os.chdir(os.path.dirname(_dh[0]));
sys.path.insert(0, os.getcwd());

from src.examples_dilations import *;
np.random.seed(7098123); # for repeatability

In [17]:

# User input:
N = 4; # dimension of the Hilbert space.
d = 4;

# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}
# + you want S_TK > 0 (strictly) for all K ≠ {1,2,...,d}:
alpha = 1/math.sqrt(d - 0.5);

# If you ensure that the failure of S_{T,K} >= 0 only occurs for K = {1,2,...,d}
# alpha = 1/math.sqrt(d - 1);

# Otherwise:
# alpha = 1;

In [18]:

# create the generators `A_i` of the marginal semigroups `T_i`:`
A = [
    generate_semigroup_generator(
        shape    = [N, N],
        rational = True,
        base     = 100,
        alpha    = alpha,
    )
    for _ in range(d)
];

data = [];
for i, A_i in enumerate(A):
    omega_Re = spec_bounds((1/2)*(A_i + A_i.T.conj()));
    omega = spec_bounds(A_i);
    data.append((i+1, omega, True if (omega_Re <= 0) else False));

repr = tabulate(
    tabular_data = data,
    headers      = ['i', 'spec bound of A_i', 'A_i dissipative (<==> T_i contractive)?'],
    showindex    = False,
    floatfmt = '.6f',
    colalign = ['center', 'center', 'center'],
    tablefmt     = 'simple',
);
print(f'\nThe marginal semigroups T_i and their generators A_i:\n\n{repr}');

The marginal semigroups T_i and their generators A_i:

 i    spec bound of A_i    A_i dissipative (<==> T_i contractive)?
---  -------------------  -----------------------------------------
 1        -1.000000                         True
 2        -1.000000                         True
 3        -1.000000                         True
 4        -1.000000                         True

In [19]:

# compute the dissipation operators `S_TK` for each `K ⊆ {1,2,...,d}``:
S, beta_T = dissipation_operators(shape=[N, N], A=A);

repr = tabulate(
    tabular_data = [
        (K, b, True if b >= -MACHINE_EPS else False)
        for K, S_TK, b in sorted(S, key=lambda x: len(x[0]))
    ],
    headers      = ['K', 'min σ(S_{T,K})', 'S_{T,K} >= 0?'],
    showindex    = False,
    floatfmt = '.6f',
    colalign = ['center', 'center', 'center'],
    tablefmt     = 'simple',
);
print(f'\nDissipation operators:\n\n{repr}');

Dissipation operators:

     K         min σ(S_{T,K})    S_{T,K} >= 0?
------------  ----------------  ---------------
     []           1.000000           True
    [3]           0.465478           True
    [2]           0.465478           True
    [1]           0.465478           True
    [0]           0.465478           True
   [2, 3]         0.207789           True
   [1, 3]         0.179513           True
   [1, 2]         0.179239           True
   [0, 3]         0.315586           True
   [0, 2]         0.205222           True
   [0, 1]         0.327619           True
 [1, 2, 3]        0.049226           True
 [0, 2, 3]        0.088339           True
 [0, 1, 3]        0.098730           True
 [0, 1, 2]        0.081731           True
[0, 1, 2, 3]     -0.133914           False

In [20]:

# Display summary:
print('')
print(f'β_T = min_K min σ(S_{{T,K}}) = {beta_T:.6f}');
if beta_T == 0:
    print('⟹ T is compeletely dissipative.');
    print('⟹ (by Thm 1.1) T has a regular unitary dilation.');
elif beta_T > 0:
    print('⟹ T is compeletely super dissipative.');
    print('⟹ (by Thm 1.1) T has a regular unitary dilation.');
else:
    print('⟹ T is not compeletely dissipative.');
    print('⟹ (by Thm 1.1) T does not have a regular unitary dilation.');

β_T = min_K min σ(S_{T,K}) = -0.133914
⟹ T is not compeletely dissipative.
⟹ (by Thm 1.1) T does not have a regular unitary dilation.