This repository contains our computational experiment for testing various instances of density matrix and measurements for violation of C2(b) coherence property. We tested different kinds of norms and this program was used as tool for checking for violations of C2(b).

Install all the requirements using

pip install -r requirements.txt

From there you can perform three kinds of computational experiment. Random sampling, test using precomputed data and meta-heuristic test. Each of those has a dedicated section below.

Numerical test of C2(b) with schatten-1 norm and L1 norm (for debugging) on one million instances.

Run

python rsample.py

Expected result

L = 2, N = 2
100%|███████████████| 100000/100000 [07:10<00:00, 232.41it/s]

total violations so far: 0

L = 2, N = 4
100%|███████████████| 100000/100000 [08:38<00:00, 192.94it/s]

total violations so far: 0

L = 4, N = 2
100%|██████████████████████████████████| 100000/100000 [14:13<00:00, 117.11it/s]

total violations so far: 0

L = 4, N = 4
100%|███████████████████████████████████| 100000/100000 [18:10<00:00, 91.72it/s]

total violations so far: 0

L = 8, N = 2
100%|███████████████████████████████████| 100000/100000 [30:34<00:00, 54.51it/s]

total violations so far: 0

L = 8, N = 4
100%|███████████████████████████████████| 100000/100000 [42:40<00:00, 39.06it/s]

total violations so far: 0

L = 16, N = 2
100%|█████████████████████████████████| 100000/100000 [1:19:11<00:00, 21.05it/s]

total violations so far: 0

L = 16, N = 4
100%|█████████████████████████████████| 100000/100000 [1:59:23<00:00, 13.96it/s]

total violations so far: 0

L = 32, N = 2
100%|█████████████████████████████████| 100000/100000 [3:53:51<00:00,  7.13it/s]

total violations so far: 0

L = 32, N = 4
100%|█████████████████████████████████| 100000/100000 [6:16:29<00:00,  4.43it/s]

total violations so far: 0


total number of instances: 1000000


total number of violated instances: 0


number of violating instances per test:

+--- norm ----+-- Pn / P --+-- B + iBd --+
| L01         |          0 |           0 |
| Schatten-01 |          0 |           0 |
+-------------+------------+-------------+

At the moment of development of this computational experiment qutip library doesn't give control of randomness, we cannot pass our own seed as parameter thus it is harder to reproduce results. As a dirty solution we implemented a script that pre-generates problem instances and stores them in files. You can feel free to generate as many as you need, during the test instances are streamed from files avoiding wasting memory.

We pre-generated some instances, look up instances directory.

Generate set of random density matrices

python density_gen.py

Generate random measurements

python x_gen.py

Once set of instances is generated you can run the test script

python fsample.py

Output is compatible with one from random sampling

In order to avoid relying entirely on random sampling we also implemented a meta-heuristic approach based on simulated annealing.

The way it works is initially it random samples for density matrix for which C2(b) is at equality, and then it uses simulated annealing to explore local neighborhood of solutions to search for violation. Simulated annealing part is based on simanneal module

Reason why we decided to implement it that way was simulated annealing performed well locally, solution was improving up to 30%, however local search was not aggressive enough to lift C2(b) to equality, even with extreme initial temperatures.

Run

python anneal.py

Output

random sampling for density matrix

0.020981039448955685
0.014407357024333434
0.013948015156652283
0.001910239006014322
0.0005186086142509811
0.0

found tightest C2(b) bound which is approximately 0.
beginning simulated annealing to find measurement violating C2(b) as local optimum

 Temperature        Energy    Accept   Improve     Elapsed   Remaining
     2.50000          0.00   100.00%     0.00%     0:01:21     0:00:00


no C2(b) violation found