Incorrect bounds for I3322

[anubhav70747]

Hey,

I tried to obtain an upper bound for the quantum value I3322 inequality introduced in https://journals.aps.org/prl/pdf/10.1103/PhysRevLett.100.210503

The code of the implementation is attached.
TestI3322.py.zip

Now this implementation returns the values with cvxopt and SDPA:
for level = 1: 0.374999
for level = 1 + AB: 0.362010076878
for level = 2: 0.260629

These values are higher than those reported in the paper:
http://iopscience.iop.org/article/10.1088/1367-2630/10/7/073013/meta
(I can also confirm these values using a separate implementation with YALMIP, sdpt3 and Octave.)
for level = 1: 0.3333
for level = 1 + AB: 0.2515
for level = 2: 0.25091

Why do you think this is happening?

Thanks

PS: I also tried to implement it define_objective_with_I(I, P) using I = [[0,-1,0,0],[-2,1,1,1], [-1,1,1,1],[0,1,-1,0]] which leads to completely unrelated results for levels=1,1+AB,2.

[apozas]

Hi,

This is an interesting problem.

First, let me tell you that I have been able to reproduce all the results you mention with the last version of ncpol2sdpa.

Next (the easy part), creating the objective with define_objective_with_I(I, P) gave me the correct objective function by using the transposed of the matrix I that you use.

And now the difficult part. I have exported the level 1 problem to SDPA format, and the problem looks like the following.

"file test.dat-s generated by ncpol2sdpa"
21 = number of vars
1 = number of blocs
(7) = BlocStructure
{1.0, 0, 0, 2.0, 1.0, 0, 0, 0, -1.0, -1.0, -1.0, 0, -1.0, -1.0, 1.0, -1.0, 1.0, 0, 0, 0, 0}
0	1	1	1	-1.0
1	1	1	2	1.0
1	1	2	2	1.0
2	1	1	3	1.0
2	1	3	3	1.0
3	1	1	4	1.0
3	1	4	4	1.0
4	1	1	5	1.0
4	1	5	5	1.0
5	1	1	6	1.0
5	1	6	6	1.0
6	1	1	7	1.0
6	1	7	7	1.0
7	1	2	3	1.0
8	1	2	4	1.0
9	1	2	5	1.0
10	1	2	6	1.0
11	1	2	7	1.0
12	1	3	4	1.0
13	1	3	5	1.0
14	1	3	6	1.0
15	1	3	7	1.0
16	1	4	5	1.0
17	1	4	6	1.0
18	1	4	7	1.0
19	1	5	6	1.0
20	1	5	7	1.0
21	1	6	7	1.0

It makes sense for me, just using orthogonality and commutation constraints. However, when solving I get 0.375 as you do. Do you think you can export the problem that you solve in Octave in a similar format, so we can compare both matrices?

[peterwittek]

Please note that Ncpol2sdpa migrated to GitLlab. Discussions should take place there, and please send all pull requests there. Thanks.

[denisrosset]

Hi @peterwittek , do you include additional nonnegativity (LP) constraints at level 1? That's how the level 1 is defined in the NPA paper.