Bell inequalities
Linear bell inequalities
The CHSH inequality:
using DynamicPolynomials
using NCTSSOS
@ncpolyvar x[1:2]
@ncpolyvar y[1:2]
f = x[1]*y[1] + x[1]*y[2] + x[2]*y[1] - x[2]*y[2]
opt,data = ncpop([-f], [x;y], 1, TS=false, partition=2, constraint="unipotent")The I_3322 inequality:
using DynamicPolynomials
using NCTSSOS
@ncpolyvar x[1:3]
@ncpolyvar y[1:3]
f = x[1]*(y[1] + y[2] + y[3]) + x[2]*(y[1] + y[2] - y[3]) + x[3]*(y[1] - y[2]) - x[1] - 2*y[1] - y[2]
opt,data = ncpop([-f], [x;y], 3, TS=false, normality=true, partition=3, constraint="projection")The I_3322 inequality solved with the sequential NPA hierarchy:
using JuMP
using MosekTools
using DynamicPolynomials
using MultivariatePolynomials
using NCTSSOS
@ncpolyvar y[1:3]
y1 = y[1]
y2 = y[2]
y3 = y[3]
### f = -1 + x1/4 + x2/4 - y1/4 - y2/4 + x1*y1/4 + x1*y2/4 - x1*y3/4 + x2*y1/4 + x2*y2/4 + x2*y3/4 - x3*y1/4 + x3*y2/4
### Objective function of I_3322 after the change of variable x -> 2x-1, y-> 2y-1 to optimize over unitaries
### The 3 observables x are not needed anymore.
### Instead, one defines 6 linear functionals L_ij satisfying
### L_0j + L_1j = L, for all j = 1,2,3
### L_j := L_0j - L_1j, for all j = 1,2,3
### L_j(yk) = <xj yk> = L_0j(yk) - L_1j(yk)
### L(1) = <1> = L_01(1) + L_11(1)
### L(yk) = <yk> = L_0j(yk) + L_1j(yk)
### L_j(1) = <xj> = L_0j(1) - L_1j(1)
### Then one can write <f> = sum_ij L_ij (fij), with fij given below:
f01 = -1 + y1/4 + y2/4 - y3/4 + 1/4 - y1/4
f11 = -1 + -y1/4 - y2/4 + y3/4 - 1/4 - y1/4
f02 = y1/4 + y2/4 + y3/4 + 1/4 - y2/4
f12 = -y1/4 - y2/4 - y3/4 - 1/4 - y2/4
f03 = -y1/4 + y2/4
f13 = y1/4 - y2/4
d = 5
model = Model(optimizer_with_attributes(Mosek.Optimizer))
set_optimizer_attribute(model, MOI.Silent(), false)
λ = @variable(model)
p = add_poly!(model, y, 2d, constraint="unipotent")
q = add_poly!(model, y, 2d, constraint="unipotent")
info = add_psatz!(model, λ - f01 + p, y, [], [], d, QUIET=true, TS=false, constraint="unipotent")
info = add_psatz!(model, λ - f11 + p, y, [], [], d, QUIET=true, TS=false, constraint="unipotent")
info = add_psatz!(model, - f02 - p + q, y, [], [], d, QUIET=true, TS=false, constraint="unipotent")
info = add_psatz!(model, - f12 - p + q, y, [], [], d, QUIET=true, TS=false, constraint="unipotent")
info = add_psatz!(model, - f03 - q, y, [], [], d, QUIET=true, TS=false, constraint="unipotent")
info = add_psatz!(model, - f13 - q, y, [], [], d, QUIET=true, TS=false, constraint="unipotent")
@objective(model, Min, λ)
optimize!(model)
objv = objective_value(model)
@show objv