Methods
The main tool we use in this work to show that antihole inequalities admit robust selftesting is Theorem 1, shown in BRVWCK19 , which provides a sufficient condition for a graph to be selftestable. This result relies crucially on the rich properties of a powerful class of mathematical optimisation models, known as Semidefinite programs (SDPs) (see Appendix A
). SDPs constitute a vast generalisation of linear optimisation models where scalar variables are replaced by vectors and the constraints and objective function are affine in terms of the inner products of the vectors. Equivalently, collecting all pairwise inner products of these vectors in matrix, known as the Gram matrix, an SDP corresponds to optimising a linear function of the Gram matrix subject to affine constraints. Analogously to linear programs, to any SDP there is an associated a dual program whose value is equal to the primal under reasonable assumptions. Next, we single out certain properties of primaldual solutions that are of relevance to his work. A pair of primal dual optimal solutions (
) with no duality gap (i.e. ), satisfies strict complementarity if the and give a direct sum decomposition of the underlying space. Furthermore, an optimal dual solution with rank is dual nondegenerate if the tangent space at of the manifold of symmetric matrices with rank equal to together with the linear space of matrices defining the SDP span the entire space of symmetric matrices. In this work we focus on the Lovász theta SDP, see () in Appendix B. The proof of our main result involves two main steps. First, we construct a dual optimal solution of () for an oddcycle graph by providing an explicit mapping between the Gram vectors of a primal () optimal solution of an oddcycle graph and the Gram vectors of a dual optimal solution of the complement graph; see Theorem 2 for details. Next, once we construct a dual optimal solution, we show in Theorem 3 that it satisfies the nondegeneracy conditions given in (7). By Theorem 1 , this shows that antihole inequalities admit robust selftesting. Details of the proofs can be found in Appendix C. Additionally, we show that not all graphs admit selftesting by providing a counter example of such a graph (see Appendix F). The overall scheme for determining whether a graph is selftestable (equivalently whether the primal optimal solution of the Lovász theta SDP corresponding to that graph is unique or not) is provided in the form of a flowchart in Figure 3.Acknowledgements.— We thank the National Research Foundation of Singapore, the Ministry of Education of Singapore, MINECO Project No. FIS201789609P with FEDER funds, and the Knut and Alice Wallenberg Foundation for financial support. A significant part of the research project Local certification of programmable quantum devices of arbitrary high dimensionalitywas carried out during “New directions in quantum information” conference organized by Nordita, the Nordic Institute for Theoretical Physics.
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Appendix A Semidefinite programming basics
A semidefinite program (SDP) is given by an optimisation problem of the following form
(P) 
where denotes the cone of real positive semidefinite matrices and . The corresponding dual problem is given by
(D) 
A pair of primaldual optimal solutions () with no duality gap (i.e. ), satisfies strict complementarity if
(4) 
Lastly, an optimal dual solution is called dual nondegenerate if the linear system in the symmetric matrix variable
only admits the trivial solution .
Central to this work is the Lovász theta SDP corresponding to a graph , whose primal formulation is:
()  
and the dual formulation we use is given by:
()  
Appendix B Robust selftesting
To prove our main result we use the following definitions from BRVWCK19 . A noncontextuality inequality is a selftest for the realisation if:

achieves the quantum supremum ;

For any other realisation that also achieves , there exists an isometry such that
(5)
Furthermore, a noncontextuality inequality is an robust selftest for if it is a selftest, and furthermore, for any other realisation satisfying
there exists an isometry such that
(6) 
The proof of our main result hinges on the following theorem (first introduced in BRVWCK19 ):
Theorem 1.
Consider a noncontextuality inequality . Assume that

There exists an optimal quantum realisation such that
and for all , and
Then, the noncontextuality inequality is an robust selftest for .
Appendix C Selftesting antihole inequalities
The antihole noncontextuality inequalities are given by for all . The quantum bound for the antihole inequalities, i.e., the Lovász theta number for is knuth1994sandwich . A canonical quantum ensemble which achieves the quantum value for the antihole inequalities corresponding to odd is given by dimensional quantum state and projectors. Explicitly, the quantum state is
(8) 
but the description of the projectors is more involved CDLP13 . Let us denote the th component of corresponding to projector as . For and ,
(9) 
(10) 
(11) 
for and
(12) 
(13) 
For the antihole noncontextuality inequalities, the ensemble described above achieves the quantum value and satisfies the first condition of Theorem 1. It remains to establish the existence of a dual optimal solution for the SDP corresponding to the Lovász theta number of antihole graphs such that the conditions in (7) are satisfied. Towards this goal, we first proceed to provide the explicit form of the dual optimal solution.
Theorem 2.
Let be the unique optimal solution for . Then,
is a dual optimal solution for Another useful expression for is given by:
(14) 
where is the vector of all ones of length ,
and maps an dimensional vector and outputs the corresponding circulant matrix.
Proof.
It was shown in BRVWCK19 that admits a unique optimal solution . For any , the map taking (modulo ) is an automorphism of (i.e., a bijective map that preserves adjacency and nonadjacency). In particular, this implies that is circulant and furthermore, constant along each band. Specifically, all diagonal entries of are equal, and as , it follows that
(15) 
Analogously, for a pair of indices with we have that . Moreover, be feasibility of we have that . Thus has the correct value and it remains to show that is feasible. Next, by feasibility of we have that , when in . Thus, by definition of we have that for all edges of . Finally we show that . Indeed,
where we used (15) and that (see Theorem 8 of lovasz1979shannon ). To finish the proof we note that
(16)  
where the second last equality follows from the constraint that for and the last equality follows by substituting . ∎
Finally, we show that the dual optimal solution satisfies the conditions in 7.
Theorem 3.
The dual optimal solution , corresponding to the complement of an oddcycle graph satisfies the conditions in 7.
Proof.
We show that for any odd the only symmetric matrix satisfying
(17) 
is the matrix , where here refers to an edge in the graph. Barring the constraint, the rest already guarantee that there are at most potentially nonzero entries in the matrix (not counting the repeated entries) corresponding to graph. Let the first row of be . We fill the rest of the potential nonzero slots in with . For example for , we have
(18) 
For notational convenience, let , where , and . In the rest of this section we use the notation to denote where is an integer. The linear equation corresponding to implies that
(19) 
For , the linear equations corresponding to and imply
(20) 
Now consider the linear equations corresponding to . These equations along with 20 imply
(21) 
Using 21 along with 19 we have
(22) 
Finally, using the equations corresponding to again and 22 we have
(23) 
implying that , as desired. ∎
Appendix D Dimension for optimal violation of antihole inequalities
Theorem 4.
Given an antihole noncontextuality inequality with an odd number of measurement events, the quantum system achieving the optimal quantum bound must be at least dimensional.
Proof.
The value of a noncontextuality inequality achievable within quantum theory is equal to the Lovász theta number of the underlying graph and admits the SDP formulation (). The lower bound on the dimension of a quantum system achieving the optimal quantum bound is the rank of the unique primal optimal matrix
where is the allones vector of length , is the circulant function that takes as input a dimension vector and outputs a matrix with the input vector as its top row and every subsequent row being one place right shifted modulo and . Since
is real, its rank over complex field is the same as over real field, and equals to the number of nonzero eigenvalues (with multiplicity). Furthermore, a lower bound on the rank of
is given by the rank of the lower right block matrix (the circulant portion). The eigenvalues of a circulant matrix can be calculated easily using the circulant vector. A few lines of algebra yields the following expression for the eigenvalues of the lower right block matrix,for and denotes the Lovász theta number for the holes with odd One can see that unless or Thus, the rank of the circulant matrix is for all odd values of Thus, the lower bound on the rank of the optimal feasible matrix is which is same as the lower bound on the dimension of the desired quantum system. ∎
Appendix E Complex versus Real SDPs
Lemma 5.
Consider a real SDP
that admits a unique optimal solution witnessed by a dual nondegenerate optimal solution . Then, the SDP considered over the complex numbers, i.e.,
() 
still admits a unique optimal solution, where and denotes the set of Hermitian positive semidefinite matrices.
Proof.
First, we show that the study of a complex SDP can be reduced to an equivalent real SDP. This fact is well known but we provide a brief argument for completeness. Indeed, for any feasible solution , the constraint is equivalent to two constraints on its real and imaginary part, namely: and . Furthermore, checking whether is Hermitian PSD is equivalent to
Based on these observations we define the realification of as the following SDP over the real numbers:
()  
s.t.  
Clearly, the solutions of are in bijection with the solutions of the realification, and thus, to show that has a unique solution it suffices to show that has a unique solution. Bringing into standard SDP form we arrive at the formulation:
(24)  
s.t.  
whose dual is to minimize the function over all satisfying
We conclude the proof by showing that is a dual nondegenerate optimal solution for the realification. First, by dual feasibility we have for appropriate scalars . Setting and all other dual variables to zero, we have established feasibility. Second, to show optimality note that is optimal for the realification, and furthermore, . Lastly, to check nondegeneracy consider a symmetric matrix satisfying
(25) 
and
(26)  
Now, constraint (25) is equivalent to
Furthermore, using (26), from the third equation we get , from the first one we get and from the second one . Summarizing, for all we have that
As is dual nondegenerate it has the property that for any :
Putting everything together we get .
∎
Appendix F Not all noncontextuality inequalities admit selftesting
We have proved that all fundamental noncontextuality inequalities admit selftesting. A natural question is whether every noncontextuality inequality with separation between corresponding noncontextual hidden variable bound and quantum bound admits selftesting. Below we provide an explicit noncontextuality inequality which shows that the answer is negative. In graph theoretical terms, we identify a nonperfect graph whose Lovász theta SDP admits multiple primal optimal solutions. We make crucial use of the following result (alizadeh, , Theorem 5) to determine the (non)uniqueness of primal optimal under strict complementarity, see also thinh2018structure .
Theorem 6.
Let be a pair of primal and dual optimal solutions satisfying strict complementarity.Then, uniqueness of implies that is dual nondegenerate.
The exclusivity graph of our counterexample is shown in Figure 4. The corresponding canonical noncontextuality inequality is given by
(27) 
whose quantum bound is equal to .
Consider the pair of primaldual optimal solutions
(28) 
where and
(29) 
where , , and . Since and , strict complementarity holds. Using Theorem 6, the uniqueness of implies dual nondegeneracy. To determine dual nondegeneracy for we (once again) resort to solving system of linear equations. The symmetric variable matrix is given by
(30) 
Solving for the linear systems of equations , we get , , , , , . For example, if we set , we can get a consistent assignment of , from to , which isn’t all zero. Hence, the dual solution is degenerate, which together with strict complementarity implies that the primal is not unique. Thus the noncontextuality inequality in (27) does not admit selftesting.
We also report that we found several other nonperfect graphs (and equivalently noncontextuality inequalities) which do not admit selftesting. Identifying the exact classes of graphs which admit selftesting will be interesting but we leave that as an open question.
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