The main tool we use in this work to show that anti-hole inequalities admit robust self-testing is Theorem 1, shown in BRVWCK19 , which provides a sufficient condition for a graph to be self-testable. 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 primal-dual 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 odd-cycle graph by providing an explicit mapping between the Gram vectors of a primal () optimal solution of an odd-cycle 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 non-degeneracy conditions given in (7). By Theorem 1 , this shows that anti-hole inequalities admit robust self-testing. Details of the proofs can be found in Appendix C. Additionally, we show that not all graphs admit self-testing by providing a counter example of such a graph (see Appendix F). The overall scheme for determining whether a graph is self-testable (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. FIS2017-89609-P 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
where denotes the cone of real positive semidefinite matrices and . The corresponding dual problem is given by
A pair of primal-dual optimal solutions () with no duality gap (i.e. ), satisfies strict complementarity if
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 self-testing
To prove our main result we use the following definitions from BRVWCK19 . A non-contextuality inequality is a self-test for the realisation if:
achieves the quantum supremum ;
For any other realisation that also achieves , there exists an isometry such that
Furthermore, a non-contextuality inequality is an -robust self-test for if it is a self-test, and furthermore, for any other realisation satisfying
there exists an isometry such that
The proof of our main result hinges on the following theorem (first introduced in BRVWCK19 ):
Consider a non-contextuality inequality . Assume that
There exists an optimal quantum realisation such that
and for all , and
Then, the non-contextuality inequality is an -robust self-test for .
Appendix C Self-testing anti-hole inequalities
The anti-hole non-contextuality inequalities are given by for all . The quantum bound for the anti-hole inequalities, i.e., the Lovász theta number for is knuth1994sandwich . A canonical quantum ensemble which achieves the quantum value for the anti-hole inequalities corresponding to odd is given by dimensional quantum state and projectors. Explicitly, the quantum state is
but the description of the projectors is more involved CDLP13 . Let us denote the -th component of corresponding to projector as . For and ,
For the anti-hole non-contextuality 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 anti-hole 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.
Let be the unique optimal solution for . Then,
is a dual optimal solution for Another useful expression for is given by:
where is the vector of all ones of length ,
and maps an -dimensional vector and outputs the corresponding circulant matrix.
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 non-adjacency). 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
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 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.
The dual optimal solution , corresponding to the complement of an odd-cycle graph satisfies the conditions in 7.
We show that for any odd the only symmetric matrix satisfying
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 non-zero 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 non-zero slots in with . For example for , we have
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
For , the linear equations corresponding to and imply
Now consider the linear equations corresponding to . These equations along with 20 imply
Finally, using the equations corresponding to again and 22 we have
implying that , as desired. ∎
Appendix D Dimension for optimal violation of anti-hole inequalities
Given an anti-hole non-contextuality inequality with an odd number of measurement events, the quantum system achieving the optimal quantum bound must be at least dimensional.
The value of a non-contextuality 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 all-ones 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 ofis 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
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.
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:
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:
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
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 non-contextuality inequalities admit self-testing
We have proved that all fundamental non-contextuality inequalities admit self-testing. A natural question is whether every non-contextuality inequality with separation between corresponding non-contextual hidden variable bound and quantum bound admits self-testing. Below we provide an explicit non-contextuality inequality which shows that the answer is negative. In graph theoretical terms, we identify a non-perfect 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 .
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 counter-example is shown in Figure 4. The corresponding canonical non-contextuality inequality is given by
whose quantum bound is equal to .
Consider the pair of primal-dual optimal solutions
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
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 non-contextuality inequality in (27) does not admit self-testing.
We also report that we found several other non-perfect graphs (and equivalently non-contextuality inequalities) which do not admit self-testing. Identifying the exact classes of graphs which admit self-testing will be interesting but we leave that as an open question.