Deriving robust noncontextuality inequalities from algebraic proofs of the Kochen–Specker theorem: the Peres–Mermin square

Abstract

When a measurement is compatible with each of two other measurements that are incompatible with one another, these define distinct contexts for the given measurement. The Kochen-Specker theorem rules out models of quantum theory that satisfy a particular assumption of context-independence: that sharp measurements are assigned outcomes both deterministically and independently of their context. This notion of noncontextuality is not suited to a direct experimental test because realistic measurements always have some degree of unsharpness due to noise. However, a generalized notion of noncontextuality has been proposed that is applicable to any experimental procedure, including unsharp measurements, but also preparations as well, and for which a quantum no-go result still holds. According to this notion, the model need only specify a probability distribution over the outcomes of a measurement in a context-independent way, rather than specifying a particular outcome. It also implies novel constraints of context-independence for the representation of preparations. In this article, we describe a general technique for translating proofs of the Kochen-Specker theorem into inequality constraints on realistic experimental statistics, the violation of which witnesses the impossibility of a noncontextual model. We focus on algebraic state-independent proofs, using the Peres-Mermin square as our illustrative example. Our technique yields the necessary and sufficient conditions for a particular set of correlations (between the preparations and the measurements) to admit a noncontextual model. The inequalities thus derived are demonstrably robust to noise. We specify how experimental data must be processed in order to achieve a test of these inequalities. We also provide a criticism of prior proposals for experimental tests of noncontextuality based on the Peres-Mermin square.

All the noncontextuality inequalities for arbitrary prepare-and-measure experiments with respect to any fixed set of operational equivalences

“…As such, it is important to understand where the operational equivalences come from in the first place. Historically, most operational equivalences for generalized noncontextuality have originated in quantum no-go theorems for noncontextuality [7,8,13,14,25,26], or in a frustrated network that can be used to build a quantum no-go theorem [27,28].…”

Hypergraph framework for irreducible noncontextuality inequalities from logical proofs of the Kochen-Specker theorem

“…hence there exists an odd k = 9 (with n k = n 9 = 3 > 0) such that 3 j=1 p(w (9) j ) = 1, an odd number. Since |F | = 118 is even, we have a KS contradiction for this hypergraph from Lemma 6.…”

“…8 Now consider an induced subscenario of 2Reg(G(K m,n \K 1,k )) or 2Reg(G(K m,n \K k,1 )) that consists entirely of singletons so that the number of hyperedges in 2Reg(G(K m,n \K 1,k )) or 2Reg(G(K m,n \K k,1 )) is twice the number of singletons in this induced subscenario. 9 Extending this induced subscenario by adding a k-hypercycle (disjoint from the subscenario) leads to an induced subscenario of 2Reg(K m,n ), namely, one that is a union of a k-hypercycle with the singletons. (See Fig.…”

“…However, the experimental testability of the theorem, and therefore its relevance for real-world physics with finite-precision measurements, has been a subject of intense controversy in the past [2][3][4][5]. Recent work [6][7][8][9][10][11] has taken the first steps towards turning the insight of the Kochen-Specker theorem into operational constraints -or noncontextuality inequalities -that are robust to noise and therefore experimentally testable. These inequalities do not presume the structure of quantum measurements -in particular, that they are projective -in their derivation, relying only on operational constraints that can be verified in an experiment and make sense, in particular, for nonprojective measurements in quantum theory.…”

“…Note that when k = n, G(Km,n\K 1,k=n )=K m−1,n , and when k = m, G(Km,n\K k=m,1 )=K m,n−1 . That is, the subgraphs are complete bipartite graphs in these cases, but not otherwise 9. This factor of two arises because the contextuality scenarios are 2-regular, i.e., every vertex appears in two hyperedges 10.…”