State-Independent Proof of Kochen—Specker Theorem with Thirty Rank-Two Projectors

Abstract

The Kochen-Specker theorem states that noncontextual hidden variable theories are incompatible with quantum mechanics. We provide a state independent proof of the Kochen-Specker theorem using the smallest number of projectors, i.e., thirty rank-2 projectors, associated with the Mermin pentagram for a three-qubit system. PACS numbers: 03.65. Aa, 03.65.Ta, 42.50.Dv Contextuality is one of the classically unattainable features of quantum mechanics (QM). The results of measurements in QM depend on context and do not reveal pre-existing values. A context is a set of maximally collection of compatible observables. The contextual QM thus means that the results of measurements in QM depend on the choice of other compatible measurements that are carried out previously or simultaneously. In contrast, classical physics demands that the properties of a system have pre-determined values which are independent of the measurement context. The contradiction between contextual QM and noncontextual classical physics is expressed via Kochen-Specker (KS) theorem. More specifically, the KS theorem states that the predictions of QM are in conflict with the noncontextual hidden-variable (NCHV) theories. The simplest system that can be used to prove KS theorem is a single qutrit. As a qutrit does not refer to nonlocality, it shows that KS theorem is a more general theorem compared to the Bell theorem that rules out the local hidden variable model of QM.The possibility of testing KS theorem experimentally was once doubted due to the finiteness in measurement times and precision [1, 2] In this work we provide a state independent proof of the KS theorem in three-qubit system using only 30 rank-2 projection operators (projectors). Rank-1 and rank-2 projectors are also called rays and planes, respectively. The rank-2 projectors used in our proof are generated from the aforementioned ten operators in three-qubit system [13]. To understand how we conceived our proof, it is useful to review, as what we start to do in the next paragraph, some of the KS theorem proofs proposed previously that based on same kind of system.The above-mentioned ten operators in three-qubit system form five measurement contexts and all the operators in the same context are mutually commute. The Mermin pentagram in Figure 1 depicts clearly the relationships between measurement contexts.

A proof of the Kochen–Specker theorem can always be converted to a state-independent noncontextuality inequality

“…As examples, we have applied our approach to two kinds of well-known proofs of the KS theorem, i.e., proofs based on rays and proofs based on parity arguments. Certainly, our approach is applicable to all kinds of proofs of the KS theorem,including those in [21,26,31,32], not limited to the two examples.…”

“…The Mermin-Peres square proposed by them involves only 9 observables in a 4-dimensional Hilbert space. Since then, a great number of proofs of the KS theorem based on general observables have been proposed [20][21][22][23][24][25][26][27][28][29][30][31][32].…”

Kochen-Specker contextuality

“…Some worked on the problem and simplified it by finding KS sets with an increasingly small number of vectors in different dimensions. For example, in dimension 3 (Belinfante, 1973;Alda, 1980;Peres and Ron, 1988;de Obaldia, Shimony, and Wittel, 1988;Peres, 1991Peres, , 1993Bub, 1996;Conway and Kochen, 2013), in dimension 4 (Peres, 1991;Zimba and Penrose, 1993;Kernaghan, 1994;Cabello, Estebaranz, and García-Alcaine, 1996a;Penrose, 2000), in dimension 6 (Lisoněk et al, 2014), and in dimension 8 (Kernaghan and Peres, 1995;Toh, 2013aToh, , 2013b. Subsequent works have identified many other examples of KS sets in different dimensions Aravind and Lee-Elkin, 1998;Pavičić et al, 2005Pavičić et al, , 2011Pavičić, 2006;Gould and Aravind, 2010;Waegell and Aravind, 2010;Arends, Ouaknine, and Wampler, 2011;Megill et al, 2011;Waegell and Aravind, 2011a, 2011bWaegell et al, 2011;Ruuge, 2012).…”