Kochen-Specker (KS) theorem denies the possibility for the noncontextual hidden variable theories to reproduce the predictions of quantum mechanics. A set of projection operators (projectors) and bases used to show the impossibility of noncontextual definite values assignment is named as the KS set. Since one KS set with a mixture of 16 rank-1 projectors and 14 rank-2 projectors proposed in 1995 [Kernaghan M and Peres A 1995 Phys. LettThe Kochen-Specker (KS) theorem is another major theorem of impossibility of hidden variables in quantum mechanics (QM) besides the Bell theorem. While Bell theorem rules out the local hidden variable theories, KS theorem refutes the noncontextual hidden variable (NCHV) theories. The NCHV theories assume that the result of a measurement of an observable is predetermined and independent of the other mutually compatible or co-measurable measurements that may perform previously or simultaneously. QM is contextual, its measurement outcomes depend on the context of measurement.To be precise, the KS theorem asserts that, in a Hilbert space of dimension d ≥ 3, it is impossible to associate definite numerical values, 1 or 0, with every projection operator (projector) P i , in such a way that, if a set of commuting P i satisfies P i = I, the corresponding value functions, namely v(P i ) = 0 or 1, also satisfy v(P i ) = v(I) [1]. Kochen and Specker gave the first proof of KS theorem using a set of 117 projectors in three-dimensional real Hilbert space (ℜ 3 ) [2]. The number of projectors reduced to 33 almost twenty-five years later [3], and so far the most economical proof due to Conway and Kochen uses only 31 projectors [4]. There are also proofs of KS theorem for higher dimensional spaces, includingMany experiments have been performed to show that quantum contextuality cannot be explained by any NCHV theories. The physical systems involved in the experiments include photons [12][13][14][15], neutrons [16,17] and trapped ions [18]. Experiment is also proposed to test the KS theorem at a macroscopic level with superconducting quantum circuit [19]. Unlike the Bell theorem, the proof of the KS theorem requires neither entanglement nor composite systems [20,21].A common theoretical technique to prove KS theorem is called parity proof. A set of R rays (rank-1 projectors) * SingPoh. Toh@nottingham.edu.my; singpoh@gmail.com and B bases will said to provide a parity proof of the KS theorem if (a) B is odd, and (b) each of the R rays occurs an even number of times among the B bases [10]. The set of rays and bases that satisfies these two conditions is called the KS set. The 13 rays used in [22] does not describe a KS set and an inequality instead of parity method is adopted to prove the KS theorem. Projectors in KS sets need not to be of rank-1 or all be of the same rank [10]. Kernaghan and Peres [9] provided a KS set in ℜ 8 with 11 bases formed by a mixture of 16 rank-1 and 14 rank-2 projectors, which is labeled as 30-11, where the first figure is due to the sum of both rank-1 and rank-2 projec...

We consider Kochen-Specker theorem for three-qubit system with eight-dimensional state space. Reexamining the proof given by Kernaghan and Peres, we make some clarifications on the orthogonality of rays and rank-two projectors found by them. Basing on their five groups of orthogonal octad, we then show a proof that requires only seventeen rays.

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 two-qubit system in quantum information theory is the simplest bipartite quantum system and its concurrence for pure and mixed states is well known. As a subset of two-qubit systems, Bell-diagonal states can be depicted by a very simple geometrical representation of a tetrahedron with sides of length 2 √ 2. Based on this geometric representation, we propose a simple approach to randomly generate four mixed Bell decomposable states in which the sum of their concurrence is equal to one.

We put forward three simple algorithms to generate the Kochen-Specker sets used for the parity proof of the Kochen-Specker theorem in three-qubit system. These algorithms enable us to generate 320, 640 and 64 Kochen-Specker sets with 36, 38 and 40 rays, respectively. No any computer calculation is required, every step in the algorithms is determined by the method of picking the rays that repeat 4 times in the KS sets.