Kochen-Specker Sets with a Mixture of 16 Rank-1 and 14 Rank-2 Projectors for a Three-Qubit System

Abstract

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...

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

“…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], and not limited to the two examples. Compared with the methods proposed in [36] and [49], which are applicable only to proofs of the KS theorem based on rays, the present approach is applicable to all kinds of proofs of the KS theorem.…”

“…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].…”

The many mathematical faces of Mermin's proof of the Kochen-Specker theorem

“…Colouring proofs can also be extracted from the Mermin system in more than one way; Kerhaghan and Peres [32], Waegell and Aravind [33], and Toh [34] have each given an example. Kernaghan and Peres note that each context generates 8 mutually orthogonal vectors, given as the simultaneous eigenvectors of the (commuting) observables appearing in the context.…”

Systematic construction of quantum contextuality scenarios with rank advantage

“…SIC sets involving only projectors of rank one have led to many fruitful results over the last years [23,24,4]. In comparison, SIC sets involving non-unit rank received only little attention [25,19,26,27] and, to the best of our knowledge, all of those results are based on the Mermin star [28]. But there is no physical reason to limit the projectors in a SIC set to be only of rank one and we expect that many interesting SIC sets involve projectors of non-unit rank.…”