Kochen-Specker set with seven contexts

Abstract

The Kochen-Specker (KS) theorem is a central result in quantum theory and has applications in quantum information. Its proof requires several yes-no tests that can be grouped in contexts or subsets of jointly measurable tests. Arguably, the best measure of simplicity of a KS set is the number of contexts. The smaller this number is, the smaller the number of experiments needed to reveal the conflict between quantum theory and noncontextual theories and to get a quantum vs classical outperformance. The original KS set had 132 contexts. Here we introduce a KS set with seven contexts and prove that this is the simplest KS set that admits a symmetric parity proof.

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

“…By applying our approach to proofs of the KS theorem based on parity arguments, we have obtained a general expression of the state-independent noncontextuality inequality. With this result, all known parity proofs, for example, those in [14, 17-20, 22, 23, 25-28, 30] can be easily converted to state-independent noncontextuality inequalities, and the inequalities in [35] and [51] are special cases of equation (10).…”

Arbitrarily exhaustive hypergraph generation of 4-, 6-, 8-, 16-, and 32-dimensional quantum contextual sets

“…Lisoněk, Badziag, Portillo, and Cabello [24] recently found a 6-dim 21-7 KS set which they drew in the form of a seven pointed star, a regular heptagram with Schläfli symbol {7/3}, as shown in Fig. 11.…”

“…The authors of [24] have made an attempt to find a bigger 6-dim set but did not find any. Waegell and Aravind appreciated the approach as the first one "in a dimension that is not of the form 2 N " [52], meaning that the 6-dim space cannot "host" qubits (recall that two qubits reside in the 2 2 dim, i.e., 4-dim Hilbert space, three qubits in the 2 3 dim, i.e., 8-dim space, etc.).…”

“…6. In particular, there are 16,20,24, and 75 such vertices (grey dots) in Bub, Conway-Kochen, Peres, and Kochen-Specker's hypergraphs in Fig. 19, respectively.…”

“…They were implemented for 4-dim systems with photons [13][14][15][16][17][18], neutrons [19][20][21] trapped ions [22], and molecular nuclear spins in the solid states [23], for 6-dim systems via six path possibilities for the photon transmission through a diffractive aperture [24,25], and for 8-dim systems by means of the linear transverse momentum of single photons transmitted by diffractive apertures addressed in spatial light modulators [26].…”