New Examples of Kochen-Specker Type Configurations on Three Qubits

A new example of a saturated Kochen-Specker (KS) type configuration of 64 rays in 8-dimensional space (the Hilbert space of a triple of qubits) is constructed. It is proven that this configuration has a tropical dimension 6 and that it contains a critical subconfiguration of 36 rays. A natural multicolored generalisation of the Kochen-Specker theory is given based on a concept of an entropy of a saturated configuration of rays.Comment: 24 page

q-Legendre Transformation: Partition Functions and Quantization of the Boltzmann Constant

In this paper we construct a q-analogue of the Legendre transformation, where q is a matrix of formal variables defining the phase space braidings between the coordinates and momenta (the extensive and intensive thermodynamic observables). Our approach is based on an analogy between the semiclassical wave functions in quantum mechanics and the quasithermodynamic partition functions in statistical physics. The basic idea is to go from the q-Hamilton-Jacobi equation in mechanics to the q-Legendre transformation in thermodynamics. It is shown, that this requires a non-commutative analogue of the Planck-Boltzmann constants (hbar and k_B) to be introduced back into the classical formulae. Being applied to statistical physics, this naturally leads to an idea to go further and to replace the Boltzmann constant with an infinite collection of generators of the so-called epoch\'e (bracketing) algebra. The latter is an infinite dimensional noncommutative algebra recently introduced in our previous work, which can be perceived as an infinite sequence of "deformations of deformations" of the Weyl algebra. The generators mentioned are naturally indexed by planar binary leaf-labelled trees in such a way, that the trees with a single leaf correspond to the observables of the limiting thermodynamic system

Parity proofs of the Bell-Kochen-Specker theorem based on the 600-cell

The set of 60 real rays in four dimensions derived from the vertices of a 600-cell is shown to possess numerous subsets of rays and bases that provide basis-critical parity proofs of the Bell-Kochen-Specker (BKS) theorem (a basis-critical proof is one that fails if even a single basis is deleted from it). The proofs vary considerably in size, with the smallest having 26 rays and 13 bases and the largest 60 rays and 41 bases. There are at least 90 basic types of proofs, with each coming in a number of geometrically distinct varieties. The replicas of all the proofs under the symmetries of the 600-cell yield a total of almost a hundred million parity proofs of the BKS theorem. The proofs are all very transparent and take no more than simple counting to verify. A few of the proofs are exhibited, both in tabular form as well as in the form of MMP hypergraphs that assist in their visualization. A survey of the proofs is given, simple procedures for generating some of them are described and their applications are discussed. It is shown that all four-dimensional parity proofs of the BKS theorem can be turned into experimental disproofs of noncontextuality.Comment: 19 pages, 11 tables, 3 figures. Email address of first author has been corrected. Ref.[5] has been corrected, as has an error in Fig.3. Formatting error in Sec.4 has been corrected and the placement of tables and figures has been improved. A new paragraph has been added to Sec.4 and another new paragraph to the end of the Appendi

Critical noncolorings of the 600-cell proving the Bell-Kochen-Specker theorem

Aravind and Lee-Elkin (1997) gave a proof of the Bell-Kochen-Specker theorem by showing that it is impossible to color the 60 directions from the center of a 600-cell to its vertices in a certain way. This paper refines that result by showing that the 60 directions contain many subsets of 36 and 30 directions that cannot be similarly colored, and so provide more economical demonstrations of the theorem. Further, these subsets are shown to be critical in the sense that deleting even a single direction from any of them causes the proof to fail. The critical sets of size 36 and 30 are shown to belong to orbits of 200 and 240 members, respectively, under the symmetries of the polytope. A comparison is made between these critical sets and other such sets in four dimensions, and the significance of these results is discussed.Comment: 2 new references added, caption to Table 9 correcte