9 research outputs found
New Class of 4-Dim Kochen-Specker Sets
We find a new highly symmetrical and very numerous class (millions of
non-isomorphic sets) of 4-dim Kochen-Specker (KS) vector sets. Due to the
nature of their geometrical symmetries, they cannot be obtained from previously
known ones. We generate the sets from a single set of 60 orthogonal spin
vectors and 75 of their tetrads (which we obtained from the 600-cell) by means
of our newly developed "stripping technique." We also consider "critical KS
subsets" and analyze their geometry. The algorithms and programs for the
generation of our KS sets are presented.Comment: 7 pages, 3 figures; to appear in J. Math. Phys. Vol.52, No. 2 (2011
Parity proofs of the Kochen-Specker theorem based on the 24 rays of Peres
A diagrammatic representation is given of the 24 rays of Peres that makes it
easy to pick out all the 512 parity proofs of the Kochen-Specker theorem
contained in them. The origin of this representation in the four-dimensional
geometry of the rays is pointed out.Comment: 14 pages, 6 figures and 3 tables. Three references have been added.
Minor typos have been correcte
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
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
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
Parity proofs of the Kochen-Specker theorem based on 60 complex rays in four dimensions
It is pointed out that the 60 complex rays in four dimensions associated with
a system of two qubits yield over 10^9 critical parity proofs of the
Kochen-Specker theorem. The geometrical properties of the rays are described,
an overview of the parity proofs contained in them is given, and examples of
some of the proofs are exhibited.Comment: 17 pages, 13 tables, 3 figures. Several new references have been
adde
The Sheaf-Theoretic Structure Of Non-Locality and Contextuality
We use the mathematical language of sheaf theory to give a unified treatment
of non-locality and contextuality, in a setting which generalizes the familiar
probability tables used in non-locality theory to arbitrary measurement covers;
this includes Kochen-Specker configurations and more. We show that
contextuality, and non-locality as a special case, correspond exactly to
obstructions to the existence of global sections. We describe a linear
algebraic approach to computing these obstructions, which allows a systematic
treatment of arguments for non-locality and contextuality. We distinguish a
proper hierarchy of strengths of no-go theorems, and show that three leading
examples --- due to Bell, Hardy, and Greenberger, Horne and Zeilinger,
respectively --- occupy successively higher levels of this hierarchy. A general
correspondence is shown between the existence of local hidden-variable
realizations using negative probabilities, and no-signalling; this is based on
a result showing that the linear subspaces generated by the non-contextual and
no-signalling models, over an arbitrary measurement cover, coincide. Maximal
non-locality is generalized to maximal contextuality, and characterized in
purely qualitative terms, as the non-existence of global sections in the
support. A general setting is developed for Kochen-Specker type results, as
generic, model-independent proofs of maximal contextuality, and a new
combinatorial condition is given, which generalizes the `parity proofs'
commonly found in the literature. We also show how our abstract setting can be
represented in quantum mechanics. This leads to a strengthening of the usual
no-signalling theorem, which shows that quantum mechanics obeys no-signalling
for arbitrary families of commuting observables, not just those represented on
different factors of a tensor product.Comment: 33 pages. Extensively revised, new results included. Published in New
Journal of Physic