Quantum Kaleidoscopes and Bell's theorem

A quantum kaleidoscope is defined as a set of observables, or states, consisting of many different subsets that provide closely related proofs of the Bell-Kochen-Specker (BKS) and Bell nonlocality theorems. The kaleidoscopes prove the BKS theorem through a simple parity argument, which also doubles as a proof of Bell's nonlocality theorem if use is made of the right sort of entanglement. Three closely related kaleidoscopes are introduced and discussed in this paper: a 15-observable kaleidoscope, a 24-state kaleidoscope and a 60-state kaleidoscope. The close relationship of these kaleidoscopes to a configuration of 12 points and 16 lines known as Reye's configuration is pointed out. The "rotations" needed to make each kaleidoscope yield all its apparitions are laid out. The 60-state kaleidoscope, whose underlying geometrical structure is that of ten interlinked Reye's configurations (together with their duals), possesses a total of 1120 apparitions that provide proofs of the two Bell theorems. Some applications of these kaleidoscopes to problems in quantum tomography and quantum state estimation are discussed.Comment: Two new references (No. 21 and 22) to related work have been adde

The generalized Kochen-Specker theorem

A proof of the generalized Kochen-Specker theorem in two dimensions due to Cabello and Nakamura is extended to all higher dimensions. A set of 18 states in four dimensions is used to give closely related proofs of the generalized Kochen-Specker, Kochen-Specker and Bell theorems that shed some light on the relationship between these three theorems.Comment: 5 pages, 1 Table. A new third paragraph and an additional reference have been adde

On small proofs of Bell-Kochen-Specker theorem for two, three and four qubits

The Bell-Kochen-Specker theorem (BKS) theorem rules out realistic {\it non-contextual} theories by resorting to impossible assignments of rays among a selected set of maximal orthogonal bases. We investigate the geometrical structure of small $v-l$ BKS-proofs involving $v$ real rays and $l$ $2n$-dimensional bases of $n$-qubits ($1< n < 5$). Specifically, we look at the parity proof 18-9 with two qubits (A. Cabello, 1996), the parity proof 36-11 with three qubits (M. Kernaghan & A. Peres, 1995 \cite{Kernaghan1965}) and a newly discovered non-parity proof 80-21 with four qubits (that improves work of P. K Aravind's group in 2008). The rays in question arise as real eigenstates shared by some maximal commuting sets (bases) of operators in the $n$-qubit Pauli group. One finds characteristic signatures of the distances between the bases, which carry various symmetries in their graphs.Comment: version to appear in European Physical Journal Plu

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

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

The Projective Line Over the Finite Quotient Ring GF(2)[xx]/<x3−x>< x^{3} - x> and Quantum Entanglement II. The Mermin "Magic" Square/Pentagram

In 1993, Mermin (Rev. Mod. Phys. 65, 803--815) gave lucid and strikingly simple proofs of the Bell-Kochen-Specker (BKS) theorem in Hilbert spaces of dimensions four and eight by making use of what has since been referred to as the Mermin(-Peres) "magic square" and the Mermin pentagram, respectively. The former is a $3 \times 3$ array of nine observables commuting pairwise in each row and column and arranged so that their product properties contradict those of the assigned eigenvalues. The latter is a set of ten observables arranged in five groups of four lying along five edges of the pentagram and characterized by similar contradiction. An interesting one-to-one correspondence between the operators of the Mermin-Peres square and the points of the projective line over the product ring ${\rm GF}(2) \otimes \rm{GF}(2)$ is established. Under this mapping, the concept "mutually commuting" translates into "mutually distant" and the distinguishing character of the third column's observables has its counterpart in the distinguished properties of the coordinates of the corresponding points, whose entries are both either zero-divisors, or units. The ten operators of the Mermin pentagram answer to a specific subset of points of the line over GF(2)[$x$]/$$. The situation here is, however, more intricate as there are two different configurations that seem to serve equally well our purpose. The first one comprises the three distinguished points of the (sub)line over GF(2), their three "Jacobson" counterparts and the four points whose both coordinates are zero-divisors; the other features the neighbourhood of the point ($1, 0$) (or, equivalently, that of ($0, 1$)). Some other ring lines that might be relevant for BKS proofs in higher dimensions are also mentioned.Comment: 6 pages, 5 figure

Entanglement Patterns in Mutually Unbiased Basis Sets for N Prime-state Particles

A few simply-stated rules govern the entanglement patterns that can occur in mutually unbiased basis sets (MUBs), and constrain the combinations of such patterns that can coexist (ie, the stoichiometry) in full complements of p^N+1 MUBs. We consider Hilbert spaces of prime power dimension (as realized by systems of N prime-state particles, or qupits), where full complements are known to exist, and we assume only that MUBs are eigenbases of generalized Pauli operators, without using a particular construction. The general rules include the following: 1) In any MUB, a particular qupit appears either in a pure state, or totally entangled, and 2) in any full MUB complement, each qupit is pure in p+1 bases (not necessarily the same ones), and totally entangled in the remaining p^N-p. It follows that the maximum number of product bases is p+1, and when this number is realized, all remaining p^N-p bases in the complement are characterized by the total entanglement of every qupit. This "standard distribution" is inescapable for two qupits (of any p), where only product and generalized Bell bases are admissible MUB types. This and the following results generalize previous results for qubits and qutrits. With three qupits there are three MUB types, and a number of combinations (p+2) are possible in full complements. With N=4, there are 6 MUB types for p=2, but new MUB types become possible with larger p, and these are essential to the realization of full complements. With this example, we argue that new MUB types, showing new entanglement characteristics, should enter with every step in N, and when N is a prime plus 1, also at critical p values, p=N-1. Such MUBs should play critical roles in filling complements.Comment: 27 pages, one figure, to be submitted to Physical Revie

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

Mesoscopic superposition and sub-Planck-scale structure in molecular wave packets

We demonstrate the possibility of realizing sub-Planck-scale structures in the mesoscopic superposition of molecular wave packets involving vibrational levels. The time evolution of the wave packet, taken here as the SU(2) coherent state of the Morse potential describing hydrogen iodide molecules, produces macroscopicquantum- superposition-like states, responsible for the above phenomenon. We investigate the phase-space dynamics of the coherent state through the Wigner function approach and identify the interference phenomena behind the sub-Planck-scale structures. The optimal parameter ranges are specified for observing these features.Comment: 4 pages, 3 figure

Tribology of Protective Hard Coatings for Use in Oil-Less, Piston-Type Compressors

A systematic analysis of prokaryotic ubiquitin-related beta-grasp fold proteins provides new insights into the Ubiquitin family functional history. BACKGROUND. Ubiquitin (Ub)-mediated signaling is one of the hallmarks of all eukaryotes. Prokaryotic homologs of Ub (ThiS and MoaD) and E1 ligases have been studied in relation to sulfur incorporation reactions in thiamine and molybdenum/tungsten cofactor biosynthesis. However, there is no evidence for entire protein modification systems with Ub-like proteins and deconjugation by deubiquitinating enzymes in prokaryotes. Hence, the evolutionary assembly of the eukaryotic Ub-signaling apparatus remains unclear. RESULTS. We systematically analyzed prokaryotic Ub-related β-grasp fold proteins using sensitive sequence profile searches and structural analysis. Consequently, we identified novel Ub-related proteins beyond the characterized ThiS, MoaD, TGS, and YukD domains. To understand their functional associations, we sought and recovered several conserved gene neighborhoods and domain architectures. These included novel associations involving diverse sulfur metabolism proteins, siderophore biosynthesis and the gene encoding the transfer mRNA binding protein SmpB, as well as domain fusions between Ub-like domains and PIN-domain related RNAses. Most strikingly, we found conserved gene neighborhoods in phylogenetically diverse bacteria combining genes for JAB domains (the primary de-ubiquitinating isopeptidases of the proteasomal complex), along with E1-like adenylating enzymes and different Ub-related proteins. Further sequence analysis of other conserved genes in these neighborhoods revealed several Ub-conjugating enzyme/E2-ligase related proteins. Genes for an Ub-like protein and a JAB domain peptidase were also found in the tail assembly gene cluster of certain caudate bacteriophages. CONCLUSION. These observations imply that members of the Ub family had already formed strong functional associations with E1-like proteins, UBC/E2-related proteins, and JAB peptidases in the bacteria. Several of these Ub-like proteins and the associated protein families are likely to function together in signaling systems just as in eukaryotes.National Library of Medicine, National Institute of Healt