N-qubit states as points on the Bloch sphere

We show how the Majorana representation can be used to express the pure states of an N-qubit system as points on the Bloch sphere. We compare this geometrical representation of N-qubit states with an alternative one, proposed recently by the present authors.Comment: 9 pages, 2 figures, contribution to CEWQO 2009 proceedings. v2: Minor changes, published versio

Optimal state for keeping reference frames aligned and the Platonic solids

The optimal N qubit states featuring highest sensitivity to small misalignment of cartesian reference frames are found using the Quantum Cramer-Rao bound. It is shown that the optimal states are supported on the symmetric subspace and hence are mathematically equivalent to a single spin J=N/2. Majorana representation of spin states is used to reveal a beautiful connection between the states optimal for aligning reference frames and the platonic solids

Fairy, tadpole, and clam shrimps (Branchiopoda) in seasonally inundated clay pans in the western Mojave Desert and effect on primary producers

Abstract Background Fairy shrimps (Anostraca), tadpole shrimps (Notostraca), clam shrimps (Spinicaudata), algae (primarily filamentous blue-green algae [cyanobacteria]), and suspended organic particulates are dominant food web components of the seasonally inundated pans and playas of the western Mojave Desert in California. We examined the extent to which these branchiopods controlled algal abundance and species composition in clay pans between Rosamond and Rogers Dry Lakes. We surveyed branchiopods during the wet season to estimate abundances and then conducted a laboratory microcosm experiment, in which dried sediment containing cysts and the overlying algal crust were inundated and cultured. Microcosm trials were run with and without shrimps; each type of trial was run for two lengths of time: 30 and 60 days. We estimated the effect of shrimps on algae by measuring chlorophyll content and the relative abundance of algal species. Results We found two species of fairy shrimps (Branchinecta mackini and B. gigas), one tadpole shrimp (Lepidurus lemmoni), and a clam shrimp (Cyzicus setosa) in our wet-season field survey. We collected Branchinecta lindahli in a pilot study, but not subsequently. The dominant taxa were C. setosa and B. mackini, but abundances and species composition varied greatly among playas. The same species found in field surveys also occurred in the microcosm experiment. There were no significant differences as a function of experimental treatments for either chlorophyll content or algal species composition (Microcoleus vaginatus dominated all treatments). Conclusions The results suggest that there was no direct effect of shrimps on algae. Although the pans harbored an apparently high abundance of branchiopods, these animals had little role in regulating primary producers in this environment

Multiqubit symmetric states with high geometric entanglement

We propose a detailed study of the geometric entanglement properties of pure symmetric N-qubit states, focusing more particularly on the identification of symmetric states with a high geometric entanglement and how their entanglement behaves asymptotically for large N. We show that much higher geometric entanglement with improved asymptotical behavior can be obtained in comparison with the highly entangled balanced Dicke states studied previously. We also derive an upper bound for the geometric measure of entanglement of symmetric states. The connection with the quantumness of a state is discussed

Observables have no value: a no-go theorem for position and momentum observables

A very simple illustration of the Bell-Kochen-Specker contradiction is presented using continuous observables in infinite dimensional Hilbert space. It is shown that the assumption of the \emph{existence} of putative values for position and momentum observables for one single particle is incompatible with quantum mechanics.Comment: 6 pages, 1 Latex figure small corrections, refference and comments adde

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

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

Isomorphism between the Peres and Penrose proofs of the BKS theorem in three dimensions

It is shown that the 33 complex rays in three dimensions used by Penrose to prove the Bell-Kochen-Specker theorem have the same orthogonality relations as the 33 real rays of Peres, and therefore provide an isomorphic proof of the theorem. It is further shown that the Peres and Penrose rays are just two members of a continuous three-parameter family of unitarily inequivalent rays that prove the theorem.Comment: 7 pages, 2 Tables. A concluding para and 9 new references have been added to the second versio

Kochen-Specker Theorem for Finite Precision Spin One Measurements

Unsharp spin 1 observables arise from the fact that a residual uncertainty about the actual orientation of the measurement device remains. If the uncertainty is below a certain level, and if the distribution of measurement errors is covariant under rotations, a Kochen-Specker theorem for the unsharp spin observables follows: There are finite sets of directions such that not all the unsharp spin observables in these directions can consistently be assigned approximate truth-values in a non-contextual way.Comment: 4 page