On the Minimum Degree up to Local Complementation: Bounds and Complexity

The local minimum degree of a graph is the minimum degree reached by means of a series of local complementations. In this paper, we investigate on this quantity which plays an important role in quantum computation and quantum error correcting codes. First, we show that the local minimum degree of the Paley graph of order p is greater than sqrt{p} - 3/2, which is, up to our knowledge, the highest known bound on an explicit family of graphs. Probabilistic methods allows us to derive the existence of an infinite number of graphs whose local minimum degree is linear in their order with constant 0.189 for graphs in general and 0.110 for bipartite graphs. As regards the computational complexity of the decision problem associated with the local minimum degree, we show that it is NP-complete and that there exists no k-approximation algorithm for this problem for any constant k unless P = NP.Comment: 11 page

The central nervous system of sea cucumbers (Echinodermata: Holothuroidea) shows positive immunostaining for a chordate glial secretion

<p>Abstract</p> <p>Background</p> <p>Echinoderms and chordates belong to the same monophyletic taxon, the Deuterostomia. In spite of significant differences in body plan organization, the two phyla may share more common traits than was thought previously. Of particular interest are the common features in the organization of the central nervous system. The present study employs two polyclonal antisera raised against bovine Reissner's substance (RS), a secretory product produced by glial cells of the subcomissural organ, to study RS-like immunoreactivity in the central nervous system of sea cucumbers.</p> <p>Results</p> <p>In the ectoneural division of the nervous system, both antisera recognize the content of secretory vacuoles in the apical cytoplasm of the radial glia-like cells of the neuroepithelium and in the flattened glial cells of the non-neural epineural roof epithelium. The secreted immunopositive material seems to form a thin layer covering the cell apices. There is no accumulation of the immunoreactive material on the apical surface of the hyponeural neuroepithelium or the hyponeural roof epithelium. Besides labelling the supporting cells and flattened glial cells of the epineural roof epithelium, both anti-RS antisera reveal a previously unknown putative glial cell type within the neural parenchyma of the holothurian nervous system.</p> <p>Conclusion</p> <p>Our results show that: a) the glial cells of the holothurian tubular nervous system produce a material similar to Reissner's substance known to be synthesized by secretory glial cells in all chordates studied so far; b) the nervous system of sea cucumbers shows a previously unrealized complexity of glial organization. Our findings also provide significant clues for interpretation of the evolution of the nervous system in the Deuterostomia. It is suggested that echinoderms and chordates might have inherited the RS-producing radial glial cell type from the central nervous system of their common ancestor, i.e., the last common ancestor of all the Deuterostomia.</p

Frustrated antiferromagnetic quantum spin chains for spin length S > 1

We investigate frustrated antiferromagnetic Heisenberg quantum spin chains at T=0 for S=3/2 and S=2 using the DMRG method. We localize disorder and Lifshitz points, confirming that quantum disorder points can be seen as quantum remnants of classical phase transitions. Both in the S=3/2 and the S=2 chain, we observe the disappearance of effectively free S=1/2 and S=1 end spins respectively. The frustrated spin chain is therefore a suitable system for clearly showing the existence of free end spins S'=[S/2] also in half-integer antiferromagnetic spin chains with S>1/2. We suggest that the first order transition found for S=1 in our previous work is present in all frustrated spin chains with S>1/2, characterized by the disappearance of effectively free end spins with S'=[S/2].Comment: 6 pages, 8 ps figures, uses RevTeX, submitted to PR

Quantum Communication and Decoherence

In this contribution we will give a brief overview on the methods used to overcome decoherence in quantum communication protocols. We give an introduction to quantum error correction, entanglement purification and quantum cryptography. It is shown that entanglement purification can be used to create ``private entanglement'', which makes it a useful tool for cryptographic protocols.Comment: 31 pages, 10 figures, LaTeX, book chapter to appear in ``Coherent Evolution in Noisy Environments'', Lecture Notes in Physics, (Springer Verlag, Berlin-Heidelberg-New York). Minor typos correcte

Synaptic processes and immune-related pathways implicated in Tourette syndrome

Tourette syndrome (TS) is a neuropsychiatric disorder of complex genetic architecture involving multiple interacting genes. Here, we sought to elucidate the pathways that underlie the neurobiology of the disorder through genome-wide analysis. We analyzed genome-wide genotypic data of 3581 individuals with TS and 7682 ancestry-matched controls and investigated associations of TS with sets of genes that are expressed in particular cell types and operate in specific neuronal and glial functions. We employed a self-contained, set-based association method (SBA) as well as a competitive gene set method (MAGMA) using individual-level genotype data to perform a comprehensive investigation of the biological background of TS. Our SBA analysis identified three significant gene sets after Bonferroni correction, implicating ligand-gated ion channel signaling, lymphocytic, and cell adhesion and transsynaptic signaling processes. MAGMA analysis further supported the involvement of the cell adhesion and trans-synaptic signaling gene set. The lymphocytic gene set was driven by variants in FLT3, raising an intriguing hypothesis for the involvement of a neuroinflammatory element in TS pathogenesis. The indications of involvement of ligand-gated ion channel signaling reinforce the role of GABA in TS, while the association of cell adhesion and trans-synaptic signaling gene set provides additional support for the role of adhesion molecules in neuropsychiatric disorders. This study reinforces previous findings but also provides new insights into the neurobiology of TS

A security proof of quantum cryptography based entirely on entanglement purification

We give a proof that entanglement purification, even with noisy apparatus, is sufficient to disentangle an eavesdropper (Eve) from the communication channel. In the security regime, the purification process factorises the overall initial state into a tensor-product state of Alice and Bob, on one side, and Eve on the other side, thus establishing a completely private, albeit noisy, quantum communication channel between Alice and Bob. The security regime is found to coincide for all practical purposes with the purification regime of a two-way recurrence protocol. This makes two-way entanglement purification protocols, which constitute an important element in the quantum repeater, an efficient tool for secure long-distance quantum cryptography.Comment: Follow-up paper to quant-ph/0108060, submitted to PRA; 24 pages, revex

Absence of string order in the anisotropic S=2 Heisenberg antiferromagnet

We study an AFM Heisenberg S=2 quantum spin chain at T=0 with both interaction and on-site anisotropy, H = \sum_{i} {1/2}(S^{+}_{i}S^{-}_{i+1}+S^{-}_{i}S^{+}_{i+1}) +J^{z}S^{z}_{i}S^{z}_{i+1}+D(S^{z}_{i})^{2}. Contradictory scenarios exist for the S=2 anisotropic phase diagram, implying different mechanisms of the emergence of the classical limit. One main AKLT-based scenario predicts the emergence of a cascade of phase transitions not seen in the S=1 case. Another scenario is in favor of an almost classical phase diagram for S=2; the S=1 case then is very special with its dominant quantum effects. Numerical studies have not been conclusive. Using the DMRG, the existence of hidden topological order in the anisotropic S=2 chain is examined, as it distinguishes between the proposed scenarios. We show that the topological order is zero in the thermodynamical limit in all disordered phases, in particular in the new phase interposed between the Haldane and large-$D$ phases. This excludes the AKLT-model based scenario in favor of an almost classical phase diagram for the $S\leq 2$ spin chains.Comment: 9 pages, 9 eps figures, uses RevTeX, submitted to PR

Rare Copy Number Variants in \u3cem\u3eNRXN1\u3c/em\u3e and \u3cem\u3eCNTN6\u3c/em\u3e Increase Risk for Tourette Syndrome

Tourette syndrome (TS) is a model neuropsychiatric disorder thought to arise from abnormal development and/or maintenance of cortico-striato-thalamo-cortical circuits. TS is highly heritable, but its underlying genetic causes are still elusive, and no genome-wide significant loci have been discovered to date. We analyzed a European ancestry sample of 2,434 TS cases and 4,093 ancestry-matched controls for rare (\u3c 1% frequency) copy-number variants (CNVs) using SNP microarray data. We observed an enrichment of global CNV burden that was prominent for large (\u3e 1 Mb), singleton events (OR = 2.28, 95% CI [1.39–3.79], p = 1.2 × 10−3) and known, pathogenic CNVs (OR = 3.03 [1.85–5.07], p = 1.5 × 10−5). We also identified two individual, genome-wide significant loci, each conferring a substantial increase in TS risk (NRXN1 deletions, OR = 20.3, 95% CI [2.6–156.2]; CNTN6 duplications, OR = 10.1, 95% CI [2.3–45.4]). Approximately 1% of TS cases carry one of these CNVs, indicating that rare structural variation contributes significantly to the genetic architecture of TS

Improved high-temperature expansion and critical equation of state of three-dimensional Ising-like systems

High-temperature series are computed for a generalized $3d$ Ising model with arbitrary potential. Two specific ``improved'' potentials (suppressing leading scaling corrections) are selected by Monte Carlo computation. Critical exponents are extracted from high-temperature series specialized to improved potentials, achieving high accuracy; our best estimates are: $\gamma=1.2371(4)$, $\nu=0.63002(23)$, $\alpha=0.1099(7)$, $\eta=0.0364(4)$, $\beta=0.32648(18)$. By the same technique, the coefficients of the small-field expansion for the effective potential (Helmholtz free energy) are computed. These results are applied to the construction of parametric representations of the critical equation of state. A systematic approximation scheme, based on a global stationarity condition, is introduced (the lowest-order approximation reproduces the linear parametric model). This scheme is used for an accurate determination of universal ratios of amplitudes. A comparison with other theoretical and experimental determinations of universal quantities is presented.Comment: 65 pages, 1 figure, revtex. New Monte Carlo data by Hasenbusch enabled us to improve the determination of the critical exponents and of the equation of state. The discussion of several topics was improved and the bibliography was update