A note on the minimum distance of quantum LDPC codes

We provide a new lower bound on the minimum distance of a family of quantum LDPC codes based on Cayley graphs proposed by MacKay, Mitchison and Shokrollahi. Our bound is exponential, improving on the quadratic bound of Couvreur, Delfosse and Z\'emor. This result is obtained by examining a family of subsets of the hypercube which locally satisfy some parity conditions

Retrodiction of Generalised Measurement Outcomes

If a generalised measurement is performed on a quantum system and we do not know the outcome, are we able to retrodict it with a second measurement? We obtain a necessary and sufficient condition for perfect retrodiction of the outcome of a known generalised measurement, given the final state, for an arbitrary initial state. From this, we deduce that, when the input and output Hilbert spaces have equal (finite) dimension, it is impossible to perfectly retrodict the outcome of any fine-grained measurement (where each POVM element corresponds to a single Kraus operator) for all initial states unless the measurement is unitarily equivalent to a projective measurement. It also enables us to show that every POVM can be realised in such a way that perfect outcome retrodiction is possible for an arbitrary initial state when the number of outcomes does not exceed the output Hilbert space dimension. We then consider the situation where the initial state is not arbitrary, though it may be entangled, and describe the conditions under which unambiguous outcome retrodiction is possible for a fine-grained generalised measurement. We find that this is possible for some state if the Kraus operators are linearly independent. This condition is also necessary when the Kraus operators are non-singular. From this, we deduce that every trace-preserving quantum operation is associated with a generalised measurement whose outcome is unambiguously retrodictable for some initial state, and also that a set of unitary operators can be unambiguously discriminated iff they are linearly independent. We then examine the issue of unambiguous outcome retrodiction without entanglement. This has important connections with the theory of locally linearly dependent and locally linearly independent operators.Comment: To appear in Physical Review

Random geometric complexes

We study the expected topological properties of Cech and Vietoris-Rips complexes built on i.i.d. random points in R^d. We find higher dimensional analogues of known results for connectivity and component counts for random geometric graphs. However, higher homology H_k is not monotone when k > 0. In particular for every k > 0 we exhibit two thresholds, one where homology passes from vanishing to nonvanishing, and another where it passes back to vanishing. We give asymptotic formulas for the expectation of the Betti numbers in the sparser regimes, and bounds in the denser regimes. The main technical contribution of the article is in the application of discrete Morse theory in geometric probability.Comment: 26 pages, 3 figures, final revisions, to appear in Discrete & Computational Geometr

Caveolins/caveolae protect adipocytes from fatty acid-mediated lipotoxicity

Mice and humans lacking functional caveolae are dyslipidemic and have reduced fat stores and smaller fat cells. To test the role of caveolins/caveolae in maintaining lipid stores and adipocyte integrity, we compared lipolysis in caveolin-1 (Cav1)-null fat cells to that in cells reconstituted for caveolae by caveolin-1 re-expression. We find that the Cav1-null cells have a modestly enhanced rate of lipolysis and reduced cellular integrity compared with reconstituted cells as determined by the release of lipid metabolites and lactic dehydrogenase, respectively, into the media. There are no apparent differences in the levels of lipolytic enzymes or hormonally stimulated phosphorylation events in the two cell lines. In addition, acute fasting, which dramatically raises circulating fatty acid levels in vivo, causes a significant upregulation of caveolar protein constituents. These results are consistent with the hypothesis that caveolae protect fat cells from the lipotoxic effects of elevated levels fatty acids, which are weak detergents at physiological pH, by virtue of the property of caveolae to form detergentresistant membrane domains

Diquat Derivatives: Highly Active, Two-Dimensional Nonlinear Optical Chromophores with Potential Redox Switchability

In this article, we present a detailed study of structure−activity relationships in diquaternized 2,2′-bipyridyl (diquat) derivatives. Sixteen new chromophores have been synthesized, with variations in the amino electron donor substituents, π-conjugated bridge, and alkyl diquaternizing unit. Our aim is to combine very large, two-dimensional (2D) quadratic nonlinear optical (NLO) responses with reversible redox chemistry. The chromophores have been characterized as their PF_6^− salts by using various techniques including electronic absorption spectroscopy and cyclic voltammetry. Their visible absorption spectra are dominated by intense π → π^* intramolecular charge-transfer (ICT) bands, and all show two reversible diquat-based reductions. First hyperpolarizabilities β have been measured by using hyper-Rayleigh scattering with an 800 nm laser, and Stark spectroscopy of the ICT bands affords estimated static first hyperpolarizabilities β_0. The directly and indirectly derived β values are large and increase with the extent of π-conjugation and electron donor strength. Extending the quaternizing alkyl linkage always increases the ICT energy and decreases the E_(1/2) values for diquat reduction, but a compensating increase in the ICT intensity prevents significant decreases in Stark-based β_0 responses. Nine single-crystal X-ray structures have also been obtained. Time-dependent density functional theory clarifies the molecular electronic/optical properties, and finite field calculations agree with polarized HRS data in that the NLO responses of the disubstituted species are dominated by ‘off-diagonal’ β_(zyy) components. The most significant findings of these studies are: (i) β_0 values as much as 6 times that of the chromophore in the technologically important material (E)-4′-(dimethylamino)-N-methyl-4-stilbazolium tosylate; (ii) reversible electrochemistry that offers potential for redox-switching of optical properties over multiple states; (iii) strongly 2D NLO responses that may be exploited for novel practical applications; (iv) a new polar material, suitable for bulk NLO behavior

On Eigenvalues of Random Complexes

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=\Omega(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of $k$-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the higher-dimensional Laplacian spectra capture the notion of coboundary expansion - a generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every $k\geq 2$ and $n\in \mathbb{N}$, there is a $k$-dimensional complex $Y^k_n$ on $n$ vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised $k$-dimensional Laplacian lie in the interval $[1-O(1/\sqrt{n}),1+O(1/\sqrt{n})]$) but whose coboundary expansion is bounded from above by $O(\log n/n)$ and so tends to zero as $n\rightarrow \infty$; moreover, $Y^k_n$ can be taken to have vanishing integer homology in dimension less than $k$.Comment: Extended full version of an extended abstract that appeared at SoCG 2012, to appear in Israel Journal of Mathematic

Conserved synteny at the protein family level reveals genes underlying Shewanella species’ cold tolerance and predicts their novel phenotypes

© The Authors 2009. This article is distributed under the terms of the Creative Commons Attribution Noncommercial License. The definitive version was published in Functional & Integrative Genomics 10 (2010): 97-110, doi:10.1007/s10142-009-0142-y.Bacteria of the genus Shewanella can thrive in different environments and demonstrate significant variability in their metabolic and ecophysiological capabilities including cold and salt tolerance. Genomic characteristics underlying this variability across species are largely unknown. In this study, we address the problem by a comparison of the physiological, metabolic, and genomic characteristics of 19 sequenced Shewanella species. We have employed two novel approaches based on association of a phenotypic trait with the number of the trait-specific protein families (Pfam domains) and on the conservation of synteny (order in the genome) of the trait-related genes. Our first approach is top-down and involves experimental evaluation and quantification of the species’ cold tolerance followed by identification of the correlated Pfam domains and genes with a conserved synteny. The second, a bottom-up approach, predicts novel phenotypes of the species by calculating profiles of each Pfam domain among their genomes and following pair-wise correlation of the profiles and their network clustering. Using the first approach, we find a link between cold and salt tolerance of the species and the presence in the genome of a Na+/H+ antiporter gene cluster. Other cold-tolerance-related genes include peptidases, chemotaxis sensory transducer proteins, a cysteine exporter, and helicases. Using the bottom-up approach, we found several novel phenotypes in the newly sequenced Shewanella species, including degradation of aromatic compounds by an aerobic hybrid pathway in Shewanella woodyi, degradation of ethanolamine by Shewanella benthica, and propanediol degradation by Shewanella putrefaciens CN32 and Shewanella sp. W3-18-1.This research was supported by the U.S. Department of Energy (DOE) Office of Biological and Environmental Research under the Genomics: GTL Program via the Shewanella Federation consortium

Rational emotions

We present here the concept of rational emotions: Emotions may be directly controlled and utilized in a conscious, analytic fashion, enabling an individual to size up a situation, to determine that a certain "mental state" is strategically advantageous and adjust accordingly. Building on the growing body of literature recognizing the vital role of emotions in determining decisions, we explore the complementary role of rational choice in choosing emotional states. Participants played the role of "recipient" in the dictator game, in which an anonymous "dictator" decides how to split an amount of money between himself and the recipient. A subset of recipients was given a monetary incentive to be angry at low-split offers. That subset demonstrated increased physiological arousal at low offers relative to high offers as well as more anger than other participants. These results provide a fresh outlook on human decision-making and contribute to the continuing effort to build more complete models of rational behavior. © 2012 Copyright 2012 Psychology Press, an imprint of the Taylor & Francis Group, an Informa business