Parts of Quantum States

It is shown that generic N-party pure quantum states (with equidimensional subsystems) are uniquely determined by their reduced states of just over half the parties; in other words, all the information in almost all N-party pure states is in the set of reduced states of just over half the parties. For N even, the reduced states in fewer than N/2 parties are shown to be an insufficient description of almost all states (similar results hold when N is odd). It is noted that Real Algebraic Geometry is a natural framework for any analysis of parts of quantum states: two simple polynomials, a quadratic and a cubic, contain all of their structure. Algorithmic techniques are described which can provide conditions for sets of reduced states to belong to pure or mixed states.Comment: 10 pages, 1 figur

Stability and Complexity of Minimising Probabilistic Automata

We consider the state-minimisation problem for weighted and probabilistic automata. We provide a numerically stable polynomial-time minimisation algorithm for weighted automata, with guaranteed bounds on the numerical error when run with floating-point arithmetic. Our algorithm can also be used for "lossy" minimisation with bounded error. We show an application in image compression. In the second part of the paper we study the complexity of the minimisation problem for probabilistic automata. We prove that the problem is NP-hard and in PSPACE, improving a recent EXPTIME-result.Comment: This is the full version of an ICALP'14 pape

Semidefinite Representation of the kk-Ellipse

The $k$-ellipse is the plane algebraic curve consisting of all points whose sum of distances from $k$ given points is a fixed number. The polynomial equation defining the $k$-ellipse has degree $2^k$ if $k$ is odd and degree $2^k{-}\binom{k}{k/2}$ if $k$ is even. We express this polynomial equation as the determinant of a symmetric matrix of linear polynomials. Our representation extends to weighted $k$-ellipses and $k$-ellipsoids in arbitrary dimensions, and it leads to new geometric applications of semidefinite programming.Comment: 16 pages, 5 figure

An update on the Hirsch conjecture

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most fundamental, basic and old problems in polytope theory, what we know is quite scarce. Most notably, no polynomial upper bound is known for the diameters that are conjectured to be linear. In contrast, very few polytopes are known where the bound $n-d$ is attained. This paper collects known results and remarks both on the positive and on the negative side of the conjecture. Some proofs are included, but only those that we hope are accessible to a general mathematical audience without introducing too many technicalities.Comment: 28 pages, 6 figures. Many proofs have been taken out from version 2 and put into the appendix arXiv:0912.423

A scaling-invariant algorithm for linear programming whose running time depends only on the constraint matrix

Following the breakthrough work of Tardos (Oper. Res. '86) in the bit-complexity model, Vavasis and Ye (Math. Prog. '96) gave the first exact algorithm for linear programming in the real model of computation with running time depending only on the constraint matrix. For solving a linear program (LP) max cx, Ax = b, x ≥ 0, A g m × n, Vavasis and Ye developed a primal-dual interior point method using a g€layered least squares' (LLS) step, and showed that O(n3.5 log(χA+n)) iterations suffice to solve (LP) exactly, where χA is a condition measure controlling the size of solutions to linear systems related to A. Monteiro and Tsuchiya (SIAM J. Optim. '03), noting that the central path is invariant under rescalings of the columns of A and c, asked whether there exists an LP algorithm depending instead on the measure χA∗, defined as the minimum χAD value achievable by a column rescaling AD of A, and gave strong evidence that this should be the case. We resolve this open question affirmatively. Our first main contribution is an O(m2 n2 + n3) time algorithm which works on the linear matroid of A to compute a nearly optimal diagonal rescaling D satisfying χAD ≤ n(χ∗)3. This algorithm also allows us to approximate the value of χA up to a factor n (χ∗)2. This result is in (surprising) contrast to that of Tunçel (Math. Prog. '99), who showed NP-hardness for approximating χA to within 2poly(rank(A)). The key insight for our algorithm is to work with ratios gi/gj of circuits of A - i.e., minimal linear dependencies Ag=0 - which allow us to approximate the value of χA∗ by a maximum geometric mean cycle computation in what we call the g€circuit ratio digraph' of A. While this resolves Monteiro and Tsuchiya's question by appropriate preprocessing, it falls short of providing either a truly scaling invariant algorithm or an improvement upon the base LLS analysis. In this vein, as our second main contribution we develop a scaling invariant LLS algorithm, which uses and dynamically maintains improving estimates of the circuit ratio digraph, together with a refined potential function based analysis for LLS algorithms in general. With this analysis, we derive an improved O(n2.5 lognlog(χA∗+n)) iteration bound for optimally solving (LP) using our algorithm. The same argument also yields a factor n/logn improvement on the iteration complexity bound of the original Vavasis-Ye algorithm

Southeast Florida large Orbicella faveolata are highly fecund without evident disease intervention effects

The recent widespread mortality and tissue loss in Florida from stony coral tissue loss disease (SCTLD) has propelled the need for assisted reproduction to restore reefs, especially for the ESA listed species Orbicella faveolata. In situ gamete collection can be challenging due to the weather and resources (boats and divers) required during the expected spawning window. In the northern portion of the Florida coral reef tract, coral spawn collection has been even more difficult due to historical inconsistency in annual spawning times and the potential for “zombie” corals, i.e. large but reproductively senescent individuals. Therefore, we examined the current reproductive potential of seven large (>2 m diameter) O. faveolata colonies from this region, quantified their fecundity, and estimated the spawning timeframe using histology. Additionally, we explored whether previous SCTLD lesion amoxycillin treatments affected reproductive metrics. Understanding the reproductive capacity and spawning timing of these large corals, given their history of disease and disease treatment, is critical to evaluate potential impacts of SCTLD treatments and the success of assisted reproduction efforts. The histological analysis coupled with in-water observations indicated a probable split-spawn in these individuals in 2020, although the dates of spawning may not be consistent with predictions for the wider Caribbean or with other colonies in Miami and the Florida Keys. All seven large O. faveolata colonies were found to contain abundant oocytes, with no obvious impact of SCTLD treatments on gamete development or fecundity

Vaccination with M2e-Based Multiple Antigenic Peptides: Characterization of the B Cell Response and Protection Efficacy in Inbred and Outbred Mice

The extracellular domain of the influenza A virus protein matrix protein 2 (M2e) is remarkably conserved between various human isolates and thus is a viable target antigen for a universal influenza vaccine. With the goal of inducing protection in multiple mouse haplotypes, M2e-based multiple antigenic peptides (M2e-MAP) were synthesized to contain promiscuous T helper determinants from the Plasmodium falciparum circumsporozoite protein, the hepatitis B virus antigen and the influenza virus hemagglutinin. Here, we investigated the nature of the M2e-MAP-induced B cell response in terms of the distribution of antibody (Ab) secreting cells (ASCs) and Ab isotypes, and tested the protective efficacy in various mouse strains.Immunization of BALB/c mice with M2e-MAPs together with potent adjuvants, CpG 1826 oligonucleotides (ODN) and cholera toxin (CT) elicited high M2e-specific serum Ab titers that protected mice against viral challenge. Subcutaneous (s.c.) and intranasal (i.n.) delivery of M2e-MAPs resulted in the induction of IgG in serum and airway secretions, however only i.n. immunization induced anti-M2e IgA ASCs locally in the lungs, correlating with M2-specific IgA in the bronchio-alveolar lavage (BAL). Interestingly, both routes of vaccination resulted in equal protection against viral challenge. Moreover, M2e-MAPs induced cross-reactive and protective responses to diverse M2e peptides and variant influenza viruses. However, in contrast to BALB/c mice, immunization of other inbred and outbred mouse strains did not induce protective Abs. This correlated with a defect in T cell but not B cell responsiveness to the M2e-MAPs.Anti-M2e Abs induced by M2e-MAPs are highly cross-reactive and can mediate protection to variant viruses. Although synthetic MAPs are promising designs for vaccines, future constructs will need to be optimized for use in the genetically heterogeneous human population