The paradox of biodiversity involves three elements, (i) mathematical models predict that species must differ in specific ways in order to coexist as stable ecological communities, (ii) such differences are difficult to identify, yet (iii) there is widespread evidence of stability in natural communities. Debate has centred on two views. The first explanation involves tradeoffs along a small number of axes, including 'colonization-competition', resource competition (light, water, nitrogen for plants, including the 'successional niche'), and life history (e.g. high-light growth vs. low-light survival and few large vs. many small seeds). The second view is neutrality, which assumes that species differences do not contribute to dynamics. Clark et al. (2004) presented a third explanation, that coexistence is inherently high dimensional, but still depends on species differences. We demonstrate that neither traditional low-dimensional tradeoffs nor neutrality can resolve the biodiversity paradox, in part by showing that they do not properly interpret stochasticity in statistical and in theoretical models. Unless sample sizes are small, traditional data modelling assures that species will appear different in a few dimensions, but those differences will rarely predict coexistence when parameter estimates are plugged into theoretical models. Contrary to standard interpretations, neutral models do not imply functional equivalence, but rather subsume species differences in stochastic terms. New hierarchical modelling techniques for inference reveal high-dimensional differences among species that can be quantified with random individual and temporal effects (RITES), i.e. process-level variation that results from many causes. We show that this variation is large, and that it stands in for species differences along unobserved dimensions that do contribute to diversity. High dimensional coexistence contrasts with the classical notions of tradeoffs along a few axes, which are often not found in data, and with 'neutral models', which mask, rather than eliminate, tradeoffs in stochastic terms. This mechanism can explain coexistence of species that would not occur with simple, low-dimensional tradeoff scenarios.

Optimal labeling schemes lead to efficient experimental protocols for quantum information processing by nuclear magnetic resonance (NMR). A systematic approach of finding optimal labeling schemes for a given computation is described here. The scheme is described for both quadrupolar systems and spin-1/2 systems. Finally, one of the optimal labeling scheme has been used to experimentally implement a quantum full-adder in a 4-qubit system by NMR, using the technique of transition selective pulses.would require a complex pulse sequence with a series of qubit selective pulses inter spaced with Hamiltonian evolutions; where as it requires only one transition selective π pulse between the states |111..10 and |111..11 .Further, it has been shown earlier, that relabeling of states simplifies the experimental protocol of certain operations [24,25]. While implementing half-adder and subtracter operations in a quadrupolar system, relabeling led to an efficient experimental scheme which requires less number of pulses than the conventional labeling [24]. The idea behind relabeling is as follows: For spin-1/2 systems, conventional labeling (CL) uses the following logic. The state in which all the spins are in identical state, such as |ααα...α are labeled as |000...0 and each spin flip is labeled as bit flip, namely |αβα...α = |010...0 . This logic labels each state with a well identified label and leads to identification of a spin as a qubit. Spin-selective pulses then act as qubit selective pulses and many pulse schemes have been developed which use spin (or qubit) selective pulses inter missioned with Hamiltonian (or exchange coupling, J) evolution periods [2,9,10]. On the other hand quantum information processing (QIP) has also been demonstrated using spins >1/2 nuclei using quadrupolar couplings in molecules partially oriented in liquid crystalline media. In such systems, spin is no more a qubit and it has been demonstrated that the 2 N energy levels of such systems can be utilized as N qubit systems. So far only spin-3/2 and 7/2 systems have been utilized respectively yielding 2 and 3 qubit systems [20][21][22][23][24][25]27]. In such systems a bit flip is not a spin-flip while it can be treated as a qubit flip. One can follow a CL scheme in which the lowest (or highest) energy level can be given the label |000 and each subsequent level can be labeled in increasing order of binary numbers (CL) or single bit flips (Gray code) as shown in Table 1. It has been conjectured earlier that all such schemes are acceptable so long as a single label is attached to one level and the scheme is retained throughout a given set of computations [24,25]. Indeed it was demonstrated that it is acceptable to search for "optimum labeling" scheme (OLS) such that a minimum number of unitary transforms are needed for a given set of computations [24].The utility of OLS is explained in the following: A transition-selective pulse has low power and small bandwidth and it excites a selected single quantum transition. That is, it can cause an operation...

We present a divide and conquer based algorithm for optimal quantum compression/decompression, using O(n(log 4 n) log log n) elementary quantum operations. Our result provides the first quasi-linear time algorithm for asymptotically optimal (in size and fidelity) quantum compression and decompression. We also outline the quantum gate array model to bring about this compression in a quantum computer. Our method uses various classical algorithmic tools to significantly improve the bound from the previous best known bound of O(n 3) for this operation.

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