How useful is a quantum dynamical operation for quantum information processing? Motivated by this question, we investigate several strength measures quantifying the resources intrinsic to a quantum operation. We develop a general theory of such strength measures, based on axiomatic considerations independent of state-based resources. The power of this theory is demonstrated with applications to quantum communication complexity, quantum computational complexity, and entanglement generation by unitary operations.

Which gates are universal for quantum computation? Although it is well known that certain gates on two-level quantum systems (qubits), such as the controlled-not, are universal when assisted by arbitrary one-qubit gates, it has only recently become clear precisely what class of two-qubit gates is universal in this sense. We present an elementary proof that any entangling two-qubit gate is universal for quantum computation, when assisted by one-qubit gates. A proof of this result for systems of arbitrary finite dimension has been provided by Brylinski and Brylinski; however, their proof relies on a long argument using advanced mathematics. In contrast, our proof provides a simple constructive procedure which is close to optimal and experimentally practical.

Axon guidance by molecular gradients plays a crucial role in wiring up the nervous system. However, the mechanisms axons use to detect gradients are largely unknown. We first develop a Bayesian ''ideal observer'' analysis of gradient detection by axons, based on the hypothesis that a principal constraint on gradient detection is intrinsic receptor binding noise. Second, from this model, we derive an equation predicting how the degree of response of an axon to a gradient should vary with gradient steepness and absolute concentration. Third, we confirm this prediction quantitatively by performing the first systematic experimental analysis of how axonal response varies with both these quantities. These experiments demonstrate a degree of sensitivity much higher than previously reported for any chemotacting system. Together, these results reveal both the quantitative constraints that must be satisfied for effective axonal guidance and the computational principles that may be used by the underlying signal transduction pathways, and allow predictions for the degree of response of axons to gradients in a wide variety of in vivo and in vitro settings.axon guidance ͉ chemotaxis ͉ growth cone ͉ nerve growth factor ͉ nerve regeneration E ndogenous chemical gradients are a key source of information used by developing axons when wiring up the nervous system. Furthermore, artificially generated gradients are a potential therapy for restoring connectivity after neural injury. Many of the molecular gradients that direct axons in the developing nervous system have recently been identified, together with some of the signaling pathways through which they operate (1-8). However, our understanding of the mechanisms by which axons actually detect gradients remains qualitative. This limits our ability to predict both the response of axons when gradients are perturbed and the optimal parameters for promoting regrowth after injury.To be guided by a gradient, axons must be able to detect small spatial variations in receptor binding. This requires integrating signals from spatially distributed receptors to make a decision as to the direction of the gradient. Resources within the growth cone can then be marshaled appropriately by this information, for instance, via the production of steep gradients of intracellular signaling molecules (8). Although there is evidence for a role for gradients of molecules such as Ca 2ϩ in this latter step (9, 10), very little is known about the computations required to accurately make the initial decision.What constrains the ability of an axon to make a decision regarding gradient direction? Both experimental and computational work addressing chemotaxis in related systems such as bacteria, leukocytes, and Dictyostelium has identified the fundamental role of noise in limiting gradient perception. Noise can arise from low numbers of ligand molecules, from the stochastic nature of receptor binding, and from intracellular signaling pathways (11-16). Such constraints must also apply to axonal gradient sensing (17...