Phenomenology of fully many-body-localized systems

We consider fully many-body localized systems, i.e. isolated quantum systems where all the many-body eigenstates of the Hamiltonian are localized. We define a sense in which such systems are integrable, with localized conserved operators. These localized operators are interacting pseudospins, and the Hamiltonian is such that unitary time evolution produces dephasing but not "flips" of these pseudospins. As a result, an initial quantum state of a pseudospin can in principle be recovered via (pseudospin) echo procedures. We discuss how the exponentially decaying interactions between pseudospins lead to logarithmic-in-time spreading of entanglement starting from nonentangled initial states. These systems exhibit multiple different length scales that can be defined from exponential functions of distance; we suggest that some of these decay lengths diverge at the phase transition out of the fully many-body localized phase while others remain finite.Comment: 5 pages. Some of this paper has already appeared in: Huse and Oganesyan, arXiv:1305.491

A Complete Axiomatization of Quantified Differential Dynamic Logic for Distributed Hybrid Systems

We address a fundamental mismatch between the combinations of dynamics that occur in cyber-physical systems and the limited kinds of dynamics supported in analysis. Modern applications combine communication, computation, and control. They may even form dynamic distributed networks, where neither structure nor dimension stay the same while the system follows hybrid dynamics, i.e., mixed discrete and continuous dynamics. We provide the logical foundations for closing this analytic gap. We develop a formal model for distributed hybrid systems. It combines quantified differential equations with quantified assignments and dynamic dimensionality-changes. We introduce a dynamic logic for verifying distributed hybrid systems and present a proof calculus for this logic. This is the first formal verification approach for distributed hybrid systems. We prove that our calculus is a sound and complete axiomatization of the behavior of distributed hybrid systems relative to quantified differential equations. In our calculus we have proven collision freedom in distributed car control even when an unbounded number of new cars may appear dynamically on the road

On the Termination of Linear and Affine Programs over the Integers

The termination problem for affine programs over the integers was left open in\cite{Braverman}. For more that a decade, it has been considered and cited as a challenging open problem. To the best of our knowledge, we present here the most complete response to this issue: we show that termination for affine programs over Z is decidable under an assumption holding for almost all affine programs, except for an extremely small class of zero Lesbegue measure. We use the notion of asymptotically non-terminating initial variable values} (ANT, for short) for linear loop programs over Z. Those values are directly associated to initial variable values for which the corresponding program does not terminate. We reduce the termination problem of linear affine programs over the integers to the emptiness check of a specific ANT set of initial variable values. For this class of linear or affine programs, we prove that the corresponding ANT set is a semi-linear space and we provide a powerful computational methods allowing the automatic generation of these $ANT$ sets. Moreover, we are able to address the conditional termination problem too. In other words, by taking ANT set complements, we obtain a precise under-approximation of the set of inputs for which the program does terminate.Comment: arXiv admin note: substantial text overlap with arXiv:1407.455

Heisenberg's uncertainty principle

Heisenberg's uncertainty principle is usually taken to express a limitation of operational possibilities imposed by quantum mechanics. Here we demonstrate that the full content of this principle also includes its positive role as a condition ensuring that mutually exclusive experimental options can be reconciled if an appropriate trade-off is accepted. The uncertainty principle is shown to appear in three manifestations, in the form of uncertainty relations: for the widths of the position and momentum distributions in any quantum state; for the inaccuracies of any joint measurement of these quantities; and for the inaccuracy of a measurement of one of the quantities and the ensuing disturbance in the distribution of the other quantity. Whilst conceptually distinct, these three kinds of uncertainty relations are shown to be closely related formally. Finally, we survey models and experimental implementations of joint measurements of position and momentum and comment briefly on the status of experimental tests of the uncertainty principle. (c) 2007 Elsevier B.V. All rights reserved

Minimizing energy below the glass thresholds

Focusing on the optimization version of the random K-satisfiability problem, the MAX-K-SAT problem, we study the performance of the finite energy version of the Survey Propagation (SP) algorithm. We show that a simple (linear time) backtrack decimation strategy is sufficient to reach configurations well below the lower bound for the dynamic threshold energy and very close to the analytic prediction for the optimal ground states. A comparative numerical study on one of the most efficient local search procedures is also given.Comment: 12 pages, submitted to Phys. Rev. E, accepted for publicatio