Conditions for compatibility of quantum state assignments

Suppose N parties describe the state of a quantum system by N possibly different density operators. These N state assignments represent the beliefs of the parties about the system. We examine conditions for determining whether the N state assignments are compatible. We distinguish two kinds of procedures for assessing compatibility, the first based on the compatibility of the prior beliefs on which the N state assignments are based and the second based on the compatibility of predictive measurement probabilities they define. The first procedure leads to a compatibility criterion proposed by Brun, Finkelstein, and Mermin [BFM, Phys. Rev. A 65, 032315 (2002)]. The second procedure leads to a hierarchy of measurement-based compatibility criteria which is fundamentally different from the corresponding classical situation. Quantum mechanically none of the measurement-based compatibility criteria is equivalent to the BFM criterion.Comment: REVTEX 4, 19 pages, 1 postscript figur

Quantum Operation Time Reversal

The dynamics of an open quantum system can be described by a quantum operation, a linear, complete positive map of operators. Here, I exhibit a compact expression for the time reversal of a quantum operation, which is closely analogous to the time reversal of a classical Markov transition matrix. Since open quantum dynamics are stochastic, and not, in general, deterministic, the time reversal is not, in general, an inversion of the dynamics. Rather, the system relaxes towards equilibrium in both the forward and reverse time directions. The probability of a quantum trajectory and the conjugate, time reversed trajectory are related by the heat exchanged with the environment.Comment: 4 page

Long-term storage and age‐biased export of fluvial organic carbon: field evidence from West Iceland

Terrestrial organic carbon (OC) plays an important role in the carbon cycle, but questions remain regarding the controls and timescale(s) over which atmospheric CO₂ remains sequestered as particulate OC (POC). Motivated by observations that terrestrial POC is physically stored within soils and other shallow sedimentary deposits, we examined the role that sediment storage plays in the terrestrial OC cycle. Specifically, we tested the hypothesis that sediment storage impacts the age of terrestrial POC. We focused on the Efri Haukadalsá River catchment in Iceland as it lacks ancient sedimentary bedrock that would otherwise bias radiocarbon‐based determinations of POC storage duration by supplying pre‐aged “petrogenic” POC. Our radiocarbon measurements of riverine suspended sediments and deposits implicated millennial‐scale storage times. Comparison between the sample types (suspended and deposits) suggested an age offset between transported (suspended sediments) and stored (deposits) POC at the time of sampling, which is predicted by theory for the sediment age distribution in floodplains. We also observed that POC in suspended sediments is younger than the predicted mean storage duration generated from independent geomorphological data, which suggested an additional role for OC cycling. Consistent with this, we observed interparticle heterogeneity in the composition of POC by imaging our samples at the microscale using X‐ray absorption spectroscopy. Specifically, we found that particles within individual samples differed in their sulfur oxidation state, which is indicative of multiple origins and/or diagenetic histories. Altogether, our results support recent coupled sediment storage and OC cycling models and indicate that the physical drivers of sediment storage are important factors controlling the cadence of carbon cycling

Zero-Reachability in Probabilistic Multi-Counter Automata

We study the qualitative and quantitative zero-reachability problem in probabilistic multi-counter systems. We identify the undecidable variants of the problems, and then we concentrate on the remaining two cases. In the first case, when we are interested in the probability of all runs that visit zero in some counter, we show that the qualitative zero-reachability is decidable in time which is polynomial in the size of a given pMC and doubly exponential in the number of counters. Further, we show that the probability of all zero-reaching runs can be effectively approximated up to an arbitrarily small given error epsilon > 0 in time which is polynomial in log(epsilon), exponential in the size of a given pMC, and doubly exponential in the number of counters. In the second case, we are interested in the probability of all runs that visit zero in some counter different from the last counter. Here we show that the qualitative zero-reachability is decidable and SquareRootSum-hard, and the probability of all zero-reaching runs can be effectively approximated up to an arbitrarily small given error epsilon > 0 (these result applies to pMC satisfying a suitable technical condition that can be verified in polynomial time). The proof techniques invented in the second case allow to construct counterexamples for some classical results about ergodicity in stochastic Petri nets.Comment: 20 page

Computing the entropy of user navigation in the web

Navigation through the web, colloquially known as "surfing", is one of the main activities of users during web interaction. When users follow a navigation trail they often tend to get disoriented in terms of the goals of their original query and thus the discovery of typical user trails could be useful in providing navigation assistance. Herein, we give a theoretical underpinning of user navigation in terms of the entropy of an underlying Markov chain modelling the web topology. We present a novel method for online incremental computation of the entropy and a large deviation result regarding the length of a trail to realize the said entropy. We provide an error analysis for our estimation of the entropy in terms of the divergence between the empirical and actual probabilities. We then indicate applications of our algorithm in the area of web data mining. Finally, we present an extension of our technique to higher-order Markov chains by a suitable reduction of a higher-order Markov chain model to a first-order one

The thermodynamics of urban population flows

Orderliness, reflected via mathematical laws, is encountered in different frameworks involving social groups. Here we show that a thermodynamics can be constructed that macroscopically describes urban population flows. Microscopic dynamic equations and simulations with random walkers underlie the macroscopic approach. Our results might be regarded, via suitable analogies, as a step towards building an explicit social thermodynamics

Algorithmic Analysis of Qualitative and Quantitative Termination Problems for Affine Probabilistic Programs

In this paper, we consider termination of probabilistic programs with real-valued variables. The questions concerned are: 1. qualitative ones that ask (i) whether the program terminates with probability 1 (almost-sure termination) and (ii) whether the expected termination time is finite (finite termination); 2. quantitative ones that ask (i) to approximate the expected termination time (expectation problem) and (ii) to compute a bound B such that the probability to terminate after B steps decreases exponentially (concentration problem). To solve these questions, we utilize the notion of ranking supermartingales which is a powerful approach for proving termination of probabilistic programs. In detail, we focus on algorithmic synthesis of linear ranking-supermartingales over affine probabilistic programs (APP's) with both angelic and demonic non-determinism. An important subclass of APP's is LRAPP which is defined as the class of all APP's over which a linear ranking-supermartingale exists. Our main contributions are as follows. Firstly, we show that the membership problem of LRAPP (i) can be decided in polynomial time for APP's with at most demonic non-determinism, and (ii) is NP-hard and in PSPACE for APP's with angelic non-determinism; moreover, the NP-hardness result holds already for APP's without probability and demonic non-determinism. Secondly, we show that the concentration problem over LRAPP can be solved in the same complexity as for the membership problem of LRAPP. Finally, we show that the expectation problem over LRAPP can be solved in 2EXPTIME and is PSPACE-hard even for APP's without probability and non-determinism (i.e., deterministic programs). Our experimental results demonstrate the effectiveness of our approach to answer the qualitative and quantitative questions over APP's with at most demonic non-determinism.Comment: 24 pages, full version to the conference paper on POPL 201

Long-Range Navigation on Complex Networks using L\'evy Random Walks

We introduce a strategy of navigation in undirected networks, including regular, random, and complex networks, that is inspired by L\'evy random walks, generalizing previous navigation rules. We obtained exact expressions for the stationary probability distribution, the occupation probability, the mean first passage time, and the average time to reach a node on the network. We found that the long-range navigation using the L\'evy random walk strategy, compared with the normal random walk strategy, is more efficient at reducing the time to cover the network. The dynamical effect of using the L\'evy walk strategy is to transform a large-world network into a small world. Our exact results provide a general framework that connects two important fields: L\'evy navigation strategies and dynamics on complex networks.Comment: 6 pages, 3 figure

Quantitative multi-objective verification for probabilistic systems

We present a verification framework for analysing multiple quantitative objectives of systems that exhibit both nondeterministic and stochastic behaviour. These systems are modelled as probabilistic automata, enriched with cost or reward structures that capture, for example, energy usage or performance metrics. Quantitative properties of these models are expressed in a specification language that incorporates probabilistic safety and liveness properties, expected total cost or reward, and supports multiple objectives of these types. We propose and implement an efficient verification framework for such properties and then present two distinct applications of it: firstly, controller synthesis subject to multiple quantitative objectives; and, secondly, quantitative compositional verification. The practical applicability of both approaches is illustrated with experimental results from several large case studies