One-dimensional many-body entangled open quantum systems with tensor network methods

We present a collection of methods to simulate entangled dynamics of open quantum systems governed by the Lindblad equation with tensor network methods. Tensor network methods using matrix product states have been proven very useful to simulate many-body quantum systems and have driven many innovations in research. Since the matrix product state design is tailored for closed one-dimensional systems governed by the Schr\"odinger equation, the next step for many-body quantum dynamics is the simulation of open quantum systems. We review the three dominant approaches to the simulation of open quantum systems via the Lindblad master equation: quantum trajectories, matrix product density operators, and locally purified tensor networks. Selected examples guide possible applications of the methods and serve moreover as a benchmark between the techniques. These examples include the finite temperature states of the transverse quantum Ising model, the dynamics of an exciton traveling under the influence of spontaneous emission and dephasing, and a double-well potential simulated with the Bose-Hubbard model including dephasing. We analyze which approach is favorable leading to the conclusion that a complete set of all three methods is most beneficial, push- ing the limits of different scenarios. The convergence studies using analytical results for macroscopic variables and exact diagonalization methods as comparison, show, for example, that matrix product density operators are favorable for the exciton problem in our study. All three methods access the same library, i.e., the software package Open Source Matrix Product States, allowing us to have a meaningful comparison between the approaches based on the selected examples. For example, tensor operations are accessed from the same subroutines and with the same optimization eliminating one possible bias in a comparison of such numerical methods.Comment: 24 pages, 8 figures. Small extension of time evolution section and moving quantum simulators to introduction in comparison to v

Open source Matrix Product States: Opening ways to simulate entangled many-body quantum systems in one dimension

Numerical simulations are a powerful tool to study quantum systems beyond exactly solvable systems lacking an analytic expression. For one-dimensional entangled quantum systems, tensor network methods, amongst them Matrix Product States (MPSs), have attracted interest from different fields of quantum physics ranging from solid state systems to quantum simulators and quantum computing. Our open source MPS code provides the community with a toolset to analyze the statics and dynamics of one-dimensional quantum systems. Here, we present our open source library, Open Source Matrix Product States (OSMPS), of MPS methods implemented in Python and Fortran2003. The library includes tools for ground state calculation and excited states via the variational ansatz. We also support ground states for infinite systems with translational invariance. Dynamics are simulated with different algorithms, including three algorithms with support for long-range interactions. Convenient features include built-in support for fermionic systems and number conservation with rotational $\mathcal{U}(1)$ and discrete $\mathbb{Z}_2$ symmetries for finite systems, as well as data parallelism with MPI. We explain the principles and techniques used in this library along with examples of how to efficiently use the general interfaces to analyze the Ising and Bose-Hubbard models. This description includes the preparation of simulations as well as dispatching and post-processing of them

Analyzing Tag Semantics Across Collaborative Tagging Systems

The objective of our group was to exploit state-of-the-art Information Retrieval methods for finding associations and dependencies between tags, capturing and representing differences in tagging behavior and vocabulary of various folksonomies, with the overall aim to better understand the semantics of tags and the tagging process. Therefore we analyze the semantic content of tags in the Flickr and Delicious folksonomies. We find that: tag context similarity leads to meaningful results in Flickr, despite its narrow folksonomy character; the comparison of tags across Flickr and Delicious shows little semantic overlap, being tags in Flickr associated more to visual aspects rather than technological as it seems to be in Delicious; there are regions in the tag-tag space, provided with the cosine similarity metric, that are characterized by high density; the order of tags inside a post has a semantic relevance

Accounting for risk of non linear portfolios: a novel Fourier approach

The presence of non linear instruments is responsible for the emergence of non Gaussian features in the price changes distribution of realistic portfolios, even for Normally distributed risk factors. This is especially true for the benchmark Delta Gamma Normal model, which in general exhibits exponentially damped power law tails. We show how the knowledge of the model characteristic function leads to Fourier representations for two standard risk measures, the Value at Risk and the Expected Shortfall, and for their sensitivities with respect to the model parameters. We detail the numerical implementation of our formulae and we emphasizes the reliability and efficiency of our results in comparison with Monte Carlo simulation.Comment: 10 pages, 12 figures. Final version accepted for publication on Eur. Phys. J.

Gain-loss based convex risk limits in discrete-time trading

Cataloged from PDF version of article.We present an approach for pricing and hedging in incomplete markets, which encompasses other recently introduced approaches for the same purpose. In a discrete time, finite space probability framework conducive to numerical computation we introduce a gain–loss ratio based restriction controlled by a loss aversion parameter, and characterize portfolio values which can be traded in discrete time to acceptability. The new risk measure specializes to a well-known risk measure (the Carr–Geman– Madan risk measure) for a specific choice of the risk aversion parameter, and to a robust version of the gain–loss measure (the Bernardo–Ledoit proposal) for a specific choice of thresholds. The result implies potentially tighter price bounds for contingent claims than the no-arbitrage price bounds. We illustrate the price bounds through numerical examples from option pricing

Anomalous light propagation and continuous lasing without inversion in an open driven VV-system

We explore a driven three-level $V$-system coupled to an environment with dynamics governed by the Lindblad master equation. We perform a transformation into superoperator space, which brings the Lindblad equation into a Schr\"{o}dinger-like, thus allowing us to obtain an exact analytical solution for the time-dependence of the density matrix in a closed form. We demonstrate a regime for continuous lasing without inversion for driving with a continuous wave laser. We show a mechanism for achieving superluminal, negative, and vanishing light pulse group velocities and provide a range of physical parameters for realizing these regimes experimentally