Positivity violations of the density operator in the Caldeira-Leggett master equation

The Caldeira-Leggett master equation as an example of Markovian master equation without Lindblad form is investigated for mathematical consistency. We explore situations both analytically and numerically where the positivity violations of the density operator occur. We reinforce some known knowledge about this problem but also find new surprising cases. Our analytical results are based on the full solution of the Caldeira-Leggett master equation obtained via the method of characteristics. The preservation of positivity is mainly investigated with the help of the density operator's purity and we give also some numerical results about the violation of the Robertson-Schr\"odinger uncertainty relation.Comment: 13 pages, 12 figure

An entropy production based method for determining the position diffusion's coefficient of a quantum Brownian motion

Quantum Brownian motion of a harmonic oscillator in the Markovian approximation is described by the respective Caldeira-Leggett master equation. This master equation can be brought into Lindblad form by adding a position diffusion term to it. The coefficient of this term is either customarily taken to be the lower bound dictated by the Dekker inequality or determined by more detailed derivations on the linearly damped quantum harmonic oscillator. In this paper, we explore the theoretical possibilities of determining the position diffusion term's coefficient by analyzing the entropy production of the master equation.Comment: 13 pages, 10 figure

Choi representation of completely positive maps: a technical introduction

This is a very brief operational introduction to the Choi representation of completely positive maps, i.e. quantum channels. It focuses on certain useful calculational techniques which are presented in full detail

Calculation of the even-odd energy difference in superfluid Fermi systems using the pseudopotential theory

The pseudopotential theory is extended to the Bogoliubov-de Gennes equations to determine the excess energy when one atom is added to the trapped superfluid Fermi system with even number of atoms. Particular attention is paid to systems being at the Feshbach resonance point. The results for relatively small particle numbers are in harmony with the Monte Carlo calculations, but are also relevant for systems with larger particle numbers. Concerning the additional one-quasiparticle state we define and determine two new universal numbers to characterize its widths. Copyright © EPLA, 2012

Analytical evaluation of the coefficients of the Hu-Paz-Zhang master equation: Ohmic spectral density, zero temperature, and consistency check

We investigate the exact master equation of Hu, Paz, and Zhang for a quantum harmonic oscillator at zero temperature with a Lorentz-Drude type Ohmic spectral density. This master equation plays an important role in the study of quantum Brownian motion and in various applications. In this paper, we give an analytical evaluation of the coefficients of this non-Markovian master equation without Lindblad form, which allows us to investigate consistencies of the solutions, the positivity of the stationary density operator, and the boundaries of the model's parameters.Comment: 17 pages, 8 figure

Quantifying and Classifying Streamflow Ensembles Using a Broad Range of Metrics for an Evidence‐Based Analysis: Colorado River Case Study

Stochastic hydrology produces ensembles of time series that represent plausible future streamflow to simulate and test the operation of water resource systems. A premise of stochastic hydrology is that ensembles should be statistically representative of what may occur in the future. In the past, the application of this premise has involved producing ensembles that are statistically equivalent to the observed or historical streamflow sequence. This requires a number of metrics or statistics that can be used to test statistical similarity. However, with climate change, the past may no longer be representative of the future. Ensembles to test future systems operations should recognize non‐stationarity and include time series representing expected changes. This poses challenges for their testing and validation. In this paper, we suggest an evidence‐based analysis in which streamflow ensembles, whether statistically similar to and representative of the past or a changing future, should be characterized and assessed using an extensive set of statistical metrics. We have assembled a broad set of metrics and applied them to annual streamflow in the Colorado River at Lees Ferry to illustrate the approach. We have also developed a tree‐based classification approach to categorize both ensembles and metrics. This approach provides a way to visualize and interpret differences between streamflow ensembles. The metrics presented, along with the classification, provide an analytical framework for characterizing and assessing the suitability of future streamflow ensembles, recognizing the presence of non‐stationarity. This contributes to better planning in large river basins, such as the Colorado, facing water supply shortages

Range of applicability of the Hu-Paz-Zhang master equation

We investigate a case of the Hu-Paz-Zhang master equation of the Caldeira-Leggett model without Lindblad form obtained in the weak-coupling limit up to the second-order perturbation. In our study, we use Gaussian initial states to be able to employ a sufficient and necessary condition, which can expose positivity violations of the density operator during the time evolution. We demonstrate that the evolution of the non-Markovian master equation has problems when the stationary solution is not a positive operator, i.e., does not have physical interpretation. We also show that solutions always remain physical for small-times of evolution. Moreover, we identify a strong anomalous behavior, when the trace of the solution is diverging. We also provide results for the corresponding Markovian master equation and show that positivity violations occur for various types of initial conditions even when the stationary solution is a positive operator. Based on our numerical results, we conclude that this non-Markovian master equation is superior to the corresponding Markovian one.Comment: 14 pages, 19 figure

Newton's identities and positivity of trace class integral operators

We provide a countable set of conditions based on elementary symmetric polynomials that are necessary and sufficient for a trace class integral operator to be positive semidefinite, which is an important cornerstone for quantum theory in phase-space representation. We also present a new, efficiently computable algorithm based on Newton's identities. Our test of positivity is much more sensitive than the ones given by the linear entropy and Robertson-Schr\"odinger's uncertainty relations; our first condition is equivalent to the non-negativity of the linear entropy.Comment: 15 pages, 6 figure

Derandomizing Codes for the Binary Adversarial Wiretap Channel of Type II

We revisit the binary adversarial wiretap channel (AWTC) of type II in which an active adversary can read a fraction $r$ and flip a fraction $p$ of codeword bits. The semantic-secrecy capacity of the AWTC II is partially known, where the best-known lower bound is non-constructive, proven via a random coding argument that uses a large number (that is exponential in blocklength $n$) of random bits to seed the random code. In this paper, we establish a new derandomization result in which we match the best-known lower bound of $1-H_2(p)-r$ where $H_2(\cdot)$ is the binary entropy function via a random code that uses a small seed of only $O(n^2)$ bits. Our random code construction is a novel application of pseudolinear codes -- a class of non-linear codes that have $k$-wise independent codewords when picked at random where $k$ is a design parameter. As the key technical tool in our analysis, we provide a soft-covering lemma in the flavor of Goldfeld, Cuff and Permuter (Trans. Inf. Theory 2016) that holds for random codes with $k$-wise independent codewords

Wetting behaviour and reactivity between liquid Gd and ZrO2 substrate

The wetting behavior and reactivity between molten pure Gd and polycrystalline 3YSZ substrate (ZrO2 stabilized with 3 wt% of Y2O3)were experimentally determined by a sessile drop method using a classical contact heating coupled with drop pushing procedure. The test was performed under an inert flowing gas atmosphere (Ar) at two temperatures of 1362°C and 1412°C. Immediately after melting (Tm=1341°C), liquid Gd did not wet the substrate forming a contact angle of θ=141°. The non-wetting to wetting transition (θ < 90°) took place after about 110 seconds of interaction and was accompanied by a sudden decrease in the contact angle value to 67°. Further heating of the couple to 1412 °C did not affect wetting (θ=67°±1°). The solidified Gd/3YSZ couple was studied by means of optical microscopy and scanning electron microscopy coupled with X-ray energy dispersive spectroscopy. Structural investigations revealed that the wettability in the Gd/3YSZ system is of a reactive nature associated with the formation of a continuous layer of a wettable reaction product Gd2Zr2O7