Entanglement entropy in restricted integrable spin-chains

In questa tesi abbiamo presentato il calcolo dell’Entropia di Entanglement di un sistema quantistico unidimensionale integrabile la cui rappresentazione statistica é data dal modello RSOS, il cui punto critico é una realizzazione su reticolo di tutti i modelli conformi minimali. Sfruttando l’integrabilitá di questi modelli, abbiamo svolto il calcolo utilizzando la tecnica delle Corner Transfer Matrices (CTM). Il risultato ottenuto si discosta leggermente dalla previsione di J. Cardy e P. Calabrese ricavata utilizzando la teoria dei campi conformi descriventi il punto critico. Questa differenza é stata imputata alla non-unitarietá del modello studiato, in quanto la tecnica CTM studia il ground state, mentre la previsione di Cardy e Calabrese si focalizza sul vuoto conforme del modello: nel caso dei sistemi non-unitari questi due stati non coincidono, ma possono essere visti come eccitazioni l’uno dell’altro. Dato che l’Entanglement é un fenomeno genuinamente quantistico e il modello RSOS descrive un sistema statistico classico bidimensionale, abbiamo proposto una Hamiltoniana quantistica unidimensionale integrabile la cui rappresentazione statistica é data dal modello RSOS.

Implementation of network entropy algorithms on hpc machines, with application to high-dimensional experimental data

Network Theory is a prolific and lively field, especially when it approaches Biology. New concepts from this theory find application in areas where extensive datasets are already available for analysis, without the need to invest money to collect them. The only tools that are necessary to accomplish an analysis are easily accessible: a computing machine and a good algorithm. As these two tools progress, thanks to technology advancement and human efforts, wider and wider datasets can be analysed. The aim of this paper is twofold. Firstly, to provide an overview of one of these concepts, which originates at the meeting point between Network Theory and Statistical Mechanics: the entropy of a network ensemble. This quantity has been described from different angles in the literature. Our approach tries to be a synthesis of the different points of view. The second part of the work is devoted to presenting a parallel algorithm that can evaluate this quantity over an extensive dataset. Eventually, the algorithm will also be used to analyse high-throughput data coming from biology.

Horizon entropy from scratch

Scopo di questo lavoro di tesi è lo studio di alcune proprietà delle teorie generali della gravità in relazione alla meccanica e la termodinamica dei buchi neri. In particolare, la trattazione che seguirà ha lo scopo di fornire un percorso autoconsistente che conduca alla nozione di entropia di un orizzonte descritta in termini delle carica di Noether associata all'invarianza del funzionale d'azione, che descrive la teoria gravitazionale in considerazione, per trasformazioni di coordinate generali. Si presterà particolare attenzione ad alcune proprietà geometriche della Lagrangiana, proprietà che sono indipendenti dalla particolare forma della teoria che si sta prendendo in considerazione; trattasi cioè non di proprietà dinamiche, legate cioè alla forma delle equazioni del moto del campo gravitazionale, ma piuttosto caratteristiche proprie di qualunque varietà rappresentante uno spaziotempo curvo. Queste caratteristiche fanno sì che ogni teoria generale della gravità possieda alcune grandezze definite localmente sullo spaziotempo, in particolare una corrente di Noether e la carica ad essa associata. La forma esplicita della corrente e della carica dipende invece dalla Lagrangiana che si sceglie di adottare per descrivere il campo gravitazionale. Il lavoro di tesi sarà orientato prima a descrivere come questa corrente di Noether emerge in qualunque teoria della gravità invariante per trasformazioni generali e come essa viene esplicitata nel caso di Lagrangiane particolari, per poi identificare la carica ad essa associata come una grandezza connessa all' entropia di un orizzonte in qualunque teoria generale della gravità.

Entanglement entropy in 1-d quantum spin chains

In questa tesi abbiamo studiato il comportamento delle entropie di Entanglement e dello spettro di Entanglement nel modello XYZ attraverso delle simulazioni numeriche. Le formule per le entropie di Von Neumann e di Renyi nel caso di una catena bipartita infinita esistevano già, ma mancavano ancora dei test numerici dettagliati. Inoltre, rispetto alla formula per l'Entropia di Entanglement di J. Cardy e P. Calabrese per sistemi non critici, tali relazioni presentano delle correzioni che non hanno ancora una spiegazione analitica: i risultati delle simulazioni numeriche ne hanno confermato la presenza. Abbiamo inoltre testato l'ipotesi che lo Schmidt Gap sia proporzionale a uno dei parametri d'ordine della teoria, e infine abbiamo simulato numericamente l'andamento delle Entropie e dello spettro di Entanglement in funzione della lunghezza della catena di spin. Ciò è stato possibile solo introducendo dei campi magnetici ''ad hoc'' nella catena, con la proprietà che l'andamento delle suddette quantità varia a seconda di come vengono disposti tali campi. Abbiamo quindi discusso i vari risultati ottenuti.

Prediction of hypertensive crisis based on average, variability and approximate entropy of 24-h ambulatory blood pressure monitoring

Approximate entropy (ApEn) of blood pressure (BP) can be easily measured based on software analysing 24-h ambulatory BP monitoring (ABPM), but the clinical value of this measure is unknown. In a prospective study we investigated whether ApEn of BP predicts, in addition to average and variability of BP, the risk of hypertensive crisis. In 57 patients with known hypertension we measured ApEn, average and variability of systolic and diastolic BP based on 24-h ABPM. Eight of these fifty-seven patients developed hypertensive crisis during follow-up (mean follow-up duration 726 days). In bivariate regression analysis, ApEn of systolic BP (P<0.01), average of systolic BP (P=0.02) and average of diastolic BP (P=0.03) were significant predictors of hypertensive crisis. The incidence rate ratio of hypertensive crisis was 14.0 (95% confidence interval (CI) 1.8, 631.5; P<0.01) for high ApEn of systolic BP as compared to low values. In multivariable regression analysis, ApEn of systolic (P=0.01) and average of diastolic BP (P<0.01) were independent predictors of hypertensive crisis. A combination of these two measures had a positive predictive value of 75%, and a negative predictive value of 91%, respectively. ApEn, combined with other measures of 24-h ABPM, is a potentially powerful predictor of hypertensive crisis. If confirmed in independent samples, these findings have major clinical implications since measures predicting the risk of hypertensive crisis define patients requiring intensive follow-up and intensified therapy.

Entropy and bispectral index for assessment of sedation, analgesia and the effects of unpleasant stimuli in critically ill patients: an observational study

INTRODUCTION: Sedative and analgesic drugs are frequently used in critically ill patients. Their overuse may prolong mechanical ventilation and length of stay in the intensive care unit. Guidelines recommend use of sedation protocols that include sedation scores and trials of sedation cessation to minimize drug use. We evaluated processed electroencephalography (response and state entropy and bispectral index) as an adjunct to monitoring effects of commonly used sedative and analgesic drugs and intratracheal suctioning. METHODS: Electrodes for monitoring bispectral index and entropy were placed on the foreheads of 44 critically ill patients requiring mechanical ventilation and who previously had no brain dysfunction. Sedation was targeted individually using the Ramsay Sedation Scale, recorded every 2 hours or more frequently. Use of and indications for sedative and analgesic drugs and intratracheal suctioning were recorded manually and using a camera. At the end of the study, processed electroencephalographical and haemodynamic variables collected before and after each drug application and tracheal suctioning were analyzed. Ramsay score was used for comparison with processed electroencephalography when assessed within 15 minutes of an intervention. RESULTS: The indications for boli of sedative drugs exhibited statistically significant, albeit clinically irrelevant, differences in terms of their association with processed electroencephalographical parameters. Electroencephalographical variables decreased significantly after bolus, but a specific pattern in electroencephalographical variables before drug administration was not identified. The same was true for opiate administration. At both 30 minutes and 2 minutes before intratracheal suctioning, there was no difference in electroencephalographical or clinical signs in patients who had or had not received drugs 10 minutes before suctioning. Among patients who received drugs, electroencephalographical parameters returned to baseline more rapidly. In those cases in which Ramsay score was assessed before the event, processed electroencephalography exhibited high variation. CONCLUSIONS: Unpleasant or painful stimuli and sedative and analgesic drugs are associated with significant changes in processed electroencephalographical parameters. However, clinical indications for drug administration were not reflected by these electroencephalographical parameters, and barely by sedation level before drug administration or tracheal suction. This precludes incorporation of entropy and bispectral index as target variables for sedation and analgesia protocols in critically ill patients.

Auditory event-related potentials, bispectral index, and entropy for the discrimination of different levels of sedation in intensive care unit patients

BACKGROUND: Sedation protocols, including the use of sedation scales and regular sedation stops, help to reduce the length of mechanical ventilation and intensive care unit stay. Because clinical assessment of depth of sedation is labor-intensive, performed only intermittently, and interferes with sedation and sleep, processed electrophysiological signals from the brain have gained interest as surrogates. We hypothesized that auditory event-related potentials (ERPs), Bispectral Index (BIS), and Entropy can discriminate among clinically relevant sedation levels. METHODS: We studied 10 patients after elective thoracic or abdominal surgery with general anesthesia. Electroencephalogram, BIS, state entropy (SE), response entropy (RE), and ERPs were recorded immediately after surgery in the intensive care unit at Richmond Agitation-Sedation Scale (RASS) scores of -5 (very deep sedation), -4 (deep sedation), -3 to -1 (moderate sedation), and 0 (awake) during decreasing target-controlled sedation with propofol and remifentanil. Reference measurements for baseline levels were performed before or several days after the operation. RESULTS: At baseline, RASS -5, RASS -4, RASS -3 to -1, and RASS 0, BIS was 94 [4] (median, IQR), 47 [15], 68 [9], 75 [10], and 88 [6]; SE was 87 [3], 46 [10], 60 [22], 74 [21], and 87 [5]; and RE was 97 [4], 48 [9], 71 [25], 81 [18], and 96 [3], respectively (all P < 0.05, Friedman Test). Both BIS and Entropy had high variabilities. When ERP N100 amplitudes were considered alone, ERPs did not differ significantly among sedation levels. Nevertheless, discriminant ERP analysis including two parameters of principal component analysis revealed a prediction probability PK value of 0.89 for differentiating deep sedation, moderate sedation, and awake state. The corresponding PK for RE, SE, and BIS was 0.88, 0.89, and 0.85, respectively. CONCLUSIONS: Neither ERPs nor BIS or Entropy can replace clinical sedation assessment with standard scoring systems. Discrimination among very deep, deep to moderate, and no sedation after general anesthesia can be provided by ERPs and processed electroencephalograms, with similar P(K)s. The high inter- and intraindividual variability of Entropy and BIS precludes defining a target range of values to predict the sedation level in critically ill patients using these parameters. The variability of ERPs is unknown.

Intra- and inter-individual variation of BIS-index and Entropy during controlled sedation with midazolam/remifentanil and dexmedetomidine/remifentanil in healthy volunteers: an interventional study

INTRODUCTION: We studied intra-individual and inter-individual variability of two online sedation monitors, BIS and Entropy, in volunteers under sedation. METHODS: Ten healthy volunteers were sedated in a stepwise manner with doses of either midazolam and remifentanil or dexmedetomidine and remifentanil. One week later the procedure was repeated with the remaining drug combination. The doses were adjusted to achieve three different sedation levels (Ramsay Scores 2, 3 and 4) and controlled by a computer-driven drug-delivery system to maintain stable plasma concentrations of the drugs. At each level of sedation, BIS and Entropy (response entropy and state entropy) values were recorded for 20 minutes. Baseline recordings were obtained before the sedative medications were administered. RESULTS: Both inter-individual and intra-individual variability increased as the sedation level deepened. Entropy values showed greater variability than BIS(R) values, and the variability was greater during dexmedetomidine/remifentanil sedation than during midazolam/remifentanil sedation. CONCLUSIONS: The large intra-individual and inter-individual variability of BIS and Entropy values in sedated volunteers makes the determination of sedation levels by processed electroencephalogram (EEG) variables impossible. Reports in the literature which draw conclusions based on processed EEG variables obtained from sedated intensive care unit (ICU) patients may be inaccurate due to this variability. TRIAL REGISTRATION: clinicaltrials.gov Nr. NCT00641563.

Twenty questions with noise: Bayes optimal policies for entropy loss

We consider the problem of twenty questions with noisy answers, in which we seek to find a target by repeatedly choosing a set, asking an oracle whether the target lies in this set, and obtaining an answer corrupted by noise. Starting with a prior distribution on the target's location, we seek to minimize the expected entropy of the posterior distribution. We formulate this problem as a dynamic program and show that any policy optimizing the one-step expected reduction in entropy is also optimal over the full horizon. Two such Bayes optimal policies are presented: one generalizes the probabilistic bisection policy due to Horstein and the other asks a deterministic set of questions. We study the structural properties of the latter, and illustrate its use in a computer vision application.

Energy-Entropy-Momentum integration of discrete thermo-visco-elastic dynamics.

A novel time integration scheme is presented for the numerical solution of the dynamics of discrete systems consisting of point masses and thermo-visco-elastic springs. Even considering fully coupled constitutive laws for the elements, the obtained solutions strictly preserve the two laws of thermo dynamics and the symmetries of the continuum evolution equations. Moreover, the unconditional control over the energy and the entropy growth have the effect of stabilizing the numerical solution, allowing the use of larger time steps than those suitable for comparable implicit algorithms. Proofs for these claims are provided in the article as well as numerical examples that illustrate the performance of the method.