Growth of entanglement of generic states under dual-unitary dynamics

Dual-unitary circuits are a class of locally-interacting quantum many-body systems displaying unitary dynamics also when the roles of space and time are exchanged. These systems have recently emerged as a remarkable framework where certain features of many-body quantum chaos can be studied exactly. In particular, they admit a class of ``solvable" initial states for which, in the thermodynamic limit, one can access the full non-equilibrium dynamics. This reveals a surprising property: when a dual-unitary circuit is prepared in a solvable state the quantum entanglement between two complementary spatial regions grows at the maximal speed allowed by the local structure of the evolution. Here we investigate the fate of this property when the system is prepared in a generic pair-product state. We show that in this case the entanglement increment during a time step is sub-maximal for finite times, however, it approaches the maximal value in the infinite-time limit. This statement is proven rigorously for dual-unitary circuits generating high enough entanglement, while it is argued to hold for the entire class.Comment: 17 pages, 3 figure

Entanglement resolution of free Dirac fermions on a torus

Whenever a system possesses a conserved charge, the density matrix splits into eigenspaces associated to the each symmetry sector and we can access the entanglement entropy in a given subspace, known as symmetry resolved entanglement (SRE). Here, we first evaluate the SRE for massless Dirac fermions in a system at finite temperature and size, i.e. on a torus. Then we add a massive term to the Dirac action and we treat it as a perturbation of the massless theory. The charge-dependent entropies turn out to be equally distributed among all the symmetry sectors at leading order. However, we find subleading corrections which depend both on the mass and on the boundary conditions along the torus. We also study the resolution of the fermionic negativity in terms of the charge imbalance between two subsystems. We show that also for this quantity, the presence of the mass alters the equipartition among the different imbalance sectors at subleading order.Comment: 45 pages, 8 Figure

Temporal Entanglement in Chaotic Quantum Circuits

The concept of space-evolution (or space-time duality) has emerged as a promising approach for studying quantum dynamics. The basic idea involves exchanging the roles of space and time and evolving the system using a space transfer matrix. The infinite-volume limit is then described by the fixed points of this operator, also known as influence matrices. To evaluate the potential of this method as a numerical scheme, it is important to understand whether the influence matrices can be efficiently encoded in a classical computer. It is then natural to wonder what is the scaling of their entanglement, dubbed temporal entanglement, as a function of time. In this work we study space evolution in chaotic quantum circuits. First, we extend the concept of space-evolution to include evolution in any generic space-like direction, which enables us to use influence matrices on a generic time-like surface, or path, to describe any two-point function. Then we study their entanglement, finding that it scales linearly with time for all paths but with two interesting marginal cases: (i) vertical paths in generic chaotic systems (ii) any path in dual-unitary circuits. In these cases R\'enyi entropies with index larger than one are sub-linear in time, while the von Neumann entanglement entropy is linear but grows at a slower rate compared to regular state entanglement. We attribute this behaviour to the existence of a product state with large overlap with the influence matrices, similarly to what has been observed for regular entanglement in systems with conservation laws.Comment: 34 pages, 17 figure

Symmetry-Resolved Entanglement Entropy of Free Dirac Fermions on a Torus

Entanglement is a fundamental concept in quantum mechanics, with applications in many different fields. There are many ways to quantify entanglement, a very convenient one for bipartite systems at zero temperature being the entanglement entropy, which is the one considered in this manuscript. The goal of this thesis is to study the behavior of entanglement in a many-body system with an internal global symmetry, more specifically a system of one dimensional free Dirac fermions with U (1) symmetry. In the first chapters, previous results in the literature regarding these topics are presented, along with basic techniques and relations used throughout the thesis. Chapters 5 and 6 are the core of the thesis, and present explicit expressions regarding the symmetry-resolved entanglement entropy of Dirac Fermions at finite temperature and size. These expressions are plotted and compared with exact lattice computation. While chapter 5 deals with massless fermions, in chapter 6 the leading massive correction is considered, extending the previous result in the vicinity of a critical point. An analysis of the charge dependence on the calculated entanglement entropy shows that this quantity is equally distributed between all the symmetry sectors at the leading order

Otogenic lateral sinus thrombosis in children: proposal of an experience-based treatment flowchart

Purpose: To describe the prevalent clinical, laboratory, and radiological features of otogenic lateral sinus thrombosis (OLST) in children; to identify clinical predictors of outcome; to propose a management algorithm derived from experience. Methods: A retrospective review was conducted of the clinical records of patients with OLST, treated in a single tertiary care referral center for pediatric disease from 2006 to 2017. The inclusion criteria were pediatric age (0–16 years) and OLST diagnosis confirmed by a pre- and post-contrast CT or venography–MRI scan. Primary outcome measures were early (1–2 months) and late (6 months) sinus recanalization assessed by means of neuroimaging. Results: Twenty-five patients (8 females and 17 males; mean age = 6 ± 3 years) were included. A genetic abnormality associated with thrombophilia was found in 24 (96%) patients. At diagnosis, anticoagulant treatment with low-molecular-weight heparin (LMWH) was started in all subjects, while surgical treatment (mastoidectomy and tympanostomy tube insertion) was performed in 16/25 (64%) patients. Follow-up neuroimaging showed lateral sinus recanalization in 12/25 (48%) patients after 1–2 months and in 17/25 (68%) after 6 months. At multivariate logistic regression analysis, no significant predictors of the early and late neuroradiological outcome were found. Conclusions: All children with OLST should be screened for thrombophilia to decide on treatment duration and to assess the need for future antithrombotic prophylaxis. Immediately after diagnosis, anticoagulant treatment with LMWH should be started according to the international guidelines. Instead, our experience suggests that surgical treatment should not be indicated in all patients, but decided on a case-to-case basis

Empagliflozin in Patients with Chronic Kidney Disease

Background The effects of empagliflozin in patients with chronic kidney disease who are at risk for disease progression are not well understood. The EMPA-KIDNEY trial was designed to assess the effects of treatment with empagliflozin in a broad range of such patients. Methods We enrolled patients with chronic kidney disease who had an estimated glomerular filtration rate (eGFR) of at least 20 but less than 45 ml per minute per 1.73 m(2) of body-surface area, or who had an eGFR of at least 45 but less than 90 ml per minute per 1.73 m(2) with a urinary albumin-to-creatinine ratio (with albumin measured in milligrams and creatinine measured in grams) of at least 200. Patients were randomly assigned to receive empagliflozin (10 mg once daily) or matching placebo. The primary outcome was a composite of progression of kidney disease (defined as end-stage kidney disease, a sustained decrease in eGFR to < 10 ml per minute per 1.73 m(2), a sustained decrease in eGFR of & GE;40% from baseline, or death from renal causes) or death from cardiovascular causes. Results A total of 6609 patients underwent randomization. During a median of 2.0 years of follow-up, progression of kidney disease or death from cardiovascular causes occurred in 432 of 3304 patients (13.1%) in the empagliflozin group and in 558 of 3305 patients (16.9%) in the placebo group (hazard ratio, 0.72; 95% confidence interval [CI], 0.64 to 0.82; P < 0.001). Results were consistent among patients with or without diabetes and across subgroups defined according to eGFR ranges. The rate of hospitalization from any cause was lower in the empagliflozin group than in the placebo group (hazard ratio, 0.86; 95% CI, 0.78 to 0.95; P=0.003), but there were no significant between-group differences with respect to the composite outcome of hospitalization for heart failure or death from cardiovascular causes (which occurred in 4.0% in the empagliflozin group and 4.6% in the placebo group) or death from any cause (in 4.5% and 5.1%, respectively). The rates of serious adverse events were similar in the two groups. Conclusions Among a wide range of patients with chronic kidney disease who were at risk for disease progression, empagliflozin therapy led to a lower risk of progression of kidney disease or death from cardiovascular causes than placebo