The conjugate gradient algorithm on a general class of spiked covariance matrices

We consider the conjugate gradient algorithm applied to a general class of spiked sample covariance matrices. The main result of the paper is that the norms of the error and residual vectors at any finite step concentrate on deterministic values determined by orthogonal polynomials with respect to a deformed Marchenko–Pastur law. The first-order limits and fluctuations are shown to be universal. Additionally, for the case where the bulk eigenvalues lie in a single interval we show a stronger universality result in that the asymptotic rate of convergence of the conjugate gradient algorithm only depends on the support of the bulk, provided the spikes are well-separated from the bulk. In particular, this shows that the classical condition number bound for the conjugate gradient algorithm is pessimistic for spiked matrices.

An improved Trapezoidal rule for numerical integration

Numerical Methods have attracted of research community for solving engineering problems. This interest is due to its practicality and the improvement of highspeed calculations done on current century processors. The increase in numerical method tools in engineering software, such as Matlab, is an example of the increased interest. In this paper, we are present a new improved numerical integration method, that is based on the well-known trapezoidal rule. The proposed method gives a great enhancement to the trapezoidal rule and overcomes the issue of the error value when dealing with some higher order functions even when solving for a single interval. After literature review, the proposed system is mathematically explained along with error analysis. Few examples are illustrated to prove improved accuracy of the proposed method over traditional trapezoidal method.

Holographic entanglement entropy and modular Hamiltonian in warped CFT in the framework of GMMG model

We study some aspects of a class of non-AdS holography where the 3D bulk gravity is given by generalized minimal massive gravity (GMMG). We consider the spacelike warped $$AdS_3$$ A d S 3 ($$WAdS_3$$ W A d S 3 ) black hole solution of this model where the 2d dual boundary theory is the warped conformal field theory (WFCT). The charge algebra of the isometries in the bulk and the charge algebra of the vacuum symmetries at the boundary are compatible and this is an evidence for the duality conjecture. Further evidence for this duality is the equality of entanglement entropy and modular Hamiltonian on both sides of the duality. So we consider the modular Hamiltonian for the single interval at the boundary in associated to the modular flow generators of the vacuum. We calculate the gravitational charge in associated to the asymptotic Killing vectors that preserve the metric boundary conditions. Assuming the first law of the entanglement entropy to be true, we introduce the matching conditions between the variables in two side of the duality and we find equality of the modular Hamiltonian variations and the gravitational charge variations in two sides of the duality. According to the results of the present paper we can say with more sure that the dual theory of the warped AdS3 black hole solution of GMMG is a Warped CFT.

Symmetry resolved relative entropies and distances in conformal field theory

We develop a systematic approach to compute the subsystem trace distances and relative entropies for subsystem reduced density matrices associated to excited states in different symmetry sectors of a 1+1 dimensional conformal field theory having an internal U(1) symmetry. We provide analytic expressions for the charged moments corresponding to the resolution of both relative entropies and distances for general integer n. For the relative entropies, these formulas are manageable and the analytic continuation to n = 1 can be worked out in most of the cases. Conversely, for the distances the corresponding charged moments become soon untreatable as n increases. A remarkable result is that relative entropies and distances are the same for all symmetry sectors, i.e. they satisfy entanglement equipartition, like the entropies. Moreover, we exploit the OPE expansion of composite twist fields, to provide very general results when the subsystem is a single interval much smaller than the total system. We focus on the massless compact boson and our results are tested against exact numerical calculations in the XX spin chain.

Free fermion entanglement with a semitransparent interface: the effect of graybody factors on entanglement islands

We study the entanglement entropy of free fermions in 2d in the presence of a partially transmitting interface that splits Minkowski space into two half-spaces. We focus on the case of a single interval that straddles the defect, and compute its entanglement entropy in three limits: Perturbing away from the fully transmitting and fully reflecting cases, and perturbing in the amount of asymmetry of the interval about the defect. Using these results within the setup of the Poincaré patch of AdS_22 statically coupled to a zero temperature flat space bath, we calculate the effect of a partially transmitting AdS_22 boundary on the location of the entanglement island region. The partially transmitting boundary is a toy model for black hole graybody factors. Our results indicate that the entanglement island region behaves in a monotonic fashion as a function of the transmission/reflection coefficient at the interface.

Target space entanglement in quantum mechanics of fermions and matrices

We consider entanglement of first-quantized identical particles by adopting an algebraic approach. In particular, we investigate fermions whose wave functions are given by the Slater determinants, as for singlet sectors of one-matrix models. We show that the upper bounds of the general Rényi entropies are N log 2 for N particles or an N × N matrix. We compute the target space entanglement entropy and the mutual information in a free one-matrix model. We confirm the area law: the single-interval entropy for the ground state scales as $$ \frac{1}{3} $$ 1 3 log N in the large N model. We obtain an analytical $$ \mathcal{O}\left({N}^0\right) $$ O N 0 expression of the mutual information for two intervals in the large N expansion.

Virasoro algebras, kinematic space and the spectrum of modular Hamiltonians in CFT2

We construct an infinite class of eigenmodes with integer eigenvalues for the Vacuum Modular Hamiltonian of a single interval N in 2d CFT and study some of its interesting properties, which includes its action on OPE blocks as well as its bulk duals. Our analysis suggests that these eigenmodes, like the OPE blocks have a natural description on the so called kinematic space of CFT2 and in particular realize the Virasoro algebra of the theory on this kinematic space. Taken together, our results hints at the possibility of an effective description of the CFT2 in the kinematic space language.

Entanglement spectrum of geometric states

The reduced density matrix of a given subsystem, denoted by ρA, contains the information on subregion duality in a holographic theory. We may extract the information by using the spectrum (eigenvalue) of the matrix, called entanglement spectrum in this paper. We evaluate the density of eigenstates, one-point and two-point correlation functions in the microcanonical ensemble state ρA,m associated with an eigenvalue λ for some examples, including a single interval and two intervals in vacuum state of 2D CFTs. We find there exists a microcanonical ensemble state with λ0 which can be seen as an approximate state of ρA. The parameter λ0 is obtained in the two examples. For a general geometric state, the approximate microcanonical ensemble state also exists. The parameter λ0 is associated with the entanglement entropy of A and Rényi entropy in the limit n → ∞. As an application of the above conclusion we reform the equality case of the Araki-Lieb inequality of the entanglement entropies of two intervals in vacuum state of 2D CFTs as conditions of Holevo information. We show the constraints on the eigenstates. Finally, we point out some unsolved problems and their significance on understanding the geometric states.

Solving a Fuzzy Linear Equation with a Variable, Using the Expected Interval of a Fuzzy Number

In this work, a fuzzy linear equation AX + B = 0, is solved, were A, B y C are triangular diffuse numbers, could also be trapezoidal. For this type of equations there are several solution methods, the classic method that does not always obtain solutions, the most used is the method of alpha cuts and arithmetic intervals that although it always finds a solution, as a value is taken closer to zero (more inaccurate), the solution satisfies less to the equation. The new method using the expected interval, allows to obtain a smaller support set where the solutions come closer to satisfying the equation, also allows to find a single interval where the best solutions for decision making are expected to be found. It is recommended to study the incorporation of the concept of the expected interval in the methods to solve systems of fuzzy linear equations