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Soumya Das

Publications and source records attributed to Soumya Das.

At least 37 records · Page 2Linked to original sources
arXiv2021

Maximum Violation of Monogamy of Entanglement for Indistinguishable Particles by Measures that are Monogamous for Distinguishable Particles

Two important results of quantum physics are the \textit{no-cloning} theorem and the \textit{monogamy of entanglement}. The former forbids the creation of an independent and identical copy of an arbitrary unknown quantum state and the latter restricts the shareability of quantum entanglement among multiple quantum systems. For distinguishable particles, one of these results imply the other. In this Letter, we show that in qubit systems with indistinguishable particles (where each particle cannot be addressed individually), a maximum violation of the monogamy of entanglement is possible by the measures that are monogamous for distinguishable particles. To derive this result, we formulate the degree of freedom trace-out rule for indistinguishable particles corresponding to a spatial location where each degree of freedom might be entangled with the other degrees of freedom. Our result removes the restriction on the shareability of quantum entanglement for indistinguishable particles, without contradicting the no-cloning theorem.

quant-ph↗
arXiv2021

Hyper-hybrid entanglement, indistinguishability, and two-particle entanglement swapping

Hyper-hybrid entanglement for two indistinguishable bosons has been recently proposed by Li \textit{et al.} [Y. Li, M. Gessner, W. Li, and A. Smerzi, \href{https://doi.org/10.1103/PhysRevLett.120.050404}{Phys. Rev. Lett. 120, 050404 (2018)}]. In the current paper, we show that this entanglement exists for two indistinguishable fermions also. Next, we establish two {\em no-go} results: no hyper-hybrid entanglement for two {\em distinguishable} particles, and no unit fidelity quantum teleportation using {\em indistinguishable} particles. If either of these is possible, then the {\em no-signaling principle} would be violated. While several earlier works have attempted extending many results on distinguishable particles to indistinguishable ones, and vice versa, the above two no-go results establish a nontrivial separation between the two domains. Finally, we propose an efficient entanglement swapping using only two indistinguishable particles, whereas a minimum number of either three distinguishable or four indistinguishable particles is necessary for existing protocols.

quant-ph↗
arXiv2020

Large Hecke eigenvalues and an Omega result for non Saito--Kurokawa lifts

We prove a result on the distribution of Hecke eigenvalues, $μ_F(p^r)$ (for $r=1,2$ or $3$) of a non Saito--Kurokawa lift $F$ of degree $2$. As a consequence, we obtain an Omega result for the Hecke eigenvalues for such an $F$, which is the best possible in terms of orders of magnitude.

math.NT↗
arXiv2020

The first negative eigenvalue of Yoshida lifts

We prove that for any given $ε>0$, the first negative eigenvalue of the Yoshida lift $F$ of a pair of elliptic cusp forms $f,g$ having square-free levels (where $g$ has weight $2$ and satisfies $(\log Q_{g})^2 \ll \log Q_f$), occurs before $c_ε \cdot Q_F^{1/2-2 θ+ ε} $; where $Q_F,Q_f,Q_g$ are the analytic conductors of $F,f,g$ respectively, $θ< 1/4$, and $c_ε$ is a constant depending only on $ε$.

math.NT↗
arXiv2020

Jacobi forms and differential operators: odd weights

We show that it is possible to remove two differential operators from the standard collection of $m$ of them used to embed the space of Jacobi forms of \textit{odd} weight $k$ and index $m$ into several pieces of elliptic modular forms. This complements the previous work of one of the authors in the case of even weights.

math.NT↗
arXiv2020

On Fourier coefficients of elliptic modular forms $\bmod \, \ell$ with applications to Siegel modular forms

We study several aspects of nonvanishing Fourier coefficients of elliptic modular forms $\bmod \ell$, partially answering a question of Bellaïche-Soundararajan concerning the asymptotic formula for the count of the number of Fourier coefficients upto $x$ which do not vanish $\bmod \ell$. We also propose a precise conjecture as a possible answer to this question. Further, we prove several results related to the nonvanishing of arithmetically interesting (e.g., primitive or fundamental) Fourier coefficients $\bmod \ell $ of a Siegel modular form with integral algebraic Fourier coefficients provided $\ell$ is large enough. We also make some efforts to make this "largeness" of $\ell$ effective.

math.NT↗
arXiv2020

Hecke-Siegel type threshold for square-free Fourier coefficients: an improvement

We prove that if $f$ is a non zero cusp form of weight $k$ on $Γ_0(N)$ with character $χ$ such that $N/(\text{conductor }χ)$ square-free, then there exists a square-free $n\ll_ε k^{3+ε}N^{7/2+ε}$ such that $a(f,n)\neq 0$. This significantly improves the already known existential and quantitative result from previous works.

math.NT↗
arXiv2020

Distinguishing Hermitian cusp forms of degree 2 by a certain subset of all Fourier coefficients

We prove that Hermitian cusp forms of weight $k$ for the Hermitian modular group of degree $2$ are determined by their Fourier coefficients indexed by matrices whose determinants are essentially square-free. Moreover, we give a quantitative version of the above result. This is a consequence of the corresponding results for integral weight elliptic cusp forms, which are also treated in this paper.

math.NT↗
arXiv2020

On Construction of Higher Order Kernels Using Fourier Transforms and Covariance Functions

In this paper, we show that a suitably chosen covariance function of a continuous time, second order stationary stochastic process can be viewed as a symmetric higher order kernel. This leads to the construction of a higher order kernel by choosing an appropriate covariance function. An optimal choice of the constructed higher order kernel that partially minimizes the mean integrated square error of the kernel density estimator is also discussed.

math.ST↗
arXiv2020

L^\infty norms of holomorphic modular forms in the case of compact quotient

We prove a sub-convex estimate for the sup-norm of $L^2$-normalized holomorphic modular forms of weight $k$ on the upper half plane, with respect to the unit group of a quaternion division algebra over $\mf Q$. More precisely we show that when the $L^2$ norm of an eigenfunction $f$ is one, | f |_\infty \ll k^{1/2 - 12/131 + \varepsilon}, for any $\varepsilon>0$ and for all $k$ sufficiently large.

math.NT↗
arXiv2019

Efficient Multi-Year Security Constrained AC Transmission Network Expansion Planning

Solution of multi-year, dynamic AC Transmission network expansion planning (TNEP) problem is gradually taking center stage of planning research owing to its potential accuracy. However, computational burden for a security constrained AC TNEP is huge compared to that with DC TNEP. For a dynamic, security constrained AC TNEP problem, the computational burden becomes so very excessive that solution for even moderately sized systems becomes almost impossible. Hence, this paper presents an efficient, four-stage solution methodology for multi-year, network N-1 contingency and voltage stability constrained, dynamic ACTNEP problems. Several intelligent logical strategies are developed and applied to reduce the computational burden of optimization algorithms. The proposed methodology is applied to Garver 6, IEEE 24 and 118 bus systems to demonstrate its efficiency and ability to solve TNEP for varying system sizes.

eess.SY↗
arXiv2019

A New Efficient Methodology for AC Transmission Network Expansion Planning in The Presence of Uncertainties

Consideration of generation, load and network uncertainties in modern transmission network expansion planning (TNEP) is gaining interest due to large-scale integration of renewable energy sources with the existing grid. However, it is a formidable task when iterative AC formulation is used. Computational burden for solving the usual ACTNEP with these uncertainties is such that, it is almost impossible to obtain a solution even for a medium-sized system within a viable time frame. In this work, a two-stage solution methodology is proposed to obtain quick, good-quality, sub-optimal solutions with reasonable computational burden. Probabilistic formulation is used to account for the different uncertainties. Probabilistic TNEP is solved by 2m+1-point estimate method along with a modified artificial bee colony (MABC) algorithm, for Garver 6 bus and IEEE 24 bus systems. In both the systems, rated wind generation is considered to be more than one-tenth of the total generation capacity. When compared with the conventional single stage and existing solution methods, the proposed methodology is able to obtain almost identical solutions with extremely low computational burdens. Therefore, the proposed method provides a tool for efficient solution of future probabilistic ACTNEP problems with greater level of complexity.

math.OC↗
arXiv2018

Petersson norms of not necessarily cuspidal Jacobi modular forms and applications

We extend the usual notion of Petersson inner product on the space of cuspidal Jacobi forms to include non-cuspidal forms as well. This is done by examining carefully the relation between certain "growth-killing" invariant differential operators on $\mathbf H_2$ and those on $\mathbf{H}_1 \times \mathbf{C}$ (here $\mathbf H_n$ denotes the Siegel upper half space of degree $n$). As applications, we can understand better the growth of Petersson norms of Fourier Jacobi coefficients of Klingen Eisenstein series, which in turn has applications to finer issues about representation numbers of quadratic forms, and as a by-product we also show that \textit{any} Siegel modular form of degree $2$ is determined by its `fundamental' Fourier coefficients.

math.NT↗
arXiv2018

Security Constrained AC Transmission Network Expansion Planning

Modern transmission network expansion planning (TNEP) is carried out with AC network model, which is able to handle voltage and voltage stability constraints. However, such a model requires optimization with iterative AC power flow model, which is computationally so demanding that most of the researchers have ignored the vital (N-1) security constraints. Therefore, the objective of this research work is to develop an efficient, two stage optimization strategy for solving this problem. In the first stage, a DC expansion planning problem is solved which provides an initial guess as well as some very good heuristics to reduce the number of power flow solutions for the second stage of AC transmission and reactive expansion planning. A modified artificial bee colony (MABC) algorithm is used to solve the resulting optimization problem. Static AC TNEP results for Garver 6 bus, IEEE 24 bus and IEEE 118 bus test systems have been obtained with the proposed and rigorous approaches and wherever possible, compared with similar results reported in literature to demonstrate the benefits of the proposed method. Also, multi-stage dynamic AC TNEP for the Garver 6 bus system is solved to show the applicability of the methodology to such problems.

math.OC↗
arXiv2013

On a convolution series attached to a Siegel Hecke cusp form of degree 2

We prove that the "naive" convolution Dirichlet series D_2(s) attached to a degree 2 Siegel Hecke cusp form F, has a pole at s=1. As an application, we write down the asymptotic formula for the partial sums of the squares of the eigenvalues of $F$ with an explicit error term. Further, as a corollary, we are able to show that the abscissa of absolute convergence of the (normalized) spinor zeta function attached to F is s = 1.

math.NT↗
arXiv2010

Note on Hermitian Jacobi Forms

We compare the spaces of Hermitian Jacobi forms (HJF) of weight $k$ and indices $1,2$ with classical Jacobi forms (JF) of weight $k$ and indices $1,2,4$. Using the embedding into JF, upper bounds for the order of vanishing of HJF at the origin is obtained. We compute the rank of HJF as a module over elliptic modular forms and prove the algebraic independence of the generators in case of index 1. Some related questions are discussed.

math.NT↗