arXiv subjects

Santanu Dey

Publications and source records attributed to Santanu Dey.

At least 37 records · Page 2Linked to original sources

arXiv2018

A Subspace of Maximal Dimension with Bounded Schmidt Rank

We study Schmidt rank for a vector (i.e., a pure state) and Schmidt number for a mixed state which are entanglement measures. We show that if a subspace of a certain bipartite system contains no vector of Schmidt rank $\leqslant k$, then any state supported on that space has Schmidt number at least $k+1$. A construction of subspace of $\mathbb{C}^m \otimes \mathbb{C}^n$ of maximal dimension, which does not contain any vector of Schmidt rank less than $3$, is given here.

quant-ph↗

arXiv2017

$t$-Frobenius Negacyclic Codes

Let $\mathbb{F}_p$ denote the finite field of order $p$, where $p$ is an odd prime. We study certain quantum negacyclic codes over $\mathbb{F}_p$ which we call $t$-Frobenius negacyclic codes. We obtain a criterion for constructing such codes from certain subspaces of $\mathbb{F}_p^n\times\mathbb{F}_p^n$ in terms of generating pairs of ideals of cyclotomic rings and we completely classify all linear $t$-Frobenius negacyclic codes. Further, the notion of BCH distance is extended for these codes and several new codes are identified.

quant-ph↗

arXiv2016

Bilayer honeycomb lattice with ultracold atoms: Multiple Fermi surfaces and incommensurate spin density wave instability

We propose an experimental setup using ultracold atoms to implement a bilayer honeycomb lattice with Bernal stacking. In presence of a potential bias between the layers and at low densities, Fermions placed in this lattice form an annular Fermi sea. The presence of two Fermi surfaces leads to interesting patterns in Friedel oscillations and RKKY interactions in presence of impurities. Furthermore, a repulsive fermion-fermion interaction leads to a Stoner instability towards an incommensurate spin-density-wave order with a wave vector equal to the thickness of the Fermi sea. The instability occurs at a critical interaction strength which goes down with the density of the fermions. We find that the instability survives interaction renormalization due to vertex corrections and discuss how this can be seen in experiments. We also track the renormalization group flows of the different couplings between the fermionic degrees of freedom, and find that there are no perturbative instabilities, and that Stoner instability is the strongest instability which occurs at a critical threshold value of the interaction. The critical interaction goes to zero as the chemical potential is tuned towards the band bottom.

cond-mat.str-el↗

arXiv2016

Strong electronic interaction and multiple quantum Hall ferromagnetic phases in trilayer graphene

There is an increasing interest in the electronic properties of few layer graphene as it offers a platform to study electronic interactions because the dispersion of bands can be tuned with number and stacking of layers in combination with electric field. However, electronic interaction becomes important only in very clean devices and so far the trilayer graphene experiments are understood within non-interacting electron picture. Here, we report evidence of strong electronic interactions and quantum Hall ferromagnetism (QHF) seen in ABA trilayer graphene (ABA-TLG). Due to high mobility $\sim$500,000 cm$^2$V$^{-1}$s$^{-1}$ in our device compared to previous studies, we find all symmetry broken states and that Landau Level (LL) gaps are enhanced by interactions; an aspect explained by our self-consistent Hartree-Fock (H-F) calculations. Moreover, we observe hysteresis as a function of filling factor ($ν$) and spikes in the longitudinal resistance which, together, signal the formation of QHF states at low magnetic field.

cond-mat.mes-hall↗

arXiv2016

Rokhlin Property for Group Actions on Hilbert $C^*$-modules

We introduce Rokhlin properties for certain discrete group actions on $C^*$-correspondences as well as on Hilbert bimodules and analyze them. It turns out that the group actions on any $C^*$-correspondence $E$ with Rokhlin property induces group actions on the associated $C^*$-algebra $\mathcal O_E$ with Rokhlin property and the group actions on any Hilbert bimodule with Rokhlin property induces group actions on the linking algebra with Rokhlin property. Permanence properties of several notions such as nuclear dimension and $\mathcal D$-absorbing property with respect to crossed product of Hilbert $C^*$-modules with groups, where group actions have Rokhlin property, are studied. We also investigate a notion of outerness for Hilbert bimodules.

math.OA↗

arXiv2016

Bures distance and transition probability for $α$-CPD-kernels

If the symmetry (fixed invertible self adjoint map) of Krein spaces is replaced by a fixed unitary, then we obtain the notion of S-spaces which was introduced by Szafraniec. Assume $α$ to be an automorphism on a $C^*$-algebra. In this article, we obtain the Kolmogorov decomposition of $α$-completely positive definite (or $α$-CPD-kernels for short) and investigate the Bures distance between $α$-CPD-kernels. We also define transition probability for these kernels and find a characterization of the transition probability.

math.OA↗

arXiv2015

KSGNS construction for $τ$-maps on S-modules and $\mathfrak{K}$-families

We introduce S-modules, generalizing the notion of Krein $C^*$-modules, where a fixed unitary replaces the symmetry of Krein $C^*$-modules. The representation theory on S-modules is explored and for a given $*$-automorphism $α$ on a $C^*$-algebra the KSGNS construction for $α$-completely positive maps is proved. An extention of this theorem for $τ$-maps is also achieved, when $τ$ is an $α$-completely positive map, along with a decomposition theorem for $\mathfrak K$-families.

math.OA↗

arXiv2014

Incorporating Views on Marginal Distributions in the Calibration of Risk Models

Entropy based ideas find wide-ranging applications in finance for calibrating models of portfolio risk as well as options pricing. The abstracted problem, extensively studied in the literature, corresponds to finding a probability measure that minimizes relative entropy with respect to a specified measure while satisfying constraints on moments of associated random variables. These moments may correspond to views held by experts in the portfolio risk setting and to market prices of liquid options for options pricing models. However, it is reasonable that in the former settings, the experts may have views on tails of risks of some securities. Similarly, in options pricing, significant literature focuses on arriving at the implied risk neutral density of benchmark instruments through observed market prices. With the intent of calibrating models to these more general stipulations, we develop a unified entropy based methodology to allow constraints on both moments as well as marginal distributions of functions of underlying securities. This is applied to Markowitz portfolio framework, where a view that a particular portfolio incurs heavy tailed losses is shown to lead to fatter and more reasonable tails for losses of component securities. We also use this methodology to price non-traded options using market information such as observed option prices and implied risk neutral densities of benchmark instruments.

q-fin.ST↗

arXiv2014

$\mathfrak{K}$-families and CPD-H-extendable families

We introduce, for any set $S$, the concept of $\mathfrak{K}$-family between two Hilbert $C^*$-modules over two $C^*$-algebras, for a given completely positive definite (CPD-) kernel $\mathfrak{K}$ over $S$ between those $C^*$-algebras and obtain a factorization theorem for such $\mathfrak{K}$-families. If $\mathfrak{K}$ is a CPD-kernel and $E$ is a full Hilbert $C^*$-module, then any $\mathfrak{K}$-family which is covariant with respect to a dynamical system $(G,η,E)$ on $E$, extends to a $\tilde{\mathfrak{K}}$-family on the crossed product $E \times_ηG$, where $\tilde{\mathfrak{K}}$ is a CPD-kernel. Several characterizations of $\mathfrak{K}$-families, under the assumption that ${E}$ is full, are obtained and covariant versions of these results are also given. One of these characterizations says that such $\mathfrak{K}$-families extend as CPD-kernels, between associated (extended) linking algebras, whose $(2,2)$-corner is a homomorphism and vice versa. We discuss a dilation theory of CPD-kernels in relation to $\mathfrak{K}$-families.

math.OA↗

arXiv2014

Functional Models and Minimal Contractive Liftings

Based on a careful analysis of functional models for contractive multi-analytic operators we establish a one-to-one correspondence between unitary equivalence classes of minimal contractive liftings of a row contraction and injective symbols of contractive multi-analytic operators. This allows an effective construction and classification of all such liftings with given defects. Popescu's theory of characteristic functions of completely non-coisometric row contractions is obtained as a special case satisfying a Szegö condition. In another special case of single contractions and defects equal to $1$ all non-zero Schur functions on the unit disk appear in the classification. It is also shown that the process of constructing liftings iteratively reflects itself in a factorization of the corresponding symbols.

math.OA↗

arXiv2014

Generalized repeated interaction model and transfer functions

Using a scheme involving a lifting of a row contraction we introduce a toy model of repeated interactions between quantum systems. In this model there is an outgoing Cuntz scattering system involving two wandering subspaces. We associate to this model an input/output linear system which leads to a transfer function. This transfer function is a multi-analytic operator, and we show that it is inner if we assume that the system is observable. Finally it is established that transfer functions coincide with characteristic functions of associated liftings.

math.OA↗

arXiv2012

Efficient simulation of density and probability of large deviations of sum of random vectors using saddle point representations

We consider the problem of efficient simulation estimation of the density function at the tails, and the probability of large deviations for a sum of independent, identically distributed, light-tailed and non-lattice random vectors. The latter problem besides being of independent interest, also forms a building block for more complex rare event problems that arise, for instance, in queueing and financial credit risk modelling. It has been extensively studied in literature where state independent exponential twisting based importance sampling has been shown to be asymptotically efficient and a more nuanced state dependent exponential twisting has been shown to have a stronger bounded relative error property. We exploit the saddle-point based representations that exist for these rare quantities, which rely on inverting the characteristic functions of the underlying random vectors. These representations reduce the rare event estimation problem to evaluating certain integrals, which may via importance sampling be represented as expectations. Further, it is easy to identify and approximate the zero-variance importance sampling distribution to estimate these integrals. We identify such importance sampling measures and show that they possess the asymptotically vanishing relative error property that is stronger than the bounded relative error property. To illustrate the broader applicability of the proposed methodology, we extend it to similarly efficiently estimate the practically important expected overshoot of sums of iid random variables.

math.PR↗

arXiv2012

Incorporating fat tails in financial models using entropic divergence measures

In the existing financial literature, entropy based ideas have been proposed in portfolio optimization, in model calibration for options pricing as well as in ascertaining a pricing measure in incomplete markets. The abstracted problem corresponds to finding a probability measure that minimizes the relative entropy (also called $I$-divergence) with respect to a known measure while it satisfies certain moment constraints on functions of underlying assets. In this paper, we show that under $I$-divergence, the optimal solution may not exist when the underlying assets have fat tailed distributions, ubiquitous in financial practice. We note that this drawback may be corrected if `polynomial-divergence' is used. This divergence can be seen to be equivalent to the well known (relative) Tsallis or (relative) Renyi entropy. We discuss existence and uniqueness issues related to this new optimization problem as well as the nature of the optimal solution under different objectives. We also identify the optimal solution structure under $I$-divergence as well as polynomial-divergence when the associated constraints include those on marginal distribution of functions of underlying assets. These results are applied to a simple problem of model calibration to options prices as well as to portfolio modeling in Markowitz framework, where we note that a reasonable view that a particular portfolio of assets has heavy tailed losses may lead to fatter and more reasonable tail distributions of all assets.

q-fin.ST↗

arXiv2009

Split rank of triangle and quadrilateral inequalities

A simple relaxation of two rows of a simplex tableau is a mixed integer set consisting of two equations with two free integer variables and non-negative continuous variables. Recently Andersen, Louveaux, Weismantel and Wolsey (2007) and Cornuejols and Margot (2008) showed that the facet-defining inequalities of this set are either split cuts or intersection cuts obtained from lattice-free triangles and quadrilaterals. Through a result by Cook, Kannan and Schrijver (1990), it is known that one particular class of facet-defining triangle inequality does not have a finite split rank. In this paper, we show that all other facet-defining triangle and quadrilateral inequalities have a finite split-rank. The proof is constructive and given a facet-defining triangle or quadrilateral inequality we present an explicit sequence of split inequalities that can be used to generate it.

math.OC↗

arXiv2009

Characteristic Functions, Liftings and Modules

We review how some multianalytic inner functions of the Beurling type theorem are associated to row contractions following works of G.Popescu. Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. Our characteristic function is a complete unitary invariant for such tuples. Thereafter we extend the analysis of characteristic functions to contractive liftings of row contractions. We apply our theory to completely positive maps and explore applications to fixed point sets of completely positive maps related to each other by a subisometric lifting. In the last two chapter we consider constrained row contractions and some module structures in this context.

math.OA↗

arXiv2007

Characteristic Functions of Liftings

We introduce characteristic functions for certain contractive liftings of row contractions. These are multi-analytic operators which classify the liftings up to unitary equivalence and provide a kind of functional model. The most important cases are subisometric and coisometric liftings. We also identify the most general setting which we call reduced liftings. We derive properties of these new characteristic functions and discuss the relation to Popescu's definition for completely non-coisometric row contractions. Finally we apply our theory to completely positive maps and prove a one-to-one correspondence between the fixed point sets of completely positive maps related to each other by a subisometric lifting.

math.OA↗

arXiv2006

Characteristic Functions for Ergodic Tuples

Motivated by a result on weak Markov dilations, we define a notion of characteristic function for ergodic and coisometric row contractions with a one-dimensional invariant subspace for the adjoints. This extends a definition given by G. Popescu. We prove that our characteristic function is a complete unitary invariant for such tuples and show how it can be computed.

math.OA↗

arXiv2005

Minimal Cuntz-Krieger Dilations and Representations of Cuntz-Krieger Algebras

Given a contractive tuple of Hilbert space operators satisfying certain $A$-relations we show that there exists a unique minimal dilation to generators of Cuntz-Krieger algebras or its extension by compact operators. This Cuntz-Krieger dilation can be obtained from the classical minimal isometric dilation as a certain maximal $A$-relation piece. We define a maximal piece more generally for a finite set of polynomials in $n$ noncommuting variables. We classify all representations of Cuntz-Krieger algebras ${\mathcal O}_A$ obtained from dilations of commuting tuples satisfying $A$-relations. The universal properties of the minimal Cuntz-Krieger dilation and the WOT-closed algebra generated by it is studied in terms of invariant subspaces.

math.OA↗