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Anilesh Mohari

Publications and source records attributed to Anilesh Mohari.

At least 19 recordsLinked to original sources
arXiv2024

$SU_2(\mathbb{C})$ symmetry in quantum spin chain ground states and Haldane's conjecture

In this paper, we prove that any translation and $SU_2(\IC)$-invariant pure state of $\IM=\otimes_{k \in \IZ}\!M^{(k)}_d(\IC)$, that is also real, lattice symmetric and reflection positive with a certain twist $r_0 \in U_d(\IC)$, is finitely correlated and its two-point spatial correlation function decays exponentially whenever $d$ is an odd integer. In particular, the Heisenberg iso-spin anti-ferromagnetic integer spin model admits unique low temperature limiting ground state and its spatial correlation function decays exponentially. The unique low temperature limiting ground state of the Hamiltonian is determined by the unique solution to Clebsch-Gordon inter-twinning isometry between two representations of $SU_2(\IC)$.

math-ph↗
arXiv2023

Unital completely positive maps and their operator systems

A vector subspace $\cls$ of $\IM_n(\IC)$ is called unital operator system if $x \in \cls$ if and only if $x^* \in \cls$ and the identity operator $I_n \in \cls$, where $n$ is any fixed positive integer. Let $C^*(\cls)$ be the $C^*$ sub-algebra of $\IM_n(\IC)$ generated by the operator system $\cls$. We prove that a unital complete order isomorphism $\cli:\cls \raro \cls'$ between two such operator systems $\cls$ and $\cls'$ of $\IM_n(\IC)$ has a unique extension to a $C^*$-isomorphism $\cli:C^*(\cls) \raro C^*(\cls')$ if and only if $\cls$ and $\cls'$ are having equal set of complete ranks. The operator system $\cls = \mbox{span}\{v_iv_j^*:1 \le i,j \le d \}$ is uniquely determined for a unital completely positive map $τ(x)=\sum_{1 \le k \le d} v_kxv_k^*$ of index $d \ge 1$. As an application of our main result, we explore this correspondence and characterize up to co-cycle conjugacy all extreme points in the convex set of unital completely positive maps on $\IM_n(\IC)$. Using the main result, we also characterize up to co-cycle conjugacy all extreme elements in the convex set of normalized trace preserving unital completely positive maps on $\IM_n(\IC)$.

math.FA↗
arXiv2023

Translation invariant state and its mean entropy-I

Let $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ be the two sided infinite tensor product $C^*$-algebra of $d$ dimensional matrices $\!M^{(n)}(\IC)=\!M_d(\IC)$ over the field of complex numbers $\IC$ and $ω$ be a translation invariant state of $\IM$. In this paper, we have proved that the mean entropy $s(ω)$ and Connes-Størmer dynamical entropy $h_{CS}(\IM,θ,ω)$ of $ω$ are equal. Furthermore, the mean entropy $s(ω)$ is equal to the Kolmogorov-Sinai dynamical entropy $h_{KS}(\ID_ω,θ,ω)$ of $ω$ when the state $ω$ is restricted to a suitable translation invariant maximal abelian $C^*$ sub-algebra $\ID_ω$ of $\IM$. Futhermore, a translation invariant factor state of $\IM$ is pure if and only if its mean entropy is zero. The last statement can be regarded as a non commutative extension of Rokhlin-Sinai positive entropy theorem for non-pure factor states.

math.OA↗
arXiv2018

Hann-Banach-Arveson extension theorem and Kadison isomorphism

Let $C^*(\cls)$ be the $C^*$ algebra generated by an operator system $\cls$ i.e. a unital $*$-closed subspace of a unital $C^*$ algebra $\cla$. We prove that any complete order isomorphism $\cli:\cls \raro \cls'$ between two such operator systems of matrix algebras has a unique extension to a $C^*$-isomorphism $\cli:C^*(\cls) \raro C^*(\cls')$. However, the same statement with more general operator systems of infinite dimensional $C^*$-algebra is false. As an application of this result, we characterise upto cocycle conjugacy the extreme points of unital completely positive maps on matrix algebra.

math.OA↗
arXiv2017

Translation invariant state and its mean entropy-II

Let $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ be the two sided infinite tensor product $C^*$-algebra of $d$ dimensional matrices $\!M^{(n)}(\IC)=\!M_d(\IC)$ over the field of complex numbers $\IC$. Let $ω$ be a translation invariant state of $\IM$. In a recent paper, we have proved that the mean entropy $s(ω)$ is a complete invariant for certain classes of translation invariant state $ω$ of $\IM$. In this paper, we have developed a general theory for dynamical entropy for an automorphism on an arbitrary $C^*$- or von-Neumann algebras based on repeated admissible measurement processes. In particular, we prove that dynamical entropy $h_ω(θ)$ for translation dynamics $(\IM,θ,ω)$ satisfies $s(ω) \le h_ω(θ) \le 2s(ω)$. In case $ω$ is an infinite tensor product state of $\IM$ then $h_ω(θ)=s(ω)$.

math.OA↗
arXiv2017

Spontaneous $SU_2(\mathbb{C})$ symmetry breaking in the ground states of quantum spin chain

In this paper, we have proved that there exists no translation invariant pure state of $\mathbb{M}=\otimes_{k \in \mathbb{Z}}\!M^{(k)}_d(\mathbb{C})$ that is real, lattice symmetric with a certain twist and $SU_2(\mathbb{C})$ invariant for any even integer $d \ge 2$. In particular, this result also says that the Heisenberg iso-spin anti-ferromagnetic model with ${1 \over 2}$-odd integer spin degrees of freedom does not admit a unique ground state.

math-ph↗
arXiv2016

Isomorphism theorem for Kolmogorov states of $\clb = \dsp{\otimes_{j \in \IZ}}\!M^{(j)}_d(\IC)$

We consider the translation dynamics on the $C^*$-algebra $\IM =\otimes_{n \in \IZ}\!M^{(n)}(\IC)$ of two sided infinite tensor product of $d$ dimensional matrices $\!M^{(n)}(\IC)=\!M_d(\IC)$ over the field of complex numbers $\IC$ and its restriction to the maximal abelian $C^*$ sub-algebra $\ID^e = \otimes_{n \in \IZ}\!D_e^{(n)}(\IC)$ of $\IM$, where each $\!D_e^{(n)}(\IC)=\!D_d(\IC)$ is the algebra of $d$ dimensional diagonal matrices with respect to an orthonormal basis $e=(e_i)$ of $\IC^d$. We prove that any two translation invariant Kolmogorov pure states of $\IM$ give unitarily equivalent dynamics in their Gelfand-Naimark-Segal spaces. Furthermore, for a class of Kolmogorov pure states of $\IM$ satisfying some additional invariance, we prove Kolmogorov states give isomorphic translation dynamics if their restrictions to the maximal abelian $C^*$ sub-algebra $\ID^e$ of $\IM$ are isomorphic. On the other hand, we prove that the translation dynamics with two infinite tensor product translation invariant faithful states of $\IM$ are isomorphic if and only if their mean entropies are equal.

math.OA↗
arXiv2015

Extremal unital completely positive normal maps and its symmetries

We consider the convex set of ( unital ) positive ( completely ) maps from a $C^*$ algebra $\cla$ to a von-Neumann sub-algebra $\clm$ of $\clb(\clh)$, the algebra of bounded linear operators on a Hilbert space $\clh$ and study its extreme points via its canonical lifting to the convex set of ( unital ) positive ( complete ) normal maps from $\hat{\cla}$ to $\clm$, where $\hat{\cla}$ is the universal enveloping von-Neumann algebra over $\cla$. If $\cla=\clm$ and a ( complete ) positive operator $τ$ is a unique sum of a normal and a singular ( complete ) positive maps. Furthermore, a unital complete positive map is a unique convex combination of unital normal and singular complete positive maps. We used a duality argument to find a criteria for extremal elements in the convex set of unital completely positive maps having a given faithful normal invariant state. In our investigation, gauge symmetry in Stinespring representation and Kadison theorem on order isomorphism played an important role.

math.OA↗
arXiv2013

Pure inductive limit state and Kolmogorov's property-II

A translation invariant state $ω$ on $C^*$-algebra $\clb=\otimes_{k \in \IZ}M^{(k)}$, where $M^{(k)}=M_d(\IC)$ is the $d-$dimensional matrices over field of complex numbers, give rises a stationary quantum Markov chain and associates canonically a unital completely positive normal map $τ$ on a von-Neumann algebra $\clm$ with a faithful normal invariant state $ϕ$. We give an asymptotic criteria on the Markov map $(\clm,τ,ϕ)$ for purity of $ω$. Such a pure $ω$ gives only type-I or type-III factor $ω_R$ once restricted to one side of the chain $\clb_R=\otimes_{\IZ_+}M^{(k)}$. In case $ω_R$ is type-I, $ω$ admits Kolmogorov's property.

math.OA↗
arXiv2013

A mean ergodic theorem in von-Neumann algebras

We explore a duality between von-Neumann's mean ergodic theorem in von-Neumann algebra and Birkhoff's mean ergodic theorem in the pre-dual Banach space of von-Neumann algebras. Besides improving known mean ergodic theorems on von-Neumann algebras, we prove Birkhoff's mean ergodic theorem for any locally compact second countable amenable group action on the pre-dual Banach space.

math.OA↗
arXiv2013

Translation invariant pure state on $\otimes_{\IZ}/!M_d(\IC)$ and Haag duality

We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}/!M_d(\IC), \;d \ge 2$, where $/!M_d(\IC)$ is the set of $d \times d$ dimensional matrices over the field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$.

math.OA↗
arXiv2013

Phase transition and split property in quantum spin chain

In this exposition we investigate further the general methodology proposed in [Mo2] to study properties of the ground states of a translation invariant Hamiltonian for one lattice dimensional quantum spin chain $\cla=\otimes_{\IZ}M_d$, where $M_d$ is the matrix of $d \times d$ complex matrices. We introduce a notion of quantum detailed balance [Mo1] for a translation invariant state on $\cla$ and prove that such a pure state is uniformly mixing [BR,Ma2] if and only if the lattice space correlation functions decay exponentially. Furthermore we also prove that a pure lattice symmetric, translation and SU(2) gauge invariant state give rise to a canonical Popescu systems acting on a finite dimensional Hilbert space and thus the lattice space correlation functions of the pure state decay exponentially. \vsp As a consequence of these results we conclude that if the ground states for an integer spin SU(2) invariant ($2s+1=d$) detailed balanced Hamiltonian is unique then the state is split. In particular if the ground state for integer spin anti-ferromagnetic Heisenberg chain is unique, then our main result says that the state is uniformly mixing and lattice space correlation functions of the ground state decay exponentially. Our main result is general enough to have application to other well known models such as Ising model, XY model and quasi-one dimensional quantum spin ladder [DR,Ma2] magnetic materials.

math-ph↗
arXiv2012

Translation invariant pure state and its split property

We prove Haag duality property of any translation invariant pure state on $\clb = \otimes_{\IZ}M_d(C), \;d \ge 2$, where $M_d(C)$ is the set of $d \times d$ dimensional matrices over field of complex numbers. We also prove a necessary and sufficient condition for a translation invariant factor state to be pure on $\clb$. This result makes it possible to study such a pure state with additional symmetry. We prove that exponentially decaying two point spacial correlation function of a real lattice symmetric reflection positive translation invariant pure state is a split state. Further there exists no translation invariant pure state on $\clb$ that is real, lattice symmetric, refection positive and $su(2)$ invariant when $d$ is an even integer. This in particular says that Heisenberg iso-spin anti-ferromagnets model for 1/2-odd integer spin degrees of freedom admits spontaneous symmetry breaking at it's ground states

math.OA↗
arXiv2007

Pure inductive limit state and Kolmogorov's property

Let $(\clb,λ_t,ψ)$ be a $C^*$-dynamical system where $(λ_t: t \in \IT_+)$ be a semigroup of injective endomorphism and $ψ$ be an $(λ_t)$ invariant state on the $C^*$ subalgebra $\clb$ and $\IT_+$ is either non-negative integers or real numbers. The central aim of this exposition is to find a useful criteria for the inductive limit state $\clb \raro^{λ_t} \clb$ canonically associated with $ψ$ to be pure. We achieve this by exploring the minimal weak forward and backward Markov processes associated with the Markov semigroup on the corner von-Neumann algebra of the support projection of the state $ψ$ to prove that Kolmogorov's property [Mo2] of the Markov semigroup is a sufficient condition for the inductive state to be pure. As an application of this criteria we find a sufficient condition for a translation invariant factor state on a one dimensional quantum spin chain to be pure. This criteria in a sense complements criteria obtained in [BJKW,Mo2] as we could go beyond lattice symmetric states.

math.OA↗
arXiv2007

Jones index of a quantum dynamical semigroup

In this paper we consider a semigroup of completely positive maps $τ=(τ_t,t \ge 0)$ with a faithful normal invariant state $ϕ$ on a type-$II_1$ factor $\cla_0$ and propose an index theory. We :achieve this via a more general Kolmogorov's type of construction for stationary Markov processes which naturally associate a nested isomorphic von-Neumann algebras. In particular this construction generalizes well known Jones construction associated with a sub-factor of type-II$_1$ factor.

math.OA↗
arXiv2007

Localized bases in L^2(0,1) and their use in the analysis of Brownian motion

Motivated by problems on Brownian motion, we introduce a recursive scheme for a basis construction in the Hilbert space L^2(0,1) which is analogous to that of Haar and Walsh. More generally, we find a new decomposition theory for the Hilbert space of square-integrable functions on the unit-interval, both with respect to Lebesgue measure, and also with respect to a wider class of self-similar measures (mu). That is, we consider recursive and orthogonal decompositions for the Hilbert space L^2(mu) where mu is some self-similar measure on [0,1]. Up to two specific reflection symmetries, our scheme produces infinite families of orthonormal bases in L^2(0,1). Our approach is as versatile as the more traditional spline constructions. But while singly generated spline bases typically do not produce orthonormal bases, each of our present algorithms does.

math.CA↗
arXiv2005

A resolution of quantum dynamical semigroups

We consider a class of quantum dissipative systems governed by a one parameter completely positive maps on a von-Neumann algebra. We introduce a notion of recurrent and metastable projections for the dynamics and prove that the unit operator can be decomposed into orthogonal projections where each projections are recurrent or metastable for the dynamics.

math.OA↗