arXiv subjects

B V Rajarama Bhat

Publications and source records attributed to B V Rajarama Bhat.

12 recordsLinked to original sources

arXiv2026

On regions of mixed unitarity for semigroups of unital quantum channels

It is established that both discrete and continuous semigroups of unital quantum channels are eventually mixed unitary. This result is novel even for the subclass of Schur maps and stands in sharp contrast to the resolution of the asymptotic quantum Birkhoff conjecture by Haagerup and Musat, who demonstrated that tensor powers of some unital quantum channels maintain a persistent positive distance from the set of mixed unitary channels. Remarkably, our results show that this gap vanishes in finite time when considering ordinary powers within a semigroup. Building on this, we define the mixed unitary index of a unital quantum channel as the minimum time (or power) beyond which all subsequent maps become mixed unitary. We demonstrate that for any fixed dimension $d \geq 3$, there is no universal upper bound for this index. Furthermore, we observe that if a continuous semigroup is not mixed unitary at some $t > 0$, it remains non-mixed unitary for all $t$ sufficiently close to the origin. Finally, we investigate quantum dynamical semigroups where mixed unitarity is restricted to specific families, such as Weyl or diagonal unitaries. We show that Schur semigroups of correlation matrices eventually become mixtures of rank-one correlation matrices, and we characterize the generators of Schur semigroups that remain within this set for all $t \geq 0$.

math.OA↗

arXiv2025

Peripheral Poisson Boundary: Extensions and Examples

The main purpose of this article is to explore the possibility of extending the notion of peripheral Poisson boundary of unital completely positive (UCP) maps to contractive completely positive (CCP) maps and to unital and non-unital contractive quantum dynamical semigroups on von Neumann algebras. We observe that the theory extends easily in the setting of von Neumann algebras and normal maps. Surprisingly, the peripheral Poisson boundary is unital, whenever it is nontrivial, even for contractive semigroups. The strong operator limit formula for computing the extended Choi-Effros product remains intact. However, there are serious obstacles in the framework of $C^*$-algebras, and we are unable to define the extended Choi-Effros product in such generality. We provide several intriguing examples to illustrate this.

math.OA↗

arXiv2025

Unitary orthonormal bases of finite dimensional inclusions

We study unitary orthonormal bases in the sense of Pimsner and Popa for inclusions $(\mathcal{B}\subseteq \mathcal{A}, E),$ where $\mathcal{A}, \mathcal{B}$ are finite dimensional von Neumann algebras and $E$ is a conditional expectation map from $\mathcal{A}$ onto $\mathcal{B}$. It is shown that existence of such bases requires that the associated inclusion matrix satisfies a spectral condition forcing dimension vectors to be Perron-Frobenius eigenvectors and the conditional expectation map preserves the Markov trace. Subject to these conditions, explicit unitary orthonormal bases are constructed if either one of the algebras is abelian or simple. They generalize complex Hadamard matrices, Weyl unitary bases, and a recent work of Crann et al which correspond to the special cases of $\mathcal{A}$ being abelian, simple, and general multi-matrix algebras respectively with $\mathcal{B}$ being the algebra of complex numbers. For the first time $\mathcal{B}$ is more general. As an application of these results it is shown that if $(\mathcal{B}\subseteq \mathcal{A}, E),$ admits a unitary orthonormal basis then the Connes-Størmer relative entropy $H(\mathcal{A}_1|\mathcal{A})$ equals the logarithm of the square of the norm of the inclusion matrix, where $\mathcal{A}_1$ denotes the Jones basic construction of the inclusion. As a further application, we prove the existence of unitary orthonormal bases for a large class of depth 2 subfactors with abelian relative commutant.

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arXiv2024

Nayak's theorem for compact operators

Let $A$ be an $m\times m$ complex matrix and let $λ_1, λ_2, \ldots , λ_m$ be the eigenvalues of $A$ arranged such that $|λ_1|\geq |λ_2|\geq \cdots \geq |λ_m|$ and for $n\geq 1,$ let $s^{(n)}_1\geq s^{(n)}_2\geq \cdots \geq s^{(n)}_m$ be the singular values of $A^n$. Then a famous theorem of Yamamoto (1967) states that $$\lim _{n\to \infty}(s^{(n)}_j )^{\frac{1}{n}}= |λ_j|, ~~\forall \,1\leq j\leq m.$$ Recently S. Nayak strengthened this result very significantly by showing that the sequence of matrices $|A^n|^{\frac{1}{n}}$ itself converges to a positive matrix $B$ whose eigenvalues are $|λ_1|,|λ_2|,$ $\ldots , |λ_m|.$ Here this theorem has been extended to arbitrary compact operators on infinite dimensional complex separable Hilbert spaces. The proof makes use of Nayak's theorem, Stone-Weirstrass theorem, Borel-Caratheodory theorem and some technical results of Anselone and Palmer on collectively compact operators. Simple examples show that the result does not hold for general bounded operators.

math.FA↗

arXiv2022

On products of symmetries in von Neumann algebras

Let $\mathscr{R}$ be a type $II_1$ von Neumann algebra. We show that every unitary in $\mathscr{R}$ may be decomposed as the product of six symmetries (that is, self-adjoint unitaries) in $\mathscr{R}$, and every unitary in $\mathscr{R}$ with finite spectrum may be decomposed as the product of four symmetries in $\mathscr{R}$. Consequently, the set of products of four symmetries in $\mathscr{R}$ is norm-dense in the unitary group of $\mathscr{R}$. Furthermore, we show that the set of products of three symmetries in a von Neumann algebra $\mathscr{M}$ is not norm-dense in the unitary group of $\mathscr{M}$. This strengthens a result of Halmos-Kakutani which asserts that the set of products of three symmetries in $\mathcal{B}(\mathscr{H})$, the ring of bounded operators on a Hilbert space $\mathscr{H}$, is not the full unitary group of $\mathcal{B}(\mathscr{H})$.

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arXiv2022

Iterative square roots of functions

An iterative square root of a function $f$ is a function $g$ such that $g(g(\cdot))=f(\cdot)$. We obtain new characterizations for detecting the non-existence of such square roots for self-maps on arbitrary sets. This is used to prove that continuous self-maps with no square roots are dense in the space of all continuous self-maps for various topological spaces. The spaces studied include those that are homeomorphic to the unit cube in ${\mathbb R}^m$ and to the whole of $\mathbb{R}^m$ for every positive integer $m.$ On the other hand, we also prove that every continuous self-map of a space homeomorphic to the unit cube in $\mathbb{R}^m$ with a fixed point on the boundary can be approximated by iterative squares of continuous self-maps.

math.DS↗

arXiv2021

$C^*$-extreme points of positive operator valued measures and unital completely positive maps

We study the quantum ($C^*$) convexity structure of normalized positive operator valued measures (POVMs) on measurable spaces. In particular, it is seen that unlike extreme points under classical convexity, $C^*$-extreme points of normalized POVMs on countable spaces (in particular for finite sets) are always spectral measures (normalized projection valued measures). More generally it is shown that atomic $C^*$-extreme points are spectral. A Krein-Milman type theorem for POVMs has also been proved. As an application it is shown that a map on any commutative unital $C^*$-algebra with countable spectrum (in particular ${\mathbb C}^n$) is $C^*$-extreme in the set of unital completely positive maps if and only if it is a unital $*$-homomorphism.

math.OA↗

arXiv2020

A caricature of dilation theory

We present a set-theoretic version of some basic dilation results of operator theory. The results we have considered are Wold decomposition, Halmos dilation, Sz. Nagy dilation, inter-twining lifting, commuting and non-commuting dilations, BCL theorem etc. We point out some natural generalizations and variations.

math.OA↗

arXiv2020

Structure of block quantum dynamical semigroups and their product systems

W. Paschke's version of Stinespring's theorem associates a Hilbert $C^*$-module along with a generating vector to every completely positive map. Building on this, to every quantum dynamical semigroup (QDS) on a $C^*$-algebra $\mathcal A$ one may associate an inclusion system $E=(E_t)$ of Hilbert $\mathcal A$-$\mathcal A$-modules with a generating unit $ξ=(ξ_t)$. Suppose $\mathcal B$ is a von Neumann algebra, consider $M_2(\mathcal B)$, the von Neumann algebra of $2\times 2$ matrices with entries from $\mathcal B$. Suppose $(Φ_t)_{t\ge 0}$ with $Φ_t=\begin{pmatrix} ϕ_t^1& ψ_t ψ_t^*&ϕ_t^2 \end{pmatrix},$ is a QDS on $M_2(B)$ which acts block-wise and let $(E^i_t)_{t\ge 0}$ be the inclusion system associated to the diagonal QDS $(ϕ^i_t)_{t\ge 0}$ with the generating unit $(ξ_t^i)_{t\ge 0}, i=1,2.$ It is shown that there is a contractive (bilinear) morphism $T=(T_t)_{t\ge0}$ from $(E^2_t)_{t\ge 0}$ to $(E^1_t)_{t\ge 0}$ such that $ψ_t(a)=\langle ξ^1_t, T_t aξ^2_t\rangle $ for all $a\in\mathcal B.$ We also prove that any contractive morphism between inclusion systems of von Neumann $\mathcal B$-$\mathcal B$-modules can be lifted as a morphism between the product systems generated by them. We observe that the $E_0$-dilation of a block quantum Markov semigroup (QMS) on a unital $C^*$-algebra is again a semigroup of block maps.

math.OA↗

arXiv2019

Real Normal Operators and Williamson's Normal Form

A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose(adjoint). A structure theorem for invertible skew-symmetric operators, which is analogous to the finite dimensional situation is also proved using elementary techniques. The second result is used to establish the main theorem of this article, which is a generalization of Williamson's normal form for bounded positive operators on infinite dimensional separable Hilbert spaces. This has applications in the study of infinite mode Gaussian states.

math.SP↗

arXiv2013

Nilpotent Completely Positive Maps

We study the structure of nilpotent completely positive maps in terms of Choi-Kraus coefficients. We prove several inequalities, including certain majorization type inequalities for dimensions of kernels of powers of nilpotent completely positive maps.

math.OA↗

arXiv2010

Stinespring's theorem for maps on Hilbert C*-modules

We strengthen Mohammad B. Asadi's analogue of Stinespring's theorem for certain maps on Hilbert C*-modules. We also show that any two minimal Stinespring representations are unitarily equivalent. We illustrate the main theorem with an example.

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