Real numerical shadow and generalized B-splines

TL;DR

This paper introduces a new concept called the restricted numerical shadow of an operator, deriving explicit formulas for Hermitian operators, relating it to Dirichlet distributions, and generalizing B-splines, with applications to tensor product spaces.

Contribution

It provides explicit formulas for the restricted numerical shadow, relates it to Dirichlet distributions, and generalizes B-splines, especially for operators on tensor product spaces.

Findings

Derived explicit formulas for real-state restricted shadows.

Connected the shadow density to Dirichlet distribution.

Generalized B-splines through the shadow framework.

Abstract

Restricted numerical shadow $P^X_A(z)$ of an operator $A$ of order $N$ is a probability distribution supported on the numerical range $W_X(A)$ restricted to a certain subset $X$ of the set of all pure states - normalized, one-dimensional vectors in ${\mathbb C}^N$. Its value at point $z \in {\mathbb C}$ equals to the probability that the inner product $< u |A| u >$ is equal to $z$, where $u$ stands for a random complex vector from the set $X$ distributed according to the natural measure on this set, induced by the unitarily invariant Fubini-Study measure. For a Hermitian operator $A$ of order $N$ we derive an explicit formula for its shadow restricted to real states, $P^{\mathbb R}_A(x)$, show relation of this density to the Dirichlet distribution and demonstrate that it forms a generalization of the $B$-spline. Furthermore, for operators acting on a space with tensor product structure, ${\cal H}_A \otimes {\cal H}_B$, we analyze the shadow restricted to the set of maximally entangled states and derive distributions for operators of order N=4.