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Restrictions of holomorphic sections to products

Manimugdha Saikia

Abstract

We associate quantum states with subsets of a product of two compact connected Kähler manifolds $M_1$ and $M_2$. To associate the quantum state with the subset, we use the map that restricts holomorphic sections of the quantum line bundle over the product of the two Kähler manifolds to the subset. We present a description of the kernel of this restriction map when the subset is a finite union of products. This in turn shows that the quantum states associated with the finite union of products are separable. Finally, for every pure state and certain mixed state, we construct subsets of $M_1\times M_2$ such that the states associated with these subsets are the original states, to begin with.

Restrictions of holomorphic sections to products

Abstract

We associate quantum states with subsets of a product of two compact connected Kähler manifolds and . To associate the quantum state with the subset, we use the map that restricts holomorphic sections of the quantum line bundle over the product of the two Kähler manifolds to the subset. We present a description of the kernel of this restriction map when the subset is a finite union of products. This in turn shows that the quantum states associated with the finite union of products are separable. Finally, for every pure state and certain mixed state, we construct subsets of such that the states associated with these subsets are the original states, to begin with.
Paper Structure (4 sections, 8 theorems, 42 equations)

This paper contains 4 sections, 8 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and setup
  3. Main results
  4. Subsets coming from states

Key Result

Proposition 3.1

For $j\in \{1,2\}$, let $\Lambda_j$ be a non-empty subset of $M_j$. If either $\Lambda_1$ or $\Lambda_2$ is a singleton set, then where We dedude that $\ker(\mathcal{R}_{\Lambda_1 \times \Lambda_2})^{\perp} = \ker(\mathcal{R}_{\Lambda_1})^{\perp} \otimes \ker(\mathcal{R}_{\Lambda_2})^{\perp}$.

Theorems & Definitions (15)

  • Proposition 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • Theorem 3.1
  • proof
  • Example 3.5
  • Corollary 3.6
  • ...and 5 more