# The Necessary and Sufficient Conditions of Separability for Bipartite Pure States in Infinite Dimensional Hilbert Spaces

## TL;DR

This paper establishes precise mathematical conditions for when bipartite pure states in infinite-dimensional Hilbert spaces are separable, linking quantum state properties to operator theory in functional analysis.

## Contribution

It provides the necessary and sufficient conditions for separability of bipartite pure states in infinite dimensions, connecting quantum information to bounded linear operator theory.

## Key findings

- Matrix of amplitudes is a compact operator.
- Separable states correspond to rank-1 bounded linear operators.
- Separable states characterized by one-dimensional image of the operator.

## Abstract

In this paper, we present the necessary and sufficient conditions of separability for bipartite pure states in infinite dimensional Hilbert spaces. Let $M$ be the matrix of the amplitudes of $\ket\psi$, we prove $M$ is a compact operator. We also prove $\ket\psi$ is separable if and only if $M$ is a bounded linear operator with rank 1, that is the image of $M$ is a one dimensional Hilbert space. So we have related the separability for bipartite pure states in infinite dimensional Hilbert spaces to an important class of bounded linear operators in Functional analysis which has many interesting properties.