Supremum of Entanglement Measure for Symmetric Gaussian States, and Entangling Capacity

TL;DR

This thesis explores the maximum entanglement achievable in symmetric Gaussian states and introduces a geometric framework for understanding the entangling capacity of quantum operations based on negativity and PPT-ness.

Contribution

It provides an analytical method to calculate bounds on entangling capacity in terms of negativity and establishes a geometric measure related to PPT-ness, along with exact results for symmetric Gaussian states.

Findings

Bounds for entangling capacity linked to PPT-ness of operations.

A geometric measure quantifying entangling capability and PPT-ness.

Analytical expression for the supremum of negativity in symmetric Gaussian states.

Abstract

In this thesis there are two topics: On the entangling capacity, in terms of negativity, of quantum operations, and on the supremum of negativity for symmetric Gaussian states. Positive partial transposition (PPT) states are an important class of states in quantum information. We show a method to calculate bounds for entangling capacity, the amount of entanglement that can be produced by a quantum operation, in terms of negativity, a measure of entanglement. The bounds of entangling capacity are found to be associated with how non-PPT (PPT preserving) an operation is. A length that quantifies both entangling capacity/entanglement and PPT-ness of an operation or state can be defined, establishing a geometry characterized by PPT-ness. The distance derived from the length bounds the relative entangling capability, endowing the geometry with more physical significance. For a system composed of permutationally symmetric Gaussian modes, by identifying the boundary of valid states and making necessary change of variables, the existence and exact value of the supremum of logarithmic negativity (and negativity likewise) between any two blocks can be shown analytically. Involving only the total number of interchangeable modes and the sizes of respective blocks, this result is general and easy to be applied for such a class of states.