Everything You Always Wanted to Know About LOCC (But Were Afraid to Ask)

TL;DR

This paper provides a detailed mathematical characterization of LOCC operations in quantum information, revealing their structure, limitations, and the subtle distinctions between different classes of local quantum operations.

Contribution

It offers a precise description of LOCC using quantum instruments, including the structure of finite and infinite round protocols, and explores their topological properties and limitations.

Findings

Finite round LOCC forms a compact subset of quantum operations.

An open ball around the depolarizing map is fully LOCC implementable.

Some maps can be approximated arbitrarily closely by LOCC but not implemented exactly.

Abstract

In this paper we study the subset of generalized quantum measurements on finite dimensional systems known as local operations and classical communication (LOCC). While LOCC emerges as the natural class of operations in many important quantum information tasks, its mathematical structure is complex and difficult to characterize. Here we provide a precise description of LOCC and related operational classes in terms of quantum instruments. Our formalism captures both finite round protocols as well as those that utilize an unbounded number of communication rounds. While the set of LOCC is not topologically closed, we show that finite round LOCC constitutes a compact subset of quantum operations. Additionally we show the existence of an open ball around the completely depolarizing map that consists entirely of LOCC implementable maps. Finally, we demonstrate a two-qubit map whose action can be approached arbitrarily close using LOCC, but nevertheless cannot be implemented perfectly.