Separable Operations, Graph Codes and the Location of Quantum Information

TL;DR

This paper explores the differences between LOCC and separable operations in quantum information, extends the no-cloning theorem under separable operations, and investigates information localization in graph states and codes with arbitrary-dimensional qudits.

Contribution

It establishes the equivalence of LOCC and separable operations for pure bipartite states, generalizes the no-cloning theorem, and develops techniques for analyzing information distribution in graph codes with arbitrary dimensions.

Findings

LOCC and separable operations coincide for pure bipartite states

Separable operations impose stricter conditions on cloning

Efficient methods to determine information localization in graph codes

Abstract

In the first part of this Dissertation, I study the differences between LOCC (local operations and classical communication) and the more general class of separable operations. I show that the two classes coincide for the case of pure bipartite state input, and derive a set of important consequences. Using similar techniques I also generalize the no-cloning theorem when restricted to separable operations and show that cloning becomes much more restrictive, by providing necessary (and sometimes sufficient) conditions. In the second part I investigate graph states and graph codes with carrier qudits of arbitrary dimensionality, and extend the notion of stabilizer to any dimension, not necessarily prime. I further study how and where information is located in the various subsets of the qudit carriers of arbitrary additive graph codes, and provide efficient techniques that can be used in deciding what types of information a subset contains.