Quantum Strategies and Local Operations

TL;DR

This thesis introduces a new formalism for quantum strategies, explores local quantum operations with shared resources, and discusses their computational complexity and properties.

Contribution

It develops a novel operator-based formalism for quantum strategies and analyzes the structure and complexity of local quantum operations with shared entanglement.

Findings

A new formalism for quantum strategies using positive semidefinite operators.

Existence of a ball of local operations within no-signaling operations.

The weak membership problem for local operations with shared entanglement is strongly NP-hard.

Abstract

This thesis is divided into two parts. In Part I we introduce a new formalism for quantum strategies, which specify the actions of one party in any multi-party interaction involving the exchange of multiple quantum messages among the parties. This formalism associates with each strategy a single positive semidefinite operator acting only upon the tensor product of the input and output message spaces for the strategy. We establish three fundamental properties of this new representation for quantum strategies and we list several applications, including a quantum version of von Neumann's celebrated 1928 Min-Max Theorem for zero-sum games and an efficient algorithm for computing the value of such a game. In Part II we establish several properties of a class of quantum operations that can be implemented locally with shared quantum entanglement or classical randomness. In particular, we establish the existence of a ball of local operations with shared randomness lying within the space spanned by the no-signaling operations and centred at the completely noisy channel. The existence of this ball is employed to prove that the weak membership problem for local operations with shared entanglement is strongly NP-hard. We also provide characterizations of local operations in terms of linear functionals that are positive and "completely" positive on a certain cone of Hermitian operators, under a natural notion of complete positivity appropriate to that cone. We end the thesis with a discussion of the properties of no-signaling quantum operations.