Quantum Strategies and Local Operations
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 multiparty 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 MinMax Theorem for zerosum 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 nosignaling 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 NPhard. 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 nosignaling quantum operations.
 Publication:

Ph.D. Thesis
 Pub Date:
 February 2010
 DOI:
 10.48550/arXiv.1003.0038
 arXiv:
 arXiv:1003.0038
 Bibcode:
 2010PhDT.......249G
 Keywords:

 Quantum Physics;
 Quantum Physics
 EPrint:
 PhD thesis, 150 pages. Chapter 5 is new. Part I (except Chapter 5) is an expanded and revised version of arXiv:quantph/0611234. Chapter 5 (with corrections) was later published as arXiv:1008.4636. Part II is nearly identical to arXiv:0805.2209. v2: added errata for Chapter 5