Quantum Strategies and Local Operations
arxiv(2010)
摘要
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.
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