Generation and Uses of Distributed Entanglement in Quantum Information

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In this thesis, we focus on the questions of how quantum entanglement can be generated between two or more spatially separated systems and, once generated, how it can be applied in quantum technology. First we will discuss a protocol, which we conjecture to be optimal in some regimes, that quickly creates entangled states across long distances in systems with power-law interactions. We will discuss how this protocol compares with currently known bounds on entangled state generation and how it might be implemented in a three-dimensional lattice of Rydberg atoms.

Next, we will turn our attention to more general questions of how the Lieb-Robinson bound and other limitations on entanglement can be used to inform the design of quantum computers. Quantum computers will be required to create entanglement if they are to realize significant advantages over classical computers, meaning that the generation of entanglement is an important question. First, we will discuss the implications of the Lieb-Robinson bound on graph descriptions of quantum computer architectures, and how the relevant graph parameter (diameter) compares to likely cost functions for architectures, such as maximum graph degree and total number of necessary connections. We will present a proposed graph architecture, the hierarchical product, which we believe provides excellent balance between these considerations.

We will then introduce new methods of evaluating graphs that allow us to include quantum architectures capable of measurement and feedback operations. After doing so, we will show that the generation of entanglement entropy becomes a limit on computation. We will show that, for several possible physical models of computation, the generation of entanglement can be bounded by simple graph properties. We demonstrate a connection between worst-case scenarios for entanglement generation and a graph quantity called the Cheeger constant or isoperimetric number, which we evaluate for several proposed quantum computing architectures.

Finally, we will look at the scenario of quantum sensing. In particular, we will examine protocols for quantum function estimation, where quantum sensors are available to measure all of the inputs to the function. We will demonstrate that entangled sensors are more capable than non-entangled ones by first deriving a new lower bound on measurement error and then presenting protocols that saturate these bounds. We will first do so for linear functions of the measured quantities and then extend this to general functions using a two-step linearizing protocol.