UC Davis

A century of concerted effort brought new levels of physical understanding and engineering capability to quantum physics that promise major advances in fundamental theory and technology applications. A case in point, quantum computation has the potential to become one of humanity’s most innovative and disruptive technologies. Advances there are pushing physics to probe quantum phenomena in systems that are increasingly complex, and in more detailed ways than ever before. It is now clearer than ever that further progress will require constructively working with noise, error, and environmental interactions. A key part of this is to understand the properties of time series of quantum states emitted by a quantum system—the main subject of this work.

This dissertation sets out to study time-series of quantum states emitted by a quantum system. The fundamental objects in this are Quantum-State Stochastic Processes (QSSPs)—sequences of stochastically generated quantum states. In particular, the focus is on the interaction of a classical observer—via quantum measurement—with these objects, and the informational and statistical characterization of the classical stochastic processes that result from that interaction.

As a classical observer uses a measurement protocol to observe a Quantum-State Stochastic Process, the outcomes form a time series. Individual time series are realizations of a stochastic process over the measurements’ classical outcomes. This dissertation studies the dependence of that stochastic process of measurement outcomes on both the QSSP and the measurement protocol. In particular, it demonstrates that regardless of the measurement protocol—and for several specific protocols explicitly discussed in this work—the output classical stochastic process is generically highly complex in two specific senses: (i) it is generically unpredictable, to a degree that depends on the measurement choice, and (ii) achieving optimal prediction for these stochastic processes will generically require an infinite amount of memory.

Inspired by the study of classical stochastic processes, this dissertation uses and adapts the theory of computational mechanics and hidden Markov models to understand and categorize these measured stochastic processes. Specifically, we identify the mechanism underlying their complicatednessas generator nonunifilarity—the degeneracy between sequences of generator states and sequences v of measurement outcomes. This makes it possible to quantitatively explore the influence that measurement choice has on a quantum process’ degrees of randomness and structural complexity using recently introduced methods from ergodic theory. This dissertation provides the explicit metrics and associated algorithms for that quantification. It demonstrates that under certain conditions, dependence of these metrics on the choice of measurements is smooth. In addition, these metrics are used to design informationally-optimal measurements of time-series of quantum states. Most approaches to quantum stochastic dynamics focus on the evolution of the quantum state of a particular system—for example, quantum collision models. In contrast, the results here lay the groundwork for the study of time series of quantum states emitted by a quantum system, which have so far been largely unexplored. Additionally, they provide a first description of what classical interaction with these time series yields.