Discrete and continuous variable systems: Properties, protocols, and applications
Dissertation Committee Chair:Â Nathan Schine
Committee: Nathan Schine, Alexey V. Gorshkov, Victor V. Albert, Alexander Barg, Mohammad Hafezi
 Abstract:  Quantum information science is a promising, interdisciplinary field focusing on both understanding and utilizing quantum systems. Two major paradigms of quantum mechanics are discrete variable (finite dimensional) systems, such as qubits and qudits, and continuous variable (infinite dimensional) systems, such as bosonic modes. In this dissertation, we explore the properties, protocols, and applications of both discrete and continuous variable systems.  In the first part of this dissertation, we study Hilbert space structures called quantum state designs, which are small ensembles of quantum states that mimic properties of the full space. While such designs are well-studied in the discrete variable setting, we show that they can also be defined and constructed in the continuous variable setting. Using specific multimode ensembles, we demonstrate continuous variable shadow tomography protocols which allow for efficient estimation of expectation values of many observables. Additionally, we use these ensembles to define notions of average and entanglement fidelities of continuous variable quantum channels, and we derive an explicit relationship between them that resembles the analogous relationship in the discrete variable setting.
Meanwhile, on the discrete variable side, we construct a theory of designs on the torus and find general methods for constructing them in arbitrary dimensions. Using these toric designs and their relationship to quantum state designs, we construct many new and explicit families of quantum state designs. Furthermore, we use toric designs to prove various structure theorems about complete sets of mutually unbiased bases.Â
Finally, in the third part of this dissertation, we examine the use of qubit systems for resolving frequency spectrums in signal processing applications. Specifically, we show that a classical signal whose spectrum contains closely spaced frequencies can be resolved by coupling the signal to a qubit and performing a superresolution protocol. We find general conditions for a protocol to exhibit superresolution and show various analytic and numerically-optimized protocols that achieve superresolution.