UC San Diego

The nonlocality of quantum information is an intrinsic consequence of the interactions in many-body systems. Quantifying it can reveal how information between parties propogates across space and time, but in general it is hard to keep track of the exponentially-large set of bipartite entanglement data. In this dissertation, we exploit statistic symmetries about local-basis invariant ensembles to efficiently characterize the bipartite entanglement data in the form of the average purity, or entanglement features. We are able to systematically study the entanglement features of a variety of locally-scrambled ensembles, and determine their entanglement velocities, volume-law coefficients, butterfly velocities, etc. Next, we propose a two-parameter, bond dimension two, matrix-product state ansatz for characterizing the full, multiregion entanglement data of typical locally-scrambled state ensembles. We are able to compare this two-parameter ansatz to a variety of different models and show that it performs well at capturing the full evolution from product to Page state. We also study the meaning of the parameters physically and how they determine different aspects of the multiregion entanglement. Next, we adapt these techniques to classical shadow tomography, where the goal is to construct efficient, classical approximations of quantum states. We show that the reconstruction map is solely a function of the entanglement features of the snapshot state ensemble and we utilize our efficient description of the entanglement data to perform shallow circuit tomography scalably and flexibly. We show that shallow circuit tomography can perform better than Pauli or Clifford measurements for predicting quasi-local operators. We also demonstrate how shallow circuit tomography can serve as a more feasible alternative to Clifford measurements for fidelity estimation, a desirable task on current quantum devices. Lastly, we study the interplay between measurements and entanglement generating unitaries by studying the entanglement features of classical snapshot states produced from hybrid circuits, and show that such schemes are optimal at the phase transition in the purity.