Dynamics of Entanglement with Applications to Quantum Metrology

Abstract

Coherently controllable quantum devices have grown so large and complex that direct study of their microscopic dynamics is extremely difficult, if not infeasible. Yet, realizing the promise of quantum technologies --- such as simulation, computation and sensing --- requires a deep understanding of the structure and evolution of entanglement in these systems. Achieving this understanding challenges us to develop theoretical methods that provide insight into the dynamics of entanglement without the crutch of detailed microscopic simulations. In this thesis, we demonstrate how the tools of effective field theory and hydrodynamics can be naturally applied to meet this challenge. First, we study monitored random quantum circuits, in which random unitary dynamics are pitted against local projective measurements. The steady-state entanglement entropy of such circuits has been shown to undergo a phase transition from highly-entangled to unentangled as the measurement rate is increased. Here, we focus on the question of how the range of the entangling interactions, parametrized as a power law $r^{-\alpha}$, affects the universality of this transition. To address this question, we develop a statistical-mechanical model of the entanglement entropy from which we derive an effective field theory governing the phase transition. Comparing to numerical simulations, we find the field theory successfully predicts the key properties of the transition, including the critical power-law exponent below which long-range interactions become relevant. Applying these same conceptual tools to finite-depth circuits, we further predict a phase transition in the teleportation fidelity between distant qubits, given projective measurements on all other qubits. We investigate several examples of this finite-time teleportation transition and demonstrate that our field-theory model accurately captures the underlying physics in each case. Second, we consider quenches of low-temperature symmetry breaking states, such as evolving the x-polarized state under an $XY$ model that manifests easy-plane magnetism. Typically such quenches are analyzed by first taking the thermodynamic limit $N \to \infty$ and then finite-time evolution, which reveals the familiar process of local equilibration. However, a more relevant limit to quantum devices is that both $N$ and $t$ become large together -- that is, one has many particles that interact coherently for a long time. We analyze this limit using a combination of extensive semiclassical numerics and analytical hydrodynamics, with particular focus on the evolution of quantum fluctuations. Remarkably, under very general conditions, we find the quantum fluctuations undergo ``squeezing'' -- a pattern of entanglement useful for quantum metrology. These results lead us to the conjecture: any Hamiltonian exhibiting finite temperature, easy-plane ferromagnetism can be used to generate scalable spin squeezing. Finally, we provide experimental evidence for this conjecture using a Rydberg atom array which approximately realizes a dipolar $XY$ model.