Complexity and Entanglement: From quantum gravity to many-body systems

Abstract

In recent years, complexity and entanglement have emerged as two fundamental computational measures and played a significant role in shaping our understanding of various phenomena, from the geometric nature of quantum gravity to the critical phenomena in many-body systems. In the first part of the thesis, we primarily focus on complexity in three different aspects. We modify Nielsen’s original arguments of traditional quantum gate counting, utilizing higher-order integrators of the Suzuki-Trotter method. This provides a volume-law scaling of complexity that is consistent with holographic proposals. Then we discuss the higher-dimension generalization of path integral complexity and its holographic interpretation using the AdS/BCFT correspondence. Later, we turn our attention to subregion complexity, which is a version of complexity that plays a significant role in understanding the black hole information problem. In the second part of the thesis, we zoom in to the entanglement for both pure states and mixed states. First, we discuss the capacity of entanglement in diverse scenarios, from operator excitations in quantum field theory to the phenomena of quantum chaos in many-body systems. We then delve into the details of the mixed state entanglement, introducing a measure known as the balance partial entanglement. In several examples, we show that it generalizes the reflected entropy and equals the entanglement wedge cross-section from the gravity perspective.