Circuit Complexity of Mixed States

Quantum information has produced fresh insights into foundational questions about the AdS/CFT correspondence. One fascinating concept, which has captured increasing attention, is quantum circuit complexity. As a natural generalization for the complexity of pure states, we investigate the circuit complexity of mixed states in this thesis. First of all, we explore the so-called purification complexity which is defined as the lowest value of the circuit complexity, optimized over all possible purifications of a given mixed state. We focus on studying the complexity of Gaussian mixed states in a free scalar field theory using the ‘purification complexity’. We argue that the optimal purifications only contain the essential number of ancillary degrees of freedom necessary in order to purify the mixed state. We also introduce the concept of ‘mode-by-mode purifications’ where each mode in the mixed state is purified separately and examine the extent to which such purifications are optimal. In order to compare with the results from using the various holographic proposals for the complexity of subregions, we explore the purification complexity for thermal states of a free scalar QFT, and for subregions of the vacuum state in two dimensions. We find a number of qualitative similarities between the two in terms of the structure of divergences and the presence of a volume law. We also examine the ‘mutual complexity’ in the various cases studied in this thesis. In addition, we propose to generalize the Fubini-Study method for pure-state complexity to generic quantum states including mixed states by taking Bures metric or quantum Fisher information metric on the space of density matrices as the complexity measure. Due to Uhlmann’s theorem, we show that the mixed-state complexity exactly equals the purification complexity measured by the Fubini-Study metric for purified states but without explicitly applying any purifications. We also find that the purification complexity is nonincreasing under any trace-preserving quantum operations. As an illustration, we study the mixed Gaussian states as an example to explicitly show our conclusions for purification complexity

Gluing AdS/CFT

In this paper, we investigate gluing together two Anti-de Sitter (AdS) geometries along a timelike brane, which corresponds to coupling two brane field theories (BFTs) through gravitational interactions in the dual holographic perspective. By exploring the general conditions for this gluing process, we show that the energy stress tensors of the BFTs backreact on the dynamical metric in a manner reminiscent of the TTbar deformation. In particular, we present explicit solutions for the three-dimensional case with chiral excitations and further construct perturbative solutions with non-chiral excitations.Comment: 35 pages, 9 figures; v2: typos fixed, references added

Complexity=Anything: Singularity Probes

We investigate how the complexity=anything observables proposed by [arXiv:2111.02429, arXiv:2210.09647] can be used to investigate the interior geometry of AdS black holes. In particular, we illustrate how the flexibility of the complexity=anything approach allows us to systematically probe the geometric properties of black hole singularities. We contrast our results for the AdS Schwarzschild and AdS Reissner-Nordstr\"om geometries, i.e., for uncharged and charged black holes, respectively. In the latter case, the holographic complexity observables can only probe the interior up to the inner horizon.Comment: 36+5 pages, 2 appendices, 7 figure

Circuit Complexity for Coherent States

We examine the circuit complexity of coherent states in a free scalar field theory, applying Nielsen's geometric approach as in [1]. The complexity of the coherent states have the same UV divergences as the vacuum state complexity and so we consider the finite increase of the complexity of these states over the vacuum state. One observation is that generally, the optimal circuits introduce entanglement between the normal modes at intermediate stages even though our reference state and target states are not entangled in this basis. We also compare our results from Nielsen's approach with those found using the Fubini-Study method of [2]. For general coherent states, we find that the complexities, as well as the optimal circuits, derived from these two approaches, are different.Comment: 68 pages, 10 figures; v2, published version, added reference

Zoo of holographic moving mirrors

We systematically study moving mirror models in two-dimensional conformal field theory (CFT). By focusing on their late-time behavior, we separate the mirror profiles into four classes, named type A (timelike) mirrors, type B (escaping) mirrors, type C (chasing) mirrors, and type D (terminated) mirrors. We analytically explore the characteristic features of the energy flux and entanglement entropy for each type and work out their physical interpretation. Moreover, we construct their gravity duals for which end-of-the-world (EOW) branes play a crucial role. Depending on the mirror type, the profiles of the EOW branes show distinct behaviors. In addition, we also provide a criterion that decides whether the replica method in CFTs computes entanglement entropy or pseudo entropy in moving mirror models