Quantum Correlations and Complexity: From Spin Chains to Holography

Abstract

At the intersection of geometry, dynamics, and information lies a central question in modern theoretical physics: how is the structure of spacetime encoded in quantum correlations and complexity? Over the past two decades, quantum information theory has emerged as both a set of practical tools and a conceptual bridge linking gravitational phenomena to condensed matter and quantum many-body systems. The work presented here is organised around two central themes: fixed-time analyses of quantum correlations in holographic theories and the study of dynamical evolution and quantum complexity in both unitary and nonunitary settings.

In the first part of our study on fixed-time quantum correlations, we pursue a discrete analogue of holography by turning to boundary theories defined on hyperbolic tilings. These are modelled by aperiodic quantum spin chains with the types of couplings determined by substitution rules that reflect the symmetries of the bulk geometry. Using strong-disorder renormalisation group techniques, we extract correlation functions, entanglement entropy, and mutual information in their aperiodic singlet phases. Notably, we find that two-point functions decay with a universal power-law exponent, while the entanglement entropy grows logarithmically. These results support a discrete Ryu–Takayanagi prescription framed in terms of minimal-length geodesics across the tiling, hinting at a geometric encoding of bulk properties within the disordered boundary theory.

In holographic CFTs, we further investigate quantum correlations beyond entanglement entropy. Although von Neumann entropy is commonly used, it captures both quantum and classical correlations in mixed states and vanishes for separable states. To isolate purely quantum effects, we study geometric quantum discord (GQD) as a diagnostic for the factorisation properties of modular partition functions. Our analytical results demonstrate that GQD vanishes if and only if the modular partition function factorises. When applied to the thermofield double (TFD) state, this analysis leads to an information-theoretic derivation of the thermomixed double (TMD) state as the optimal classical approximation to the TFD. Moreover, our approach links the persistence of GQD to the presence of black hole microstates and their imprint on the gravitational path integral, thereby offering deeper insight into the factorisation problem in AdS/CFT.

The second part of the thesis addresses dynamics. We begin with quantum triangular billiards, whose spectral and dynamical properties interpolate between integrability and chaos depending on internal angular parameters. By analysing level spacing ratios, spectral complexity, and the evolution of wavefunctions in the Krylov basis, we observe oscillatory dynamics in integrable triangles contrasted with a late-time saturation of complexity in chaotic cases—a distinction that reflects differences in the topology of the underlying classical phase space. Furthermore, the variance of Lanczos coefficients and the delocalisation of eigenstates in the Krylov basis trace a sharp transition from order to chaos, establishing a novel connection between classical topology and quantum information dynamics.

We then extend these insights to non-Hermitian systems, where processes such as continuous measurements, open system evolution, or PT-symmetric Hamiltonians induce nonunitary dynamics. By adapting the Lanczos algorithm to non-Hermitian and complex-symmetric Hamiltonians, we examine wavefunction spreading and information flow using Krylov-based measures. Repeated measurements induce a quantum Zeno regime in which time evolution is halted and complexity remains static. In a quench scenario—where the system is driven from Hermitian to non-Hermitian evolution via varying measurement frequencies—we find that the onset of complexity growth is progressively delayed, tending to infinity as the interval between measurements approaches zero. In PT-broken phases, we find localisation phenomena associated with the non-Hermitian skin effect, accompanied by suppressed growth of both complexity and entropy.