Quantum entanglement has become the key notion bridging originally distinct fields of research over the last decade, namely, quantum information and computation, condensed matter physics, and quantum gravity. Previous studies on quantum entanglement have largely focused on the entanglement entropy, which quantifies the amount of entanglement. However, a natural question arises: is there additional information of a quantum state that is not captured by the entanglement entropy alone? For ground states of gapped Hamiltonians, this question has been answered in the affirmative. In this dissertation, I extend this idea to study highly entangled states typically having volume law entropy, and demonstrate that there is indeed much richer information on the complexity of a quantum state beyond the entanglement entropy. \n \nIn the first part, I study the entanglement spectrum of highly entangled states corresponding to highly excited eigenstates of non-integrable Hamiltonians, time-evolved states after a quantum quench with Hamiltonians exhibiting different dynamical phases, and random unitary circuits consisting of random braids of non-Abelian anyons. I demonstrate that the entanglement spectrum is able to capture the degree of randomness of a quantum state, which we call the entanglement complexity. In the context of scrambling, this quantifies the degree of randomness produced by scrambling beyond entropic diagnostics. \n \nOur understanding of quantum entanglement in condensed matter systems and high energy physics have largely benefited from the field of quantum computation. In the second part of the dissertation, I present two examples of novel platforms for quantum computation using state-of-the-art experimental technologies. I demonstrate how one can use hybrid quantum-classical architecture to solve computational problems based on an optimal variational ansatz of the evolution protocol. I also present a hierarchical architecture of constructing logical Majorana zero modes which can be used for demonstrating non-Abelian braiding statistics experimentally.
Quantum entanglement has become the key notion bridging originally distinct fields of research over the last decade, namely, quantum information and computation, condensed matter physics, and quantum gravity. Previous studies on quantum entanglement have largely focused on the entanglement entropy, which quantifies the amount of entanglement. However, a natural question arises: is there additional information of a quantum state that is not captured by the entanglement entropy alone? For ground states of gapped Hamiltonians, this question has been answered in the affirmative. In this dissertation, I extend this idea to study highly entangled states typically having volume law entropy, and demonstrate that there is indeed much richer information on the complexity of a quantum state beyond the entanglement entropy. \n \nIn the first part, I study the entanglement spectrum of highly entangled states corresponding to highly excited eigenstates of non-integrable Hamiltonians, time-evolved states after a quantum quench with Hamiltonians exhibiting different dynamical phases, and random unitary circuits consisting of random braids of non-Abelian anyons. I demonstrate that the entanglement spectrum is able to capture the degree of randomness of a quantum state, which we call the entanglement complexity. In the context of scrambling, this quantifies the degree of randomness produced by scrambling beyond entropic diagnostics. \n \nOur understanding of quantum entanglement in condensed matter systems and high energy physics have largely benefited from the field of quantum computation. In the second part of the dissertation, I present two examples of novel platforms for quantum computation using state-of-the-art experimental technologies. I demonstrate how one can use hybrid quantum-classical architecture to solve computational problems based on an optimal variational ansatz of the evolution protocol. I also present a hierarchical architecture of constructing logical Majorana zero modes which can be used for demonstrating non-Abelian braiding statistics experimentally.
Entanglement and circuit complexity are fundamental concepts in quantum information theory which turned out to have considerable applications in various contexts ranging from many-body systems and condensed matter theory to quantum gravity and black hole physics. In order to find connections between these aspects, an insightful approach consists in comparing the complexity with some entanglement measures of a given bipartition, as for instance the Von Neumann entanglement entropy. Since, for a system in a given quantum state, the bipartite entanglement can be determined once a spatial subsystem and its complement are chosen, this comparison has to be made considering the complexity of reduced density matrices. This motivates the development of techniques for determining the circuit complexity of mixed states. In this thesis, focussing on free bosonic systems, we study the circuit complexity of mixed Gaussian states by employing the Fisher information geometry for the covariance matrices. This approach allows computing the complexity between reduced density matrices (subsystem complexity), as well as the complexity between thermal states. It also leads to a precise prescription for computing the spectrum and the basis complexity and the temporal evolution of the subsystem complexity after global and local quantum quenches. The study of the subsystem complexity of a given bipartition and a complete characterisation of its entanglement requires the detailed knowledge of the corresponding reduced density matrix, or, equivalently, of the entanglement Hamiltonian. A part of this thesis is devoted to determining the entanglement Hamiltonians in free theories. At equilibrium, we consider gapless systems, with the aim of recovering the underlying CFT results, and gapped lattice models. The evolution of entanglement Hamiltonians in quantum chains after a global quench is also studied. In addition, other entanglement quantifiers, as the entanglement spectrum and the contour for the entanglement entropies, are discussed in some of these settings. Finally, motivated by theoretical and experimental advances, we also address the issue of how the entanglement splits into the different charge sectors of a system endowed with a U(1) symmetry. We focus in particular on massive theories, either continuous or on the lattice, and we try to understand whether the entanglement equipartition, which is known to characterise gapless systems, survives away from criticality.
Abstract Scrambling is a process by which the state of a quantum system is effectively randomized due to the global entanglement that “hides” initially localized quantum information. Closely related notions include quantum chaos and thermalization. Such phenomena play key roles in the study of quantum gravity, many-body physics, quantum statistical mechanics, quantum information etc. Scrambling can exhibit different complexities depending on the degree of randomness it produces. For example, notice that the complete randomization implies scrambling, but the converse does not hold; in fact, there is a significant complexity gap between them. In this work, we lay the mathematical foundations of studying randomness complexities beyond scrambling by entanglement properties. We do so by analyzing the generalized (in particular Rényi) entanglement entropies of designs, i.e. ensembles of unitary channels or pure states that mimic the uniformly random distribution (given by the Haar measure) up to certain moments. A main collective conclusion is that the Rényi entanglement entropies averaged over designs of the same order are almost maximal. This links the orders of entropy and design, and therefore suggests Rényi entanglement entropies as diagnostics of the randomness complexity of corresponding designs. Such complexities form a hierarchy between information scrambling and Haar randomness. As a strong separation result, we prove the existence of (state) 2-designs such that the Rényi entanglement entropies of higher orders can be bounded away from the maximum. However, we also show that the min entanglement entropy is maximized by designs of order only logarithmic in the dimension of the system. In other words, logarithmic-designs already achieve the complexity of Haar in terms of entanglement, which we also call max-scrambling. This result leads to a generalization of the fast scrambling conjecture, that max-scrambling can be achieved by physical dynamics in time roughly linear in the number of degrees of freedom. This paper is an extended version of Phys. Rev. Lett. 120 (2018) 130502 [1].
Out-of-equilibrium quantum many-body systems stand at the forefront of modern theoretical physics, addressing fundamental questions on thermalization, transport, and universal phenomena. These fields, already well established in condensed matter, statistical physics, quantum optics, and quantum information theory, are progressively gaining even greater relevance with the rapid development of quantum technologies and simulation, which inherently operate in dynamical regimes. In recent years, the traditional paradigm of unitary quantum evolution has been expanded to include measurements, opening new directions in out-of-equilibrium physics. At the core of these advances lie measurement-induced phase transitions (MIPTs), which have emerged as a new class of dynamical critical phenomena characterizing the general behavior of monitored quantum dynamics. When external monitoring intertwines with unitary evolution, many-body quantum correlations change their structure, giving rise to distinct entanglement phases of matter. This discovery has sparked enormous interest in MIPTs, leading to significant advances in open quantum systems, entanglement theory, and more broadly quantum complexity. Despite much progress, a full understanding of monitored many-body dynamics is far from complete, leaving several open questions on the nature of MIPTs, their experimental observability, and the possibilities offered by measurements to enhance control over synthetic quantum matter. These issues persist due to the intrinsic complexity of the problem and the lack of efficient tools to study it, mainly caused by the stochastic character of monitored evolution. This thesis addresses these challenges by expanding the investigation of measurement-induced phenomena in new settings and introducing innovative probes of entanglement and many-body quantum complexity for MIPTs. A core question we investigate is the role of symmetries, non-ergodicity, and especially integrability in measurement-induced criticality, which dramatically affect the non-equilibrium phases. We further explore how these phenomena extend beyond bipartite quantum correlations to multipartite entanglement and quantum non-stabilizerness, highlighting the non-trivial interplay between measurements and complexity notions rooted in quantum information theory. Finally, we focus on the compelling problem of decoherence, modeling how noise spoils entanglement structures. These findings, supported by advanced numerical simulations and theoretical analysis, deepen the current understanding of entanglement, complexity, and integrability in monitored quantum many-body systems, offering new perspectives on their rich behavior. In parallel, we address the experimental problem of dissipation in MIPTs, which is of key relevance for practical implementations. We anticipate the present investigation to foster future research on the nature of monitored dynamical critical phenomena and, more broadly, the applications of measurements in quantum state evolution.
As Schroedinger already recognised one century ago, entanglement is at the core of quantum mechanics. Nowadays it turns out to be the fundamental notion behind many quantum phenomena, from quantum algorithms to gravity, passing by critical phenomena and topological phases of matter, triggering unexpected connections between apparently far branches of physics. At the center of all these ideas, we find the (Rényi) entanglement entropies, which are powerful entanglement measures that provide fundamental insights into the investigated system or theory. This motivates the development of techniques for determining it, such as the replica trick: its implementation via the path-integral finds a wide place in this thesis. For example, we develop an efficient strategy to compute a generalised version of the Rényi entropies for all the eigenstates of a (1+1)-dimensional conformal field theory. This represents the starting point for a simulation scheme ideal to compute the entanglement in more generic (1+1)-dimensional quantum field theories (QFTs), e.g. after a quench in the sine-Gordon field theory. The study of entanglement also intertwines with another pillar of modern physics, i.e. symmetries and how their presence influences the properties of a system. Given the interest in this connection, this thesis addresses the question of how the entanglement splits into the different sectors of an internal symmetry. We approach the problem first in the QFT context, both for the free Dirac and complex scalar fields in two-dimensional spacetime, which have an abelian conserved charge, and systems having an internal Lie group symmetry to tackle the non-abelian case. Another typical framework in which we study the symmetry resolution of entanglement is lattice models, where different techniques can be exploited in order to derive exact results, ranging from the corner transfer matrix for gapped integrable systems to the connection between quadratic lattice Hamiltonians and their two-point correlation functions. The symmetry resolution also concerns other entanglement measures, namely, we analyse the behaviour of the operator entanglement, i.e. a key quantifier of the complexity of an operator, the symmetry-resolved mutual information, the effect of symmetries on entanglement negativity. The latter quantity is a genuine measure of quantum correlations in mixed states and a consistent part of the thesis is about this subject. For example, we study its time evolution after a quench and we provide an operatorial characterisation for entanglement in mixed states, which we dub negativity Hamiltonian.
Abstract Entanglement entropy has long served as a central diagnostic of complexity and information scrambling in quantum many-body systems, and its behavior in random quantum circuits is by now well characterized. However, as a single scalar, entanglement entropy captures only the total uncertainty of the reduced density matrix spectrum while discarding the detailed structure of that spectrum — including the shape of the eigenvalue distribution, level repulsion signatures, and the transition from integrable to chaotic behavior. In this work, we propose spectral entropy — the Shannon entropy of the eigenvalue distribution of the reduced density matrix — as a structured diagnostic that goes beyond standard entanglement entropy. Using numerical simulations, we compute the spectral entropy of subsystem reduced states in Haar-random and Clifford circuits as a function of circuit depth, and compare its behavior against von Neumann entanglement entropy, Rényi-2 entropy, and entanglement spectrum level-spacing statistics. We expect to find three key results: (i) spectral entropy exhibits a growth timescale distinct from that of entanglement entropy, potentially reflecting spectral restructuring more sensitively; (ii) in the depth regime where entanglement entropy has already saturated, spectral entropy may continue to evolve, revealing late-time complexification that standard entanglement measures cannot resolve; (iii) spectral entropy may display a distinctive response to measurement-induced or noise-induced disentanglement that differs qualitatively from entanglement entropy. This work positions spectral entropy as a complementary — not replacement — tool to entanglement entropy, offering a finer spectral lens on information scrambling, thermalization, and complexity growth in random quantum circuits.
Quantum computing is the offspring of quantum mechanics and computer science, two great scientific fields founded in the 20th century. Quantum computing is a relatively young field and is recognized as having the potential to revolutionize science and technology in the coming century. The primary question in this field is essentially to ask which problems are feasible with potential quantum computers and which are not. In this dissertation, we study this question with a physical bent of mind. We apply tools from computer science and mathematical physics to study the complexity of simulating quantum systems. In general, our goal is to identify parameter regimes under which simulating quantum systems is easy (efficiently solvable) or hard (not efficiently solvable). This study leads to an understanding of the features that make certain problems easy or hard to solve. We also get physical insight into the behavior of the system being simulated. In the first part of this dissertation, we study the classical complexity of simulating quantum dynamics. In general, the systems we study transition from being easy to simulate at short times to being harder to simulate at later times. We argue that the transition timescale is a useful measure for various Hamiltonians and is indicative of the physics behind the change in complexity. We illustrate this idea for a specific bosonic system, obtaining a complexity phase diagram that delineates the system into easy or hard for simulation. We also prove that the phase diagram is robust, supporting our statement that the phase diagram is indicative of the underlying physics. In the next part, we study open quantum systems from the point of view of their potential to encode hard computational problems. We study a class of fermionic Hamiltonians subject to Markovian noise described by Lindblad jump operators and illustrate how, sometimes, certain Lindblad operators can induce computational complexity into the problem. Specifically, we show that these operators can implement entangling gates, which can be used for universal quantum computation. We also study a system of bosons with Gaussian initial states subject to photon loss and detected using photon-number-resolving measurements. We show that such systems can remain hard to simulate exactly and retain a relic of the "quantumness" present in the lossless system. Finally, in the last part of this dissertation, we study the complexity of simulating a class of equilibrium states, namely ground states. We give complexity-theoretic evidence to identify two structural properties that can make ground states easier to simulate. These are the existence of a spectral gap and the existence of a classical description of the ground state. Our findings complement and guide efforts in the search for efficient algorithms.