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The brain's computational functions, from information synthesis and memory recall to consciousness, remain largely enigmatic. Despite great scientific effort, the mechanisms behind information storage, computation, and readout within the brain are still not well understood. Because of the warm, crowded nature of biological systems, it has traditionally been thought that quantum states would rapidly decohere and be unsuitable for biological information storage. Recently, the "Quantum Brain theory" has proposed the potential for a quantum information network in the brain that could maintain and process quantum information on timescales relevant to brain function. Within the framework of this theory, quantum information is stored in the collective phosphorus spin states of nanometric calcium phosphate clusters known as Posner molecules. The pair binding of these Posner molecules is postulated to be impacted by the collective phosphorus spin states, allowing for a unique quantum state to biochemical signal transduction mechanism. Given that this theory puts forward a uniquely possible method for quantum information processing in the brain, experimental validation or refutation of the theory would be an important step towards an understanding of brain function. This thesis presents experimental results aimed at testing the possibility of this Quantum Brain theory. We show that Posner molecules are stable in simulated body fluids, and uncover a differential lithium isotope effect on the growth of calcium phosphate. This isotope effect provides a link between nuclear spin states and calcium phosphate binding dynamics, and can be connected to in vivo lithium isotope effects in mitochondria. We also uncover spectroscopically 'dark state' phosphate assemblies after measuring unexpected 31P NMR signals, revealing that phosphate solution behavior and 31P nuclear spin dynamics are much more complex than previously predicted. These results verify key predictions of the Quantum Br

This thesis is devoted to the study of quantum dynamics interspersed with quantum measurements. Focusing on a one-dimensional “hybrid” quantum circuit model consisting of random unitary gates and local projective measurements, we provide extensive evidence for a stable “weakly measured phase” that exhibits volume law entanglement entropy, and a novel continuous quantum dynamical phase transition between the weakly measured phase and a strongly measured “quantum Zeno phase” that exhibits area law entanglement entropy. We further study consequences of conformal symmetry at the critical point, as well as properties of the weakly measured phase. In the latter case, we develop an effective domain wall theory which correctly accounts for an unusual subvolume powerlaw correction to the entanglement entropy, and draw close connections between the domain wall theory and an emergent quantum error correcting code. In the last Chapter, we discuss an experimental protocol that can avoid the so- called “postselection problem” and allow scalable experimental observations of such transitions on near term quantum processors.

We begin with an introduction to some of the important numerical tools in the field of condensed matter theory:exact diagonalization, quantum Monte Carlo, and tensor network algorithms. We also introduce interesting problemsto which they can be applied, including holographic duals (for tensor networks), many-body localization,and thermalization.We explore, using the previously discussed quantum Monte Carlo methods, a model of itinerant interacting fermionswith relevance to the mysterious pseudogap phase of the cuprate high temperature superconductors. We provide tentativeevidence for a non-Fermi liquid phase believed to support a violation of area law entanglement scaling. We hope tosettle numerical questions about this work in furtherance of the goal of incorporating it into a future publication.We then explore the stability of a system predisposed towards localization coupled to a system of similar size which would,on its own, thermalize. The stability of localized systems interacting with environmental baths is both experimentallyrelevant and theoretically interesting, and we explore a large parameter space using exact diagonalization techniques.We find that for a small set of parameters, localization may survive the presence of a similarly sized bath, but thetendency of the bath to delocalize the entire system is difficult to overcome. We also investigate the dynamics ofthese coupled systems, making connections with previous theoretical studies. This work is adapted from previously publishedmaterial.We take steps towards developing a new tensor network technique with which to study non-equilibrium quantum systems.We develop a disentangling circuit generation algorithm, drawing on previous algorithms for representing a variety ofinteresting wavefunctions, which can disentangle generic states without reference to a Hamiltonian. This is a novel departurefrom many previous disentangling approaches or tensor network optimization algorithms. We apply this technique to

We begin with an introduction to some of the important numerical tools in the field of condensed matter theory:exact diagonalization, quantum Monte Carlo, and tensor network algorithms. We also introduce interesting problemsto which they can be applied, including holographic duals (for tensor networks), many-body localization,and thermalization.We explore, using the previously discussed quantum Monte Carlo methods, a model of itinerant interacting fermionswith relevance to the mysterious pseudogap phase of the cuprate high temperature superconductors. We provide tentativeevidence for a non-Fermi liquid phase believed to support a violation of area law entanglement scaling. We hope tosettle numerical questions about this work in furtherance of the goal of incorporating it into a future publication.We then explore the stability of a system predisposed towards localization coupled to a system of similar size which would,on its own, thermalize. The stability of localized systems interacting with environmental baths is both experimentallyrelevant and theoretically interesting, and we explore a large parameter space using exact diagonalization techniques.We find that for a small set of parameters, localization may survive the presence of a similarly sized bath, but thetendency of the bath to delocalize the entire system is difficult to overcome. We also investigate the dynamics ofthese coupled systems, making connections with previous theoretical studies. This work is adapted from previously publishedmaterial.We take steps towards developing a new tensor network technique with which to study non-equilibrium quantum systems.We develop a disentangling circuit generation algorithm, drawing on previous algorithms for representing a variety ofinteresting wavefunctions, which can disentangle generic states without reference to a Hamiltonian. This is a novel departurefrom many previous disentangling approaches or tensor network optimization algorithms. We apply this technique to

A very fundamental problem in quantum statistical mechanics involves whether—and how—an isolated quantum system will reach thermal equilibrium after waiting a long time. In quantum systems that do thermalize, the long-time expectation value of any “reasonable” operator will match its predicted value in the canonical ensemble. The Eigenstate Thermalization Hypothesis (ETH) posits that this thermalization occurs at the level of each individual energy eigenstate; in fact, any single eigenstate in a microcanonical energy window will predict the expectation values of such operators exactly. In the first part of this dissertation, we identify, for a generic model system, precisely which operators satisfy ETH, as well as limits to the information contained in a single eigenstate. Remarkably, our results strongly suggest that a single eigenstate can contain information about energy densities—and therefore temperatures—far away from the energy density of the eigenstate.Additionally, we study the possible breakdown of quantum thermalization in a model of itinerant electrons on a one-dimensional chain, with both spin and charge degrees of freedom. This model exhibits peculiar properties in the entanglement entropy, the apparent scaling of which is modified from a “volume law” to an “area law” after performing a partial, site-wise measurement on the system. These properties and others suggest that this model realizes a new, non-thermal phase of matter, known as a Quantum Disentangled Liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.

A very fundamental problem in quantum statistical mechanics involves whether—and how—an isolated quantum system will reach thermal equilibrium after waiting a long time. In quantum systems that do thermalize, the long-time expectation value of any “reasonable” operator will match its predicted value in the canonical ensemble. The Eigenstate Thermalization Hypothesis (ETH) posits that this thermalization occurs at the level of each individual energy eigenstate; in fact, any single eigenstate in a microcanonical energy window will predict the expectation values of such operators exactly. In the first part of this dissertation, we identify, for a generic model system, precisely which operators satisfy ETH, as well as limits to the information contained in a single eigenstate. Remarkably, our results strongly suggest that a single eigenstate can contain information about energy densities—and therefore temperatures—far away from the energy density of the eigenstate.Additionally, we study the possible breakdown of quantum thermalization in a model of itinerant electrons on a one-dimensional chain, with both spin and charge degrees of freedom. This model exhibits peculiar properties in the entanglement entropy, the apparent scaling of which is modified from a “volume law” to an “area law” after performing a partial, site-wise measurement on the system. These properties and others suggest that this model realizes a new, non-thermal phase of matter, known as a Quantum Disentangled Liquid (QDL). The putative existence of this phase has striking implications for the foundations of quantum statistical mechanics.