From communication complexity to an entanglement spread area law in the ground state of gapped local Hamiltonians

In this work, we make a connection between two seemingly different problems. The first problem involves characterizing the properties of entanglement in the ground state of gapped local Hamiltonians, which is a central topic in quantum many-body physics. The second problem is on the quantum communication complexity of testing bipartite states with EPR assistance, a well-known question in quantum information theory. We construct a communication protocol for testing (or measuring) the ground state and use its communication complexity to reveal a new structural property for the ground state entanglement. This property, known as the entanglement spread, roughly measures the ratio between the largest and the smallest Schmidt coefficients across a cut in the ground state. Our main result shows that gapped ground states possess limited entanglement spread across any cut, exhibiting an "area law" behavior. Our result quite generally applies to any interaction graph with an improved bound for the special case of lattices. This entanglement spread area law includes interaction graphs constructed in [Aharonov et al., FOCS'14] that violate a generalized area law for the entanglement entropy. Our construction also provides evidence for a conjecture in physics by Li and Haldane on the entanglement spectrum of lattice Hamiltonians [Li and Haldane, PRL'08]. On the technical side, we use recent advances in Hamiltonian simulation algorithms along with quantum phase estimation to give a new construction for an approximate ground space projector (AGSP) over arbitrary interaction graphs.Comment: 29 pages, 1 figur

Entanglement area law for 1D gauge theories and bosonic systems

We prove an entanglement area law for a class of 1D quantum systems involving infinite-dimensional local Hilbert spaces. This class of quantum systems include bosonic models such as the Hubbard-Holstein model, and both U(1) and SU(2) lattice gauge theories in one spatial dimension. Our proof relies on new results concerning the robustness of the ground state and spectral gap to the truncation of Hilbert space, applied within the approximate ground state projector (AGSP) framework from previous work. In establishing this area law, we develop a system-size independent bound on the expectation value of local observables for Hamiltonians without translation symmetry, which may be of separate interest. Our result provides theoretical justification for using tensor network methods to study the ground state properties of quantum systems with infinite local degrees of freedom

Circuit Lower Bounds for Low-Energy States of Quantum Code Hamiltonians

The No Low-energy Trivial States (NLTS) conjecture of Freedman and Hastings [Freedman and Hastings, 2014] - which posits the existence of a local Hamiltonian with a super-constant quantum circuit lower bound on the complexity of all low-energy states - identifies a fundamental obstacle to the resolution of the quantum PCP conjecture. In this work, we provide new techniques, based on entropic and local indistinguishability arguments, that prove circuit lower bounds for all the low-energy states of local Hamiltonians arising from quantum error-correcting codes. For local Hamiltonians arising from nearly linear-rate or nearly linear-distance LDPC stabilizer codes, we prove super-constant circuit lower bounds for the complexity of all states of energy o(n). Such codes are known to exist and are not necessarily locally-testable, a property previously suspected to be essential for the NLTS conjecture. Curiously, such codes can also be constructed on a two-dimensional lattice, showing that low-depth states cannot accurately approximate the ground-energy even in physically relevant systems

Lecture Notes of Tensor Network Contractions

Tensor network (TN), a young mathematical tool of high vitality and great potential, has been undergoing extremely rapid developments in the last two decades, gaining tremendous success in condensed matter physics, atomic physics, quantum information science, statistical physics, and so on. In this lecture notes, we focus on the contraction algorithms of TN as well as some of the applications to the simulations of quantum many-body systems. Starting from basic concepts and definitions, we first explain the relations between TN and physical problems, including the TN representations of classical partition functions, quantum many-body states (by matrix product state, tree TN, and projected entangled pair state), time evolution simulations, etc. These problems, which are challenging to solve, can be transformed to TN contraction problems. We present then several paradigm algorithms based on the ideas of the numerical renormalization group and/or boundary states, including density matrix renormalization group, time-evolving block decimation, coarse-graining/corner tensor renormalization group, and several distinguished variational algorithms. Finally, we revisit the TN approaches from the perspective of multi-linear algebra (also known as tensor algebra or tensor decompositions) and quantum simulation. Despite the apparent differences in the ideas and strategies of different TN algorithms, we aim at revealing the underlying relations and resemblances in order to present a systematic picture to understand the TN contraction approaches.Comment: 134 pages, 68 figures. In this version, the manuscript has been changed into the format of book; new sections about tensor network and quantum circuits have been adde

Entanglement and Bell correlations in strongly correlated many-body quantum systems

During the past two decades, thanks to the mutual fertilization of the research in quantum information and condensed matter, new approaches based on purely quantum features without any classical analog turned out to be very useful in the characterization of many-body quantum systems (MBQS). A peculiar role is obviously played by the study of purely quantum correlations, manifesting in the “spooky” properties of entanglement and nonlocality (or Bell correlations), which ultimately discriminate classical from quantum regimes. It is, in fact, such kind of correlations that give rise to the plethora of intriguing emergent behaviors of MBQS, which cannot be reduced to a mere sum of the behaviors of individual components, the most important example being the quantum phase transitions. However, despite being indeed closely related concepts, entanglement and nonlocality are actually two different resources. With regard to the entanglement, we will use it to characterize several instances of MBQS, to exactly locate and characterize quantum phase transitions in spin-lattices and interacting fermionic systems, to classify different gapped quantum phases according to their topological features and to provide a purely quantum signature of chaos in dynamical systems. Our approach will be mainly numerical and for simulating the ground states of several one-dimensional lattice systems we draw heavily on the celebrated “density matrix renormalization group” (DMRG) algorithm in the “matrix product state” (MPS) ansatz. A MPS is a one-dimensional tensor network (TN) representation for quantum states and occupies a pivotal position in what we have gained in thinking MBQS from an entanglement perspective. In fact, the success of TNs states mainly relies on their fulfillment, by construction, of the so called “entanglement area law”. This is a feature shared by the ground states of gapped Hamiltonians with short-range interactions among the components and consists of a sub-extensive entanglement entropy, which grows only with the surface of the bipartition. This property translates in a reduced complexity of such systems, allowing affordable simulations, with an exponential reduction of computational costs. Besides the use of already existing TN-based algorithms, an effort will be done to develop a new one suitable for high-dimensional lattices. While many useful results are available for the entanglement in many different contexts, less is known about the role of nonlocality. Formally, a state of a multi-party system is defined nonlocal if its correlations violate some “Bell inequality” (BI). The derivation of the BIs for systems consisting of many parties is a formidable task and only recently, a class of them, relevant for nontrivial states, has been proposed. In an important chapter of the thesis, we apply these BIs to fully characterize the phase transition of a long-range ferromagnetic Ising model, doing a comparison with entanglement-based results and then making one of the first efforts in the study of MBQS from a nonlocality perspective.Durante las dos últimas décadas, gracias al enriquecimiento mutuo entre las investigaciones en información cuántica y materia condensada, se han desarrollado nuevos enfoques que han resultado muy útiles en la caracterización de los sistemas cuánticos de muchos cuerpos (SCMC), basados en características puramente cuánticas sin ningún análogo clásico. El estudio de las correlaciones puramente cuánticas juega obviamente un papel fundamental. Estas correlaciones se manifiestan en las propiedades del entrelazamiento cuántico (“entanglement”) y no-localidad (o correlaciones de Bell), que en última instancia discriminan los regímenes clásicos de los regímenes cuánticos. Este tipo de correlaciones son, de hecho, las que dan lugar a la plétora de comportamientos emergentes enigmáticos de los SCMC, que no pueden reducirse a una mera suma de los comportamientos de los componentes individuales, siendo el ejemplo más importante siendo las transiciones de fase cuánticas (TFC). Sin embargo, a pesar de ser conceptos estrechamente relacionados, el entrelazamiento y la no-localidad son en realidad dos recursos diferentes. Con respecto al entrelazamiento, lo utilizaremos para caracterizar varios ejemplos de SCMC, para localizar y caracterizar exactamente las TFC en retículos de espines y de sistemas de fermiones interactuantes, para clasificar las diferentes fases cuánticas de acuerdo con su topología y para proporcionar una señal puramente cuántica del caos en los sistemas dinámicos. Nuestro enfoque será principalmente numérico y para simular los estados fundamentales de varios sistemas unidimensionales nos basamos en gran medida en el célebre algoritmo “density matrix renormalization group” (DMRG), formulado en el ansatz de los “matrix product states” (MPS). Un MPS es un “retículos de tensores” (“tensor networks”, TN) unidimensional que representa estados cuánticos y ocupa una posición central entre los mayores logros obtenidos al estudiar los SCMC desde la perspectiva del entrelazamiento cuántico. De hecho, el éxito de los TN depende principalmente de su cumplimiento, por construcción, de una “ley del área” (“area-law”) de la entropía de entrelazamiento. Esta es una característica compartida por los estados fundamentales de los Hamiltonianos con interacciones de corto alcance entre los componentes del sistema y con una brecha (“gap”) entre el estado fundamental y los niveles excitados, que consiste en una entropía de entrelazamiento subextensiva, que crece sólo con la superficie de la bipartición. Esta propiedad se traduce en una menor complejidad de dichos sistemas, permitiendo simulaciones asequibles, con una reducción exponencial de los costes computacionales. Además del uso de los algoritmos ya existentes basados en TN, se desarrollará uno nuevo adecuado para sistemas en dimensiones altas. Si bien se dispone de muchos resultados útiles para el entrelazamiento en muchos contextos diferentes, se sabe menos sobre el papel jugado por la no-localidad. Formalmente, un estado de un sistema compuesto de muchas partes, se define como no-local si sus correlaciones violan alguna “desigualdad de Bell” (“Bell inequality”, BI). La derivación de dichas desigualdades para sistemas compuestos de muchas partes es un reto y sólo recientemente se ha propuesto una clase de ellas, relevante para estados no triviales. En un capítulo importante de la tesis, aplicamos estas BIs para caracterizar completamente la transición de fase de un modelo de Ising ferromagnético con interacciones de largo alcance, haciendo una comparación con los resultados basados en el entrelazamiento y luego haciendo uno de los primeros esfuerzos en el estudio de los SCMC desde una perspectiva de la no-localidad

Quantum Spin Liquids

Quantum spin liquids may be considered "quantum disordered" ground states of spin systems, in which zero point fluctuations are so strong that they prevent conventional magnetic long range order. More interestingly, quantum spin liquids are prototypical examples of ground states with massive many-body entanglement, of a degree sufficient to render these states distinct phases of matter. Their highly entangled nature imbues quantum spin liquids with unique physical aspects, such as non-local excitations, topological properties, and more. In this review, we discuss the nature of such phases and their properties based on paradigmatic models and general arguments, and introduce theoretical technology such as gauge theory and partons that are conveniently used in the study of quantum spin liquids. An overview is given of the different types of quantum spin liquids and the models and theories used to describe them. We also provide a guide to the current status of experiments to study quantum spin liquids, and to the diverse probes used therein.Comment: 60 pages, 8 figures, 1 tabl

Entanglement and Bell correlations in strongly correlated many-body quantum systems

During the past two decades, thanks to the mutual fertilization of the research in quantum information and condensed matter, new approaches based on purely quantum features without any classical analog turned out to be very useful in the characterization of many-body quantum systems (MBQS). A peculiar role is obviously played by the study of purely quantum correlations, manifesting in the “spooky” properties of entanglement and nonlocality (or Bell correlations), which ultimately discriminate classical from quantum regimes. It is, in fact, such kind of correlations that give rise to the plethora of intriguing emergent behaviors of MBQS, which cannot be reduced to a mere sum of the behaviors of individual components, the most important example being the quantum phase transitions. However, despite being indeed closely related concepts, entanglement and nonlocality are actually two different resources. With regard to the entanglement, we will use it to characterize several instances of MBQS, to exactly locate and characterize quantum phase transitions in spin-lattices and interacting fermionic systems, to classify different gapped quantum phases according to their topological features and to provide a purely quantum signature of chaos in dynamical systems. Our approach will be mainly numerical and for simulating the ground states of several one-dimensional lattice systems we draw heavily on the celebrated “density matrix renormalization group” (DMRG) algorithm in the “matrix product state” (MPS) ansatz. A MPS is a one-dimensional tensor network (TN) representation for quantum states and occupies a pivotal position in what we have gained in thinking MBQS from an entanglement perspective. In fact, the success of TNs states mainly relies on their fulfillment, by construction, of the so called “entanglement area law”. This is a feature shared by the ground states of gapped Hamiltonians with short-range interactions among the components and consists of a sub-extensive entanglement entropy, which grows only with the surface of the bipartition. This property translates in a reduced complexity of such systems, allowing affordable simulations, with an exponential reduction of computational costs. Besides the use of already existing TN-based algorithms, an effort will be done to develop a new one suitable for high-dimensional lattices. While many useful results are available for the entanglement in many different contexts, less is known about the role of nonlocality. Formally, a state of a multi-party system is defined nonlocal if its correlations violate some “Bell inequality” (BI). The derivation of the BIs for systems consisting of many parties is a formidable task and only recently, a class of them, relevant for nontrivial states, has been proposed. In an important chapter of the thesis, we apply these BIs to fully characterize the phase transition of a long-range ferromagnetic Ising model, doing a comparison with entanglement-based results and then making one of the first efforts in the study of MBQS from a nonlocality perspective.Durante las dos últimas décadas, gracias al enriquecimiento mutuo entre las investigaciones en información cuántica y materia condensada, se han desarrollado nuevos enfoques que han resultado muy útiles en la caracterización de los sistemas cuánticos de muchos cuerpos (SCMC), basados en características puramente cuánticas sin ningún análogo clásico. El estudio de las correlaciones puramente cuánticas juega obviamente un papel fundamental. Estas correlaciones se manifiestan en las propiedades del entrelazamiento cuántico (“entanglement”) y no-localidad (o correlaciones de Bell), que en última instancia discriminan los regímenes clásicos de los regímenes cuánticos. Este tipo de correlaciones son, de hecho, las que dan lugar a la plétora de comportamientos emergentes enigmáticos de los SCMC, que no pueden reducirse a una mera suma de los comportamientos de los componentes individuales, siendo el ejemplo más importante siendo las transiciones de fase cuánticas (TFC). Sin embargo, a pesar de ser conceptos estrechamente relacionados, el entrelazamiento y la no-localidad son en realidad dos recursos diferentes. Con respecto al entrelazamiento, lo utilizaremos para caracterizar varios ejemplos de SCMC, para localizar y caracterizar exactamente las TFC en retículos de espines y de sistemas de fermiones interactuantes, para clasificar las diferentes fases cuánticas de acuerdo con su topología y para proporcionar una señal puramente cuántica del caos en los sistemas dinámicos. Nuestro enfoque será principalmente numérico y para simular los estados fundamentales de varios sistemas unidimensionales nos basamos en gran medida en el célebre algoritmo “density matrix renormalization group” (DMRG), formulado en el ansatz de los “matrix product states” (MPS). Un MPS es un “retículos de tensores” (“tensor networks”, TN) unidimensional que representa estados cuánticos y ocupa una posición central entre los mayores logros obtenidos al estudiar los SCMC desde la perspectiva del entrelazamiento cuántico. De hecho, el éxito de los TN depende principalmente de su cumplimiento, por construcción, de una “ley del área” (“area-law”) de la entropía de entrelazamiento. Esta es una característica compartida por los estados fundamentales de los Hamiltonianos con interacciones de corto alcance entre los componentes del sistema y con una brecha (“gap”) entre el estado fundamental y los niveles excitados, que consiste en una entropía de entrelazamiento subextensiva, que crece sólo con la superficie de la bipartición. Esta propiedad se traduce en una menor complejidad de dichos sistemas, permitiendo simulaciones asequibles, con una reducción exponencial de los costes computacionales. Además del uso de los algoritmos ya existentes basados en TN, se desarrollará uno nuevo adecuado para sistemas en dimensiones altas. Si bien se dispone de muchos resultados útiles para el entrelazamiento en muchos contextos diferentes, se sabe menos sobre el papel jugado por la no-localidad. Formalmente, un estado de un sistema compuesto de muchas partes, se define como no-local si sus correlaciones violan alguna “desigualdad de Bell” (“Bell inequality”, BI). La derivación de dichas desigualdades para sistemas compuestos de muchas partes es un reto y sólo recientemente se ha propuesto una clase de ellas, relevante para estados no triviales. En un capítulo importante de la tesis, aplicamos estas BIs para caracterizar completamente la transición de fase de un modelo de Ising ferromagnético con interacciones de largo alcance, haciendo una comparación con los resultados basados en el entrelazamiento y luego haciendo uno de los primeros esfuerzos en el estudio de los SCMC desde una perspectiva de la no-localidad.Postprint (published version

Tensor Network Methods for Quantum Phases

The physics that emerges when large numbers of particles interact can be complex and exotic. The collective behaviour may not re ect the underlying constituents, for example fermionic quasiparticles can emerge from models of interacting bosons. Due to this emergent complexity, manybody phenomena can be very challenging to study, but also very useful. A theoretical understanding of such systems is important for robust quantum information storage and processing. The emergent, macroscopic physics can be classi ed using the idea of a quantum phase. All models within a given phase exhibit similar low-energy emergent physics, which is distinct from that displayed by models in di erent phases. In this thesis, we utilise tensor networks to study many-body systems in a range of quantum phases. These include topologically ordered phases, gapless symmetry-protected phases, and symmetry-enriched topological phases

Entanglement in Many-Body Systems

The recent interest in aspects common to quantum information and condensed matter has prompted a prosperous activity at the border of these disciplines that were far distant until few years ago. Numerous interesting questions have been addressed so far. Here we review an important part of this field, the properties of the entanglement in many-body systems. We discuss the zero and finite temperature properties of entanglement in interacting spin, fermionic and bosonic model systems. Both bipartite and multipartite entanglement will be considered. At equilibrium we emphasize on how entanglement is connected to the phase diagram of the underlying model. The behavior of entanglement can be related, via certain witnesses, to thermodynamic quantities thus offering interesting possibilities for an experimental test. Out of equilibrium we discuss how to generate and manipulate entangled states by means of many-body Hamiltonians.Comment: 61 pages, 29 figure