Entanglement capabilities of the spin representation of (3+1)D-conformal transformations

Relying on a mathematical analogy of the pure states of the two-qubit system of quantum information theory with four-component spinors we introduce the concept of the intrinsic entanglement of spinors. To explore its physical sense we study the entanglement capabilities of the spin representation of (pseudo-) conformal transformations in (3+1)-dimensional Minkowski space-time. We find that only those tensor product structures can sensibly be introduced in spinor space for which a given spinor is not entangled.Comment: 15 pages LaTeX (v2: minor changes, headings introduced; v3: secs. 1, 5, 6 extended, references [2], [3], [10]-[17], [37], [38] added, final version to appear in Quant. Inf. Comput.

Many body physics from a quantum information perspective

The quantum information approach to many body physics has been very successful in giving new insight and novel numerical methods. In these lecture notes we take a vertical view of the subject, starting from general concepts and at each step delving into applications or consequences of a particular topic. We first review some general quantum information concepts like entanglement and entanglement measures, which leads us to entanglement area laws. We then continue with one of the most famous examples of area-law abiding states: matrix product states, and tensor product states in general. Of these, we choose one example (classical superposition states) to introduce recent developments on a novel quantum many body approach: quantum kinetic Ising models. We conclude with a brief outlook of the field.Comment: Lectures from the Les Houches School on "Modern theories of correlated electron systems". Improved version new references adde

Quantum information and physics: Some future directions

I consider some promising future directions for quantum information theory that could influence the development of 21st century physics. Advances in the theory of the distinguishability of superoperators may lead to new strategies for improving the precision of quantum-limited measurements. A better grasp of the properties of multi-partite quantum entanglement may lead to deeper understanding of strongly-coupled dynamics in quantum many-body systems, quantum field theory, and quantum gravity

Entanglement entropy in homogeneus, fermionic chains: some results and some conjectures

El objetivo de esta tesis es el estudio de la entropía de entrelazamiento de Rényi en los estados estacionarios de cadenas de fermiones sin spin descritas por un Hamiltoniano cuadrático general con invariancia translacional y posibles acoplos a larga distancia.Nuestra investigación se basa en la relación que existe entre la matriz densidad de los estados estacionarios y la correspondiente matriz de correlaciones entre dos puntos. Esta propiedad reduce la complejidad de calcular numéricamente la entropía de entrelazamiento y permite expresar esta magnitud en términos del determinante del resolvente de la matriz de correlaciones.Dado que la cadena es invariante translacional, la matriz de correlaciones es una matriz block Toeplitz. En vista de este hecho, la filosofía que seguimos en esta tesis es la de aprovecharnos de las propiedades asintóticas de este tipo de determinantes para investigar la entropía de entrelazamiento de Rényi en el límite termodinámico. Un aspecto interesante es que los resultados conocidos sobre el comportamiento asintótico de los determinantes block Toeplitz no son válidos para algunas de las matrices de correlaciones que consideraremos. Intentando llenar esta laguna, obtenemos algunos resultados originales sobre el comportamiento asintótico de los determinantes de matrices de Toeplitz y block Toeplitz.Estos nuevos resultados combinados con los ya previamente conocidos nos permiten obtener analíticamente el término dominante en la expansión de la entropía de entrelazamiento, tanto para un único intervalo de puntos o sites contiguos de la cadena como para subsistemas formados por varios intervalos disjuntos. En particular, descubrimos que los acoplos de largo alcance dan lugar a nuevas propiedades del comportamiento asintótico de la entropía tales como la aparición de un término logarítmico no universal fuera de los puntos críticos cuando los términos de pairing decaen siguiendo una ley de potencias o un crecimiento sublogarítmico cuando dichos acoplos decaen logarítmicamente. El estudio de la entropía de entrelazamiento a través de los determinantes block Toeplitz también nos ha llevado a descubrir una nueva simetría de la entropía de entrelazamiento bajo transformaciones de Möbius que pueden verse como transformaciones de los acoplos de la teoría. En particular, encontramos que para teorías críticasesta simetría presenta un intrigante paralelismo con las transformaciones conformes en el espacio-tiempo. <br /

Real-space renormalization group methods in the age of tensor network states

This dissertation contributes to the ongoing effort of understanding the origins and applications of real-space renormalization group methods in tensor network representations of classical and quantum many-body systems. First, we construct a matrix product operator ansatz to coarse-grain real-space transfer matrices of matrix product state descriptions of one-dimensional quantum spin chains. By treating the physical spin as an impurity, we unravel the virtual entanglement degrees of freedom of matrix product states into a layered structure to reveal an inherent renormalization group scale. Secondly, we rephrase tensor network renormalization for two-dimensional classical lattice models in a manifestly nonnegative way. The resulting real-space renormalization group flow preserves positivity and hence yields an interpretation in terms of Hamiltonian flows, reconciling modern real-space tensor network renormalization methods with traditional block-spin approaches. Thirdly, we study non-local symmetries in tensor networks by expressing two-dimensional classical partition functions in terms of strange correlators of judiciously chosen product states and string-net wave functions. We exhibit and exploit the emerging non-local symmetries of the partition function at criticality and highlight parallels between topological sectors and conformal primary fields in the shared framework of matrix product operator symmetries. Additionally, we provide a complementary perspective on real-space renormalization by recognizing known tensor network renormalization methods as the approximate truncation of an exactly coarse-grained strange correlator