Factorización y criticalidad en sistemas de espines

El objetivo general de esta tesis es examinar el comportamiento crítico y las correlaciones cuánticas de sistemas cuánticos interactuantes -en particular de sistemas de espines- en función de un parámetro de control. Típicamente, los autoestados exactos de estos sistemas son estados entrelazados -cuanticamente correlacionados- aun en presencia de campos magnéticos externos. No obstante, bajo ciertas condiciones de los valores y orientaciones del campo aplicado, estos sistemas pueden poseer un estado fundamental exacto completamente separable. Este notable fenómeno es conocido como "factorización". En esta tesis se determinan y estudian las condiciones de existencia de campos factorizantes en sistemas finitos de espines con acoplamientos cuadráticos generales inmersos en campos no necesariamente uniformes o transversos. En particular, se demuestra que las ecuaciones derivadas permiten descubrir nuevos fenómenos críticos en sistemas con interacciones de tipo XYZ y XXZ, entre otros. Mediante un estudio riguroso del entrelazamiento en la vecindad de los puntos de factorización se muestra que estos se corresponden con verdaderos puntos críticos cuánticos en sistemas finitos. Finalmente, se discute cómo estos resultados posibilitan inducir fenómenos críticos y nuevas formas de frustración; así como permiten diseñar esquemas de ingeniería de estados separables, aptos para ser utilizados como estados iniciales en algoritmos cuánticos.The main goal of this thesis is to study the critical behavior and quantum correlation properties of interacting quantum systems -in particular of spin systems- when a control parameter is varied. The exact ground state (GS) of interacting spin systems in an external magnetic field is typically an entangled state, i.e., a state with quantum correlations. However, under certain conditions for an applied field these systems can possess a completely separable exact GS. This remarkable phenomenon is known as "factorization". In this thesis, we derive and study the conditions for the existence of factorization in finite spin arrays with general quadratic couplings immersed in magnetic fields, not necessarily uniform or transverse. In particular, it is shown that these equations allow for the determination of novel critical phenomena in systems with XYZ and XXZ-type couplings, among others. By rigorously studying the entanglement in the vicinity of factorization points it is proven that they correspond to quantum critical points in finite systems. We then discuss how these results enable the possibility to induce critical phenomena and new types of frustration. The present results also open the way for separable ground-state engineering, which can be useful for quantum information applications.Facultad de Ciencias Exacta

Factorization in spin systems under general fields and separable ground-state engineering

We discuss ground-state factorization schemes in spin S arrays with general quadratic couplings under general magnetic fields, not necessarily uniform or transverse. It is shown that, given arbitrary spin alignment directions at each site, nonzero XYZ couplings between any pair and fields at each site always exist such that the ensuing Hamiltonian has an exactly separable eigenstate with the spins pointing along the specified directions. Furthermore, by suitable tuning of the fields this eigenstate can always be cooled down to a nondegenerate ground state. It is also shown that in open one-dimensional systems with fixed arbitrary first-neighbor couplings at least one separable eigenstate compatible with an arbitrarily chosen spin direction at one site is always feasible if the fields at each site can be tuned. We demonstrate as well that in the vicinity of factorization, i.e., for small perturbations in the fields or couplings, pairwise entanglement reaches full range. Some noticeable examples of factorized eigenstates are unveiled. The present results open the way for separable ground-state engineering. A notation to quantify the complexity of a given type of solution according to the required control on the system couplings and fields is introduced.Instituto de Física La Plat

Factorization in spin systems under general fields and separable ground-state engineering

We discuss ground-state factorization schemes in spin S arrays with general quadratic couplings under general magnetic fields, not necessarily uniform or transverse. It is shown that, given arbitrary spin alignment directions at each site, nonzero XYZ couplings between any pair and fields at each site always exist such that the ensuing Hamiltonian has an exactly separable eigenstate with the spins pointing along the specified directions. Furthermore, by suitable tuning of the fields this eigenstate can always be cooled down to a nondegenerate ground state. It is also shown that in open one-dimensional systems with fixed arbitrary first-neighbor couplings at least one separable eigenstate compatible with an arbitrarily chosen spin direction at one site is always feasible if the fields at each site can be tuned. We demonstrate as well that in the vicinity of factorization, i.e., for small perturbations in the fields or couplings, pairwise entanglement reaches full range. Some noticeable examples of factorized eigenstates are unveiled. The present results open the way for separable ground-state engineering. A notation to quantify the complexity of a given type of solution according to the required control on the system couplings and fields is introduced.Instituto de Física La Plat

Non-transverse factorizing fields and entanglement in finite spin systems

We determine the conditions for the existence of non-transverse factorizing magnetic fields in general spin arrays with anisotropic XY Z couplings of arbitrary range. It is first shown that a uniform maximally aligned completely separable eigenstate can exist just for fields hs parallel to a principal plane and forming four straight lines in field space, with the alignment direction different from that of hs and determined by the anisotropy. Such state always becomes a non-degenerate ground state (GS) for sufficiently strong (yet finite) fields along these lines, in both ferromagnetic (FM) and antiferromagnetic (AFM) type systems. In AFM chains, this field coexists with the nontransverse factorizing field h′ s associated with a degenerate N´eel-type separable GS, which is shown to arise at a level crossing in a finite chain. It is also demonstrated for arbitrary spin that pairwise entanglement reaches full range in the vicinity of both hs and h′ s, vanishing at hs but approaching small yet finite side-limits at h′ s, which are analytically determined. The behavior of the block entropy and entanglement spectrum in their vicinity is also analyzed.Facultad de Ciencias ExactasInstituto de Física La Plat

Non-transverse factorizing fields and entanglement in finite spin systems

We determine the conditions for the existence of non-transverse factorizing magnetic fields in general spin arrays with anisotropic XY Z couplings of arbitrary range. It is first shown that a uniform maximally aligned completely separable eigenstate can exist just for fields hs parallel to a principal plane and forming four straight lines in field space, with the alignment direction different from that of hs and determined by the anisotropy. Such state always becomes a non-degenerate ground state (GS) for sufficiently strong (yet finite) fields along these lines, in both ferromagnetic (FM) and antiferromagnetic (AFM) type systems. In AFM chains, this field coexists with the nontransverse factorizing field h′ s associated with a degenerate N´eel-type separable GS, which is shown to arise at a level crossing in a finite chain. It is also demonstrated for arbitrary spin that pairwise entanglement reaches full range in the vicinity of both hs and h′ s, vanishing at hs but approaching small yet finite side-limits at h′ s, which are analytically determined. The behavior of the block entropy and entanglement spectrum in their vicinity is also analyzed.Facultad de Ciencias ExactasInstituto de Física La Plat

Factorization and criticality in finite XXZ systems of arbitrary spin

We analyze ground state (GS) factorization in general arrays of spinss_iwithXXZcouplings immersed in nonuniform fields. It is shown that an exceptionally degenerate set of completely separable symmetry-breaking GS's can arise for a wide range of field configurations, at a quantum critical point where all GS magnetization plateaus merge. Such configurations include alternating fields as well as zero bulk field solutions with edge fields only and intermediate solutions with zero field at specific sites, valid ford-dimensional arrays. The definite magnetization projected GS's at factorization can be analytically determined and depend only on the exchange anisotropies, exhibiting critical entanglement properties. We also show that some factorization compatible field configurations may result in field-induced frustration and nontrivial behavior at strong fields.Facultad de Ciencias Exacta

Non-transverse factorizing fields and entanglement in finite spin systems

We determine the conditions for the existence of non-transverse factorizing magnetic fields in general spin arrays with anisotropic XY Z couplings of arbitrary range. It is first shown that a uniform maximally aligned completely separable eigenstate can exist just for fields hs parallel to a principal plane and forming four straight lines in field space, with the alignment direction different from that of hs and determined by the anisotropy. Such state always becomes a non-degenerate ground state (GS) for sufficiently strong (yet finite) fields along these lines, in both ferromagnetic (FM) and antiferromagnetic (AFM) type systems. In AFM chains, this field coexists with the nontransverse factorizing field h′ s associated with a degenerate N´eel-type separable GS, which is shown to arise at a level crossing in a finite chain. It is also demonstrated for arbitrary spin that pairwise entanglement reaches full range in the vicinity of both hs and h′ s, vanishing at hs but approaching small yet finite side-limits at h′ s, which are analytically determined. The behavior of the block entropy and entanglement spectrum in their vicinity is also analyzed.Facultad de Ciencias ExactasInstituto de Física La Plat

Factorization and criticality in finite XXZ systems of arbitrary spin

We analyze ground state (GS) factorization in general arrays of spinss_iwithXXZcouplings immersed in nonuniform fields. It is shown that an exceptionally degenerate set of completely separable symmetry-breaking GS's can arise for a wide range of field configurations, at a quantum critical point where all GS magnetization plateaus merge. Such configurations include alternating fields as well as zero bulk field solutions with edge fields only and intermediate solutions with zero field at specific sites, valid ford-dimensional arrays. The definite magnetization projected GS's at factorization can be analytically determined and depend only on the exchange anisotropies, exhibiting critical entanglement properties. We also show that some factorization compatible field configurations may result in field-induced frustration and nontrivial behavior at strong fields.Facultad de Ciencias Exacta

System-time entanglement in a discrete time model

We present a model of discrete quantum evolution based on quantum correlations between the evolving system and a reference quantum clock system. A quantum circuit for the model is provided, which in the case of a constant Hamiltonian is able to represent the evolution over2ntime steps in terms of justntime qubits andncontrol gates. We then introduce the concept of system-time entanglement as a measure of distinguishable quantum evolution, based on the entanglement between the system and the reference clock. This quantity vanishes for stationary states and is maximum for systems jumping onto a new orthogonal state at each time step. In the case of a constant Hamiltonian leading to a cyclic evolution it is a measure of the spread over distinct energy eigenstates, and satisfies an entropic energy-time uncertainty relation. The evolution of mixed states is also examined. Analytical expressions for the basic case of a qubit clock, as well as for the continuous limit in the evolution between two states, are provided.Instituto de Física La Plat

Quantum discord and entropic measures of quantum correlations: Optimization and behavior in finite XY spin chains

We discuss a generalization of the conditional entropy and one-way information deficit in quantum systems, based on general entropic forms. The formalism allows to consider simple entropic forms for which a closed evaluation of the associated optimization problem in qudit-qubit systems is shown to become feasible, allowing to approximate that of the quantum discord. As application, we examine quantum correlations of spin pairs in the exact ground state of finiteXYspin chains in a magnetic field through the quantum discord and information deficit. While these quantities show a similar behavior, their optimizing measurements exhibit significant differences, which can be understood and predicted through the previous approximations. The remarkable behavior of these quantities in the vicinity of transverse and non-transverse factorizing fields is also discussed.Facultad de Ciencias Exacta