Quantum State Reduction: Generalized Bipartitions from Algebras of Observables

Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity

Quantum State Reduction: Generalized Bipartitions from Algebras of Observables

Reduced density matrices are a powerful tool in the analysis of entanglement structure, approximate or coarse-grained dynamics, decoherence, and the emergence of classicality. It is straightforward to produce a reduced density matrix with the partial-trace map by tracing out part of the quantum state, but in many natural situations this reduction may not be achievable. We investigate the general problem of identifying how the quantum state is reduced given a restriction on the observables. For example, in an experimental setting, the set of observables that can actually be measured is usually modest (compared to the set of all possible observables) and their resolution is limited. In such situations, the appropriate state-reduction map can be defined via a generalized bipartition, which is associated with the structure of irreducible representations of the algebra generated by the restricted set of observables. One of our main technical results is a general, not inherently numeric, algorithm for finding irreducible representations of matrix algebras. We demonstrate the viability of this approach with two examples of limited-resolution observables. The definition of quantum state reductions can also be extended beyond algebras of observables. To accomplish this task we introduce a more flexible notion of bipartition, the partial bipartition, which describes coarse grainings preserving information about a limited set (not necessarily algebra) of observables. We describe a variational method to choose the coarse grainings most compatible with a specified Hamiltonian, which exhibit emergent classicality in the reduced state space. We apply this construction to the concrete example of the one-dimensional Ising model. Our results have relevance for quantum information, bulk reconstruction in holography, and quantum gravity

Principal component pyramids using image blurring for nonlinearity reduction in hand shape recognition

The thesis presents four algorithms using a multistage hierarchical strategy for hand shape recognition. The proposed multistage hierarchy analyzes new patterns by projecting them into the different levels of a data pyramid, which consists of different principal component spaces. Image blurring is used to reduce the nonlinearity in manifolds generated by a set of example images. Flattening the space helps in classifying different hand shapes more accurately. Four algorithms using different pattern recognition techniques are proposed. The first algorithm is based on using perpendicular distance to measure the distance between new patterns and the nearest manifold. The second algorithm is based on using supervised multidimensional grids. The third algorithm uses unsupervised multidimensional grids to cluster the space into cells of similar objects. The fourth algorithm is based on training a set of simple architecture multi-layer neural networks at the different levels of the pyramid to map new patterns to the closest class. The proposed algorithms are categorized as example-based approaches where a large set of computer generated images are used to densely sample the space. Experimental results are presented to examine the accuracy and performance of the proposed algorithms. The effect of image blurring on reducing the nonlinearity in manifolds is examined. The results are compared with the exhaustive search scenario. The experimental results show that the proposed algorithms are applicable for real time applications with high accuracy measures. They can achieve frame rates of more than 10 frames per second and accuracies of up to 98% on test data

Advances in Monocular Exemplar-based Human Body Pose Analysis: Modeling, Detection and Tracking

Esta tesis contribuye en el análisis de la postura del cuerpo humano a partir de secuencias de imágenes adquiridas con una sola cámara. Esta temática presenta un amplio rango de potenciales aplicaciones en video-vigilancia, video-juegos o aplicaciones biomédicas. Las técnicas basadas en patrones han tenido éxito, sin embargo, su precisión depende de la similitud del punto de vista de la cámara y de las propiedades de la escena entre las imágenes de entrenamiento y las de prueba. Teniendo en cuenta un conjunto de datos de entrenamiento capturado mediante un número reducido de cámaras fijas, paralelas al suelo, se han identificado y analizado tres escenarios posibles con creciente nivel de dificultad: 1) una cámara estática paralela al suelo, 2) una cámara de vigilancia fija con un ángulo de visión considerablemente diferente, y 3) una secuencia de video capturada con una cámara en movimiento o simplemente una sola imagen estática