12 research outputs found
A complete graphical calculus for Spekkens' toy bit theory
While quantum theory cannot be described by a local hidden variable model, it
is nevertheless possible to construct such models that exhibit features
commonly associated with quantum mechanics. These models are also used to
explore the question of {\psi}-ontic versus {\psi}-epistemic theories for
quantum mechanics. Spekkens' toy theory is one such model. It arises from
classical probabilistic mechanics via a limit on the knowledge an observer may
have about the state of a system. The toy theory for the simplest possible
underlying system closely resembles stabilizer quantum mechanics, a fragment of
quantum theory which is efficiently classically simulable but also non-local.
Further analysis of the similarities and differences between those two theories
can thus yield new insights into what distinguishes quantum theory from
classical theories, and {\psi}-ontic from {\psi}-epistemic theories.
In this paper, we develop a graphical language for Spekkens' toy theory.
Graphical languages offer intuitive and rigorous formalisms for the analysis of
quantum mechanics and similar theories. To compare quantum mechanics and a toy
model, it is useful to have similar formalisms for both. We show that our
language fully describes Spekkens' toy theory and in particular, that it is
complete: meaning any equality that can be derived using other formalisms can
also be derived entirely graphically. Our language is inspired by a similar
graphical language for quantum mechanics called the ZX-calculus. Thus Spekkens'
toy bit theory and stabilizer quantum mechanics can be analysed and compared
using analogous graphical formalisms.Comment: Major revisions for v2. 22+7 page
A Functorial Construction of Quantum Subtheories
We apply the geometric quantization procedure via symplectic groupoids
proposed by E. Hawkins to the setting of epistemically restricted toy theories
formalized by Spekkens. In the continuous degrees of freedom, this produces the
algebraic structure of quadrature quantum subtheories. In the odd-prime finite
degrees of freedom, we obtain a functor from the Frobenius algebra in
\textbf{Rel} of the toy theories to the Frobenius algebra of stabilizer quantum
mechanics.Comment: 19 page
Geometric Quantization and Epistemically Restricted Theories: The Continuous Case
It is possible to reproduce the quantum features of quantum states, starting
from a classical statistical theory and then limiting the amount of knowledge
that an agent can have about an individual system [5, 18].These are so called
epistemic restrictions. Such restrictions have been recently formulated in
terms of the symplectic geometry of the corresponding classical theory [19].
The purpose of this note is to describe, using this symplectic framework, how
to obtain a C*-algebraic formulation for the epistemically restricted theories.
In the case of continuous variables, following the groupoid quantization recipe
of E. Hawkins, we obtain a twisted group C*-algebra which is the usual Moyal
quantization of a Poisson vector space [12].Comment: In Proceedings QPL 2016, arXiv:1701.00242. 10 page
Cohomology of Toroidal Orbifold Quotients
Let denote an integral representation of the cyclic
group of prime order . This induces a -action on the torus
. The goal of this paper is to explicitly compute the
cohomology groups for any such representation. As a
consequence we obtain an explicit calculation of the integral cohomology of the
classifying space associated to the family of finite subgroups for any
crystallographic group with prime holonomy.Comment: Final version. Accepted for publication in the Journal of Algebr
Fusion algebras and cohomology of toroidal orbifolds
In this thesis we exhibit an explicit non-trivial example of the twisted fusion algebra for a particular finite group. The product is defined for the third power of modulo two group via the pairing of projective representations where the three cocycles are chosen using the inverse transgression map. We find the rank of the fusion algebra as well as the relation between its basis elements. We also give some applications to topological gauge theories.
We next show that the twisted fusion algebra of the third power of modulo p group is isomorphic to the non-twisted fusion algebra of the extraspecial p-group of order p³ and exponent p.
The final point of my thesis is to explicitly compute the cohomology groups of toroidal orbifolds which are the quotient spaces obtained by the action of modulo p group on the k-dimensional torus. We compute the particular case where the action is induced by the n-th power of augmentation ideal.Science, Faculty ofMathematics, Department ofGraduat
Gene Coexpression Network Comparison via Persistent Homology
Persistent homology, a topological data analysis (TDA) method, is applied to microarray data sets. Although there are a few papers referring to TDA methods in microarray analysis, the usage of persistent homology in the comparison of several weighted gene coexpression networks (WGCN) was not employed before to the very best of our knowledge. We calculate the persistent homology of weighted networks constructed from 38 Arabidopsis microarray data sets to test the relevance and the success of this approach in distinguishing the stress factors. We quantify multiscale topological features of each network using persistent homology and apply a hierarchical clustering algorithm to the distance matrix whose entries are pairwise bottleneck distance between the networks. The immunoresponses to different stress factors are distinguishable by our method. The networks of similar immunoresponses are found to be close with respect to bottleneck distance indicating the similar topological features of WGCNs. This computationally efficient technique analyzing networks provides a quick test for advanced studies
Texture Analysis of Hydrophobic Polycarbonate and Polydimethylsiloxane Surfaces via Persistent Homology
Due to recent climate change-triggered, regular dust storms in the Middle East, dust mitigation has become the critical issue for solar energy harvesting devices. One of the methods to minimize and prevent dust adhesion and create self-cleaning abilities is to generate hydrophobic characteristics on surfaces. The purpose of this study is to explore the topological features of hydrophobic surfaces. We use non-standard techniques from topological data analysis to extract morphological features from the AFM images. Our method recovers most of the previous qualitative observations in a robust and quantitative way. Persistence diagrams, which is a summary of topological structures, witness quantitatively that the crystallized polycarbonate (PC) surface possesses spherulites, voids, and fibrils, and the texture height and spherulite concentration increases with the increased immersion period. The approach also shows that the polydimethylsiloxane (PDMS) exactly copied the structures at the PC surface but 80 to 90 percent of the nanofibrils were not copied at PDMS surface. We next extract a feature vector from each persistence diagram to show which experiments hold features with similar variance using principal component analysis (PCA). The K-means clustering algorithm is applied to the matrix of feature vectors to support the PCA result, grouping experiments with similar features
Uncovering Dynamic Brain Reconfiguration in MEG Working Memory n-Back Task Using Topological Data Analysis
The increasing availability of high temporal resolution neuroimaging data has increased the efforts to understand the dynamics of neural functions. Until recently, there are few studies on generative models supporting classification and prediction of neural systems compared to the description of the architecture. However, the requirement of collapsing data spatially and temporally in the state-of-the art methods to analyze functional magnetic resonance imaging (fMRI), electroencephalogram (EEG) and magnetoencephalography (MEG) data cause loss of important information. In this study, we addressed this issue using a topological data analysis (TDA) method, called Mapper, which visualizes evolving patterns of brain activity as a mathematical graph. Accordingly, we analyzed preprocessed MEG data of 83 subjects from Human Connectome Project (HCP) collected during working memory n-back task. We examined variation in the dynamics of the brain states with the Mapper graphs, and to determine how this variation relates to measures such as response time and performance. The application of the Mapper method to MEG data detected a novel neuroimaging marker that explained the performance of the participants along with the ground truth of response time. In addition, TDA enabled us to distinguish two task-positive brain activations during 0-back and 2-back tasks, which is hard to detect with the other pipelines that require collapsing the data in the spatial and temporal domain. Further, the Mapper graphs of the individuals also revealed one large group in the middle of the stimulus detecting the high engagement in the brain with fine temporal resolution, which could contribute to increase spatiotemporal resolution by merging different imaging modalities. Hence, our work provides another evidence to the effectiveness of the TDA methods for extracting subtle dynamic properties of high temporal resolution MEG data without the temporal and spatial collapse