Information-Theoretic Princple Entails Orthomodularity of a Lattice

No Heading Quantum logical axiomatic systems for quantum theory usually include a postulate that a lattice under consideration is orthomodular. We propose a derivation of orthomodularity from an information-theoretic axiom. This provides conceptual clarity and removes a long-standing puzzle about the meaning of orthomodularity.     

Local Automorphisms of Some Quantum Mechanical Structures

Let H be a separable infinite-dimensional complex Hilbert space. We prove that every continuous 2-local automorphism of the poset (that is, partially ordered set) of all idempotents on H is an automorphism. Similar results concerning the orthomodular poset of all projections and the Jordan ring of all selfadjoint operators on H without the assumption on continuity are also presented.     

Automorphisms of An Orthomodular Poset of Projections

By using a lattice characterization of continuous projections defined on a topological vector space E arising from a dual pair, we determine the automorphism group of their orthomodular poset Proj(E) by means of automorphisms and anti-automorphisms of the lattice L of all closed subspaces of E. A connection between the automorphism group of the ring of all continuous linear mappings defined on E and the automorphism group of the orthoposet Proj(E) is established.     

Glafka-2004: Categorical Quantum Gravity

A synopsis-cum-update of work in the past half-decade or so on applying the algebraico-categorical concepts, technology and general philosophy of Abstract Differential Geometry (ADG) to various issues in current classical and quantum gravity research is presented. The exposition is mainly discursive, with conceptual, interpretational and philosophical matters emphasized throughout, while their formal technical-mathematical underpinnings have been left to the original papers. The general position is assumed that Quantum Gravity is in need of a new mathematical, novel physical concepts and principles introducing, framework in which old and current problems can be reformulated, readdressed and potentially retackled afresh. It is suggested that ADG can qualify as such a theoretical framework.Paper version of a talk given at the Glafka–2004: Iconoclastic Approaches to Quantum Gravity international theoretical physics conference, held in Athens, Greece (summer 2004).     

A Link between Quantum Logic and Categorical Quantum Mechanics

Abramsky and Coecke (Proceedings of the 19th Annual IEEE Symposium on Logic in Computer Science, pp. 415–425, IEEE Comput. Soc., New York, 2004) have recently introduced an approach to finite dimensional quantum mechanics based on strongly compact closed categories with biproducts. In this note it is shown that the projections of any object A in such a category form an orthoalgebra Proj  A. Sufficient conditions are given to ensure this orthoalgebra is an orthomodular poset. A notion of a preparation for such an object is given by Abramsky and Coecke, and it is shown that each preparation induces a finitely additive map from Proj  A to the unit interval of the semiring of scalars for this category. The tensor product for the category is shown to induce an orthoalgebra bimorphism Proj  A×Proj  B→Proj (A ⊗ B) that shares some of the properties required of a tensor product of orthoalgebras. These results are established in a setting more general than that of strongly compact closed categories. Many are valid in dagger biproduct categories, others require also a symmetric monoidal tensor compatible with the dagger and biproducts. Examples are considered for several familiar strongly compact closed categories.     

Tensor Products of Quantum Structures and Their Applications in Quantum Measurements

Tensor products of quantum logics and effect algebras with some known problems are reviewed. It is noticed that although tensor products of effect algebras having at least one state exist, in the category of complex Hilbert space effect algebras similar problems as with tensor products of projection lattices occur. Nevertheless, if one of the two coupled physical systems is classical, tensor product exists and can be considered as a Boolean power. Some applications of tensor products (in the form of Boolean powers) to quantum measurements are reviewed.     

Representation Theorem of Observables on a Quantum System

An orthomodular lattice (OML) with a conditional state can be used as a model for noncompatible events (a quantum system). In this paper we will study some properties of a conditional state and an s-map which are defined on an OML. We show conditions when a quantum system has the same properties as the classical probability space.     

A Comparison of Partial Information Decompositions Using Data from Real and Simulated Layer 5b Pyramidal Cells

Partial information decomposition allows the joint mutual information between an output and a set of inputs to be divided into components that are synergistic or shared or unique to each input. We consider five different decompositions and compare their results using data from layer 5b pyramidal cells in two different studies. The first study was on the amplification of somatic action potential output by apical dendritic input and its regulation by dendritic inhibition. We find that two of the decompositions produce much larger estimates of synergy and shared information than the others, as well as large levels of unique misinformation. When within-neuron differences in the components are examined, the five methods produce more similar results for all but the shared information component, for which two methods produce a different statistical conclusion from the others. There are some differences in the expression of unique information asymmetry among the methods. It is significantly larger, on average, under dendritic inhibition. Three of the methods support a previous conclusion that apical amplification is reduced by dendritic inhibition. The second study used a detailed compartmental model to produce action potentials for many combinations of the numbers of basal and apical synaptic inputs. Decompositions of the entire data set produce similar differences to those in the first study. Two analyses of decompositions are conducted on subsets of the data. In the first, the decompositions reveal a bifurcation in unique information asymmetry. For three of the methods, this suggests that apical drive switches to basal drive as the strength of the basal input increases, while the other two show changing mixtures of information and misinformation. Decompositions produced using the second set of subsets show that all five decompositions provide support for properties of cooperative context-sensitivity—to varying extents.     

Noncommutative Symmetric Differences in Orthomodular Lattices

We deal with the following question: What is the proper way to introduce symmetric difference in orthomodular lattices? Imposing two natural conditions on this operation, six possibilities remain: the two (commutative) normal forms of the symmetric difference in Boolean algebras and four noncommutative terms. It turns out that in many respects the noncommutative forms, though more complex with respect to the lattice operations, in their properties are much nearer to the symmetric difference in Boolean algebras than the commutative terms. As application we demonstrate the usefulness of noncommutative symmetric differences in the context of congruence relations.     

Scopes and limits of modality in quantum mechanics

We develop an algebraic frame for the simultaneous treatment of actual and possible properties of quantum systems. We show that, in spite of the fact that the language is enriched with the addition of a modal operator to the orthomodular structure, contextuality remains a central feature of quantum systems.     

The Orthomodular Poset of Projections of A Symmetric Lattices

In a Hilbert space, there exists a natural correspondence between continuous projections and particular pairs of closed subspaces. In this paper, we generalize this situation and associate to a symmetric lattice L a subset P(L) of L× L, called its projection poset. If L is the lattice of closed subspaces of a topological vector space then elements of P(L) correspond to continuous projections and we prove that automorphisms of P(L) are determined by automorphisms of the lattice L when this lattice satisfies some basic properties of lattices of closed subspaces. Primary: 06C15, Secondary: 03G12 81P10.     

Automorphism Groups of Orthomodular Lattices Obtained from Quadratic Spaces

We study the automorphism group of some orthomodular lattices, obtained from a quadratic space over a field K. We show how this group is linked to the semi-orthogonal group and with the group of all similarity transformations of the quadratic space. When the field K is finite, the cardinality of the automorphism group is given. AMS subject classification (1991): 06C15, 15A63, 20D45.     

Decidability in Orthomodular Lattices

We discuss the possibility of automatic simplification of formulas in orthomodular lattices. We describe the principles of a program which decides the validity of equalities and inequalities, as well as implications between them and other important relations significant in quantum mechanics. PACS: 02.10.-v, 02.10.Ab, 02.10.De, 03.65.Fd. AMS Subject classification: 06C15, 03G12, 06B10, 06B25, 81P10.     

Equations Holding in Hilbert Lattices

We produce and study several sequences of equations, in the language of orthomodular lattices, which hold in the ortholattice of closed subspaces of any classical Hilbert space, but not in all orthomodular lattices. Most of these equations hold in any orthomodular lattice admitting a strong set of states whose values are in a real Hilbert space. For some of these equations, we give conditions under which they hold in the ortholattice of closed subspaces of a generalised Hilbert space. These conditions are relative to the dimension of the Hilbert space and to the characteristic of its division ring of scalars. In some cases, we show that these equations cannot be deduced from the already known equations, and we study their mutual independence. To conclude, we suggest a new method for obtaining such equations, using the tensorial product. PACS numbers: 02.10, 03.65, 03.67     

Categorical dependence of vowel detection in long-term speech-shaped noise

The goal of this study was to measure detection thresholds for 12 isolated American English vowels naturally spoken by three male and three female talkers for young normal-hearing listeners in the presence of a long-term speech-shaped (LTSS) noise, which was presented at 70 dB sound pressure level. The vowel duration was equalized to 170 ms and the spectrum of the LTSS noise was identical to the long-term average spectrum of 12-talker babble. Given the same duration, detection thresholds for vowels differed by 19 dB across the 72 vowels. Thresholds for vowel detection showed a roughly U-shaped pattern as a function of the vowel category across talkers with lowest thresholds at /i/ and /ae/ vowels and highest thresholds at /u/ vowel in general. Both vowel category and talker had a significant effect on vowel detectability. Detection thresholds predicted from three excitation pattern metrics by using a simulation model were well matched with thresholds obtained from human listeners, suggesting that listeners could use a constant metric in the excitation pattern of the vowel to detect the signal in noise independent of the vowel category and talker. Application of the simulation model to predict thresholds of vowel detection in noise was also discussed.     

Measuring Interactions in Categorical Datasets Using Multivariate Symmetrical Uncertainty

Interaction between variables is often found in statistical models, and it is usually expressed in the model as an additional term when the variables are numeric. However, when the variables are categorical (also known as nominal or qualitative) or mixed numerical-categorical, defining, detecting, and measuring interactions is not a simple task. In this work, based on an entropy-based correlation measure for n nominal variables (named as Multivariate Symmetrical Uncertainty (MSU)), we propose a formal and broader definition for the interaction of the variables. Two series of experiments are presented. In the first series, we observe that datasets where some record types or combinations of categories are absent, forming patterns of records, which often display interactions among their attributes. In the second series, the interaction/non-interaction behavior of a regression model (entirely built on continuous variables) gets successfully replicated under a discretized version of the dataset. It is shown that there is an interaction-wise correspondence between the continuous and the discretized versions of the dataset. Hence, we demonstrate that the proposed definition of interaction enabled by the MSU is a valuable tool for detecting and measuring interactions within linear and non-linear models.     

On extending and optimising the direct product decomposition

Abstract

The concept of the direct product decomposition (DPD) is extended to arbitrary tensors while maintaining the same theoretical reduction in storage and computation. Additionally, the structure of the DPD as introduced by Gauss and Stanton is shown to be but one of a family of direct product decompositions which may be visualised using graphs. One particular member of this family is also shown to be critically important in relating the DPD and symmetry blocking approaches. Lastly, an implementation of tensor contraction using this extended DPD based on recent work in dense tensor contraction is presented, showing how the particular DPD used to represent the tensors in memory or on disk may be divorced from the optimal DPD used for a particular tensor contraction. The performance of the new algorithm is benchmarked by interfacing with the CFOUR programme suite, where significant speedups for CCSD calculations are observed.