QUANTUM LOGIC ASSOCIATED TO FINITE DIMENSIONAL INTERVALS OF MODULAR ORTHOLATTICES

2016 ◽  
Vol 81 (2) ◽  
pp. 629-640
Author(s):  
R. GIUNTINI ◽  
H. FREYTES ◽  
G. SERGIOLI
Keyword(s):  
Quantum Logic ◽  
Finite Dimensional ◽  
Modular Ortholattice ◽  
Abstract Formulation

AbstractIn this work we study an abstract formulation of a problem posed by J.M. Dunn, T.J. Hagge et al. about the inclusion of varieties generated by the modular ortholattice of subspaces of ℂn. We shall prove that, this abstract formulation is equivalent to the direct irreducibility for atomic complete modular ortholattices.

The center of an orthologic

1972 ◽  
Vol 37 (4) ◽  
pp. 641-645 ◽  
Author(s):  
Barbara Jeffcott
Keyword(s):  
Quantum Mechanics ◽  
Boolean Algebra ◽  
Quantum Logic ◽  
Decisive Role ◽  
Orthomodular Poset ◽  
Fundamental Relation ◽  
Von Neumann ◽  
Modular Ortholattice ◽  
Probability And Statistics ◽  
Kolmogorov Theory

Since 1933, when Kolmogorov laid the foundations for probability and statistics as we know them today [1], it has been recognized that propositions asserting that such and such an event occurred as a consequence of the execution of a particular random experiment tend to band together and form a Boolean algebra. In 1936, Birkhoff and von Neumann [2] suggested that the so-called logic of quantum mechanics should not be a Boolean algebra, but rather should form what is now called a modular ortholattice [3]. Presumably, the departure from Boolean algebras encountered in quantum mechanics can be attributed to the fact that in quantum mechanics, one must consider more than one physical experiment, e.g., an experiment measuring position, an experiment measuring charge, an experiment measuring momentum, etc., and, because of the uncertainty principle, these experiments need not admit a common refinement in terms of which the Kolmogorov theory is directly applicable.Mackey's Axioms I–VI for quantum mechanics [4] imply that the logic of quantum mechanics should be a σ-orthocomplete orthomodular poset [5]. Most contemporary practitioners of quantum logic seem to agree that a quantum logic is (at least) an orthomodular poset [6], [7], [8], [9], [10] or some variation thereof [11]. P. D. Finch [12] has shown that every completely orthomodular poset is the logic arising from sets of Boolean logics, where these sets have a structure similar to the structures generally given to quantum logic. In all of these versions of quantum logic, a fundamental relation, the relation of compatibility or commutativity, plays a decisive role.

A spectral order for infinite-dimensional quantum spaces

2012 ◽  
Vol 23 (1) ◽  
pp. 95-130
Author(s):  
JOE MASHBURN
Keyword(s):  
Quantum Logic ◽  
Information Content ◽  
Shannon Entropy ◽  
Closed Subset ◽  
Quantum States ◽  
Spectral Order ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Finite Dimensional Case

In this paper we extend the spectral order of Coecke and Martin to infinite-dimensional quantum states. Many properties present in the finite-dimensional case are preserved, but some of the most important are lost. The order is constructed and its properties analysed. Most of the useful measurements of information content are lost. Shannon entropy is defined on only a part of the model, and that part is not a closed subset of the model. The finite parts of the lattices used by Birkhoff and von Neumann as models for classical and quantum logic appear as subsets of the models for infinite classical and quantum states.

scholarly journals A simple and quantum-mechanically motivated characterization of the formally real Jordan algebras

2020 ◽  
Vol 476 (2233) ◽  
pp. 20190604 ◽  
Author(s):  
Gerd Niestegge
Keyword(s):  
Hilbert Space ◽  
Quantum Logic ◽  
Jordan Algebras ◽  
Two Dimensional ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Dimensional Version

Quantum theory’s Hilbert space apparatus in its finite-dimensional version is nearly reconstructed from four simple and quantum-mechanically motivated postulates for a quantum logic. The recon- struction process is not complete, since it excludes the two-dimensional Hilbert space and still includes the exceptional Jordan algebras, which are not part of the Hilbert space apparatus. Options for physically meaningful potential generalizations of the apparatus are discussed.

scholarly journals Quantum logic as motivated by quantum computing

2005 ◽  
Vol 70 (2) ◽  
pp. 353-359 ◽  
Author(s):  
J. Michael Dunn ◽  
Tobias J. Hagge ◽  
Lawrence S. Moss ◽  
Zhenghan Wang
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Computer Science ◽  
Quantum Logic ◽  
Theoretical Computer Science ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Definition Of

§1. Introduction. Our understanding of Nature comes in layers, so should the development of logic. Classic logic is an indispensable part of our knowledge, and its interactions with computer science have recently dramatically changed our life. A new layer of logic has been developing ever since the discovery of quantum mechanics. G. D. Birkhoff and von Neumann introduced quantum logic in a seminal paper in 1936 [1]. But the definition of quantum logic varies among authors (see [2]). How to capture the logic structure inherent in quantum mechanics is very interesting and challenging. Given the close connection between classical logic and theoretical computer science as exemplified by the coincidence of computable functions through Turing machines, recursive function theory, and λ-calculus, we are interested in how to gain some insights about quantum logic from quantum computing. In this note we make some observations about quantum logic as motivated by quantum computing (see [5]) and hope more people will explore this connection.The quantum logic as envisioned by Birkhoff and von Neumann is based on the lattice of closed subspaces of a Hilbert space, usually an infinite dimensional one. The quantum logic of a fixed Hilbert space ℍ in this note is the variety of all the true equations with finitely many variables using the connectives meet, join and negation. Quantum computing is theoretically based on quantum systems with finite dimensional Hilbert spaces, especially the states space of a qubit ℂ2. (Actually the qubit is merely a convenience.

Non-Linear Dynamics of Cardiovascular Variability Signals

1994 ◽  
Vol 33 (01) ◽  
pp. 81-84 ◽  
Author(s):  
S. Cerutti ◽  
S. Guzzetti ◽  
R. Parola ◽  
M.G. Signorini
Keyword(s):  
Dimensional Space ◽  
Severe Heart Failure ◽  
Parametric Models ◽  
Deterministic Systems ◽  
Finite Dimensional ◽  
Return Maps ◽  
Pathological Conditions ◽  
Non Linear ◽  
Space State

Abstract:Long-term regulation of beat-to-beat variability involves several different kinds of controls. A linear approach performed by parametric models enhances the short-term regulation of the autonomic nervous system. Some non-linear long-term regulation can be assessed by the chaotic deterministic approach applied to the beat-to-beat variability of the discrete RR-interval series, extracted from the ECG. For chaotic deterministic systems, trajectories of the state vector describe a strange attractor characterized by a fractal of dimension D. Signals are supposed to be generated by a deterministic and finite dimensional but non-linear dynamic system with trajectories in a multi-dimensional space-state. We estimated the fractal dimension through the Grassberger and Procaccia algorithm and Self-Similarity approaches of the 24-h heart-rate variability (HRV) signal in different physiological and pathological conditions such as severe heart failure, or after heart transplantation. State-space representations through Return Maps are also obtained. Differences between physiological and pathological cases have been assessed and generally a decrease in the system complexity is correlated to pathological conditions.

Ternary quantum logic

2000 ◽  
Author(s):  
Normen Giesecke
Keyword(s):  
Quantum Logic

A closer look at the stable completion

2017 ◽  
Author(s):  
Ehud Hrushovski ◽  
François Loeser
Keyword(s):  
Inverse Limit ◽  
Unit Vector ◽  
Vector Spaces ◽  
Dimensional Vector ◽  
Compact Sets ◽  
Linear Topology ◽  
Topological Embedding ◽  
Definable Set ◽  
Finite Dimensional ◽  
The One

This chapter introduces the concept of stable completion and provides a concrete representation of unit vector Mathematical Double-Struck Capital A superscript n in terms of spaces of semi-lattices, with particular emphasis on the frontier between the definable and the topological categories. It begins by constructing a topological embedding of unit vector Mathematical Double-Struck Capital A superscript n into the inverse limit of a system of spaces of semi-lattices L(Hsubscript d) endowed with the linear topology, where Hsubscript d are finite-dimensional vector spaces. The description is extended to the projective setting. The linear topology is then related to the one induced by the finite level morphism L(Hsubscript d). The chapter also considers the condition that if a definable set in L(Hsubscript d) is an intersection of relatively compact sets, then it is itself relatively compact.

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