Properties of implication in effect algebras

2021 ◽  
Vol 71 (3) ◽  
pp. 523-534
Author(s):  
Ivan Chajda ◽  
Helmut Länger
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
State Space ◽  
Effect Algebra ◽  
Modus Ponens ◽  
Effect Algebras ◽  
Algebraic Semantics ◽  
Deductive Systems ◽  
Algebraic Description ◽  
Residuated Poset

Abstract Effect algebras form a formal algebraic description of the structure of the so-called effects in a Hilbert space which serve as an event-state space for effects in quantum mechanics. This is why effect algebras are considered as logics of quantum mechanics, more precisely as an algebraic semantics of these logics. Because every productive logic is equipped with implication, we introduce here such a concept and demonstrate its properties. In particular, we show that this implication is connected with conjunction via a certain “unsharp” residuation which is formulated on the basis of a strict unsharp residuated poset. Though this structure is rather complicated, it can be converted back into an effect algebra and hence it is sound. Further, we study the Modus Ponens rule for this implication by means of so-called deductive systems and finally we study the contraposition law.

scholarly journals Coexistency on Hilbert Space Effect Algebras and a Characterisation of Its Symmetry Transformations

2020 ◽  
Vol 379 (3) ◽  
pp. 1077-1112 ◽  
Author(s):  
György Pál Gehér ◽  
Peter Šemrl
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Mathematical Structure ◽  
Quantum Measurements ◽  
The Other ◽  
Effect Algebras ◽  
Natural Question ◽  
Fuzzy Event ◽  
Space Effect ◽  
Symmetry Transformations

Abstract The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig’s theorem. As a byproduct of our methods we also strengthen a theorem of Molnár.

Dynamic effect algebras

2012 ◽  
Vol 62 (3) ◽  
Author(s):  
Ivan Chajda ◽  
Miroslav Kolařík
Keyword(s):  
Quantum Mechanics ◽  
Effect Algebra ◽  
Dynamic Effect ◽  
Effect Algebras ◽  
Lattice Effect Algebra ◽  
Tense Operators ◽  
Lattice Effect

AbstractWe introduce the so-called tense operators in lattice effect algebras. Tense operators express the quantifiers “it is always going to be the case that” and “it has always been the case that” and hence enable us to express the dimension of time in the logic of quantum mechanics. We present an axiomatization of these tense operators and prove that every lattice effect algebra whose underlying lattice is complete can be equipped with tense operators. Such an effect algebra is called dynamic since it reflects changes of quantum events from past to future.

Qubits and the Framework of Quantum Mechanics

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Quantum Information ◽  
State Space ◽  
Quantum Physics ◽  
Complex Hilbert Space ◽  
The State ◽  
Mathematical Framework ◽  
Infinite Dimensional

In this section we introduce the framework of quantum mechanics as it pertains to the types of systems we will consider for quantum computing. Here we also introduce the notion of a quantum bit or ‘qubit’, which is a fundamental concept for quantum computing. At the beginning of the twentieth century, it was believed by most that the laws of Newton and Maxwell were the correct laws of physics. By the 1930s, however, it had become apparent that these classical theories faced serious problems in trying to account for the observed results of certain experiments. As a result, a new mathematical framework for physics called quantum mechanics was formulated, and new theories of physics called quantum physics were developed in this framework. Quantum physics includes the physical theories of quantum electrodynamics and quantum field theory, but we do not need to know these physical theories in order to learn about quantum information. Quantum information is the result of reformulating information theory in this quantum framework. We saw in Section 1.6 an example of a two-state quantum system: a photon that is constrained to follow one of two distinguishable paths. We identified the two distinguishable paths with the 2-dimensional basis vectors and then noted that a general ‘path state’ of the photon can be described by a complex vector with |α0|2 +|α1|2 = 1. This simple example captures the essence of the first postulate, which tells us how physical states are represented in quantum mechanics. Depending on the degree of freedom (i.e. the type of state) of the system being considered, H may be infinite-dimensional. For example, if the state refers to the position of a particle that is free to occupy any point in some region of space, the associated Hilbert space is usually taken to be a continuous (and thus infinite-dimensional) space. It is worth noting that in practice, with finite resources, we cannot distinguish a continuous state space from one with a discrete state space having a sufficiently small minimum spacing between adjacent locations. For describing realistic models of quantum computation, we will typically only be interested in degrees of freedom for which the state is described by a vector in a finite-dimensional (complex) Hilbert space.

The Mathematics of Quantum Mechanics

2019 ◽  
pp. 30-41
Author(s):  
Jeffrey A. Barrett
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Physical Properties ◽  
State Space ◽  
Linear Algebra ◽  
Linear Operators ◽  
Mathematical Representation ◽  
Quantum Mechanical ◽  
Standard Formulation ◽  
Schrödinger Picture

Quantum mechanics is written in the language of linear algebra. On the Schrodinger picture the theory represents quantum-mechanical states using the elements of a Hilbert space and represents observable physical properties and the standard dynamics using the linear operators on the state space. We consider the mathematical notions for understanding and working with the standard formulation of quantum mechanics. Each mathematical notion is characterized geometrically, algebraically, and physically. The mathematical representation of quantum-mechanical superpositions is discussed.

scholarly journals The logic induced by effect algebras

2020 ◽  
Vol 24 (19) ◽  
pp. 14275-14286 ◽  
Author(s):  
Ivan Chajda ◽  
Radomír Halaš ◽  
Helmut Länger
Keyword(s):  
Quantum Mechanics ◽  
Chain Condition ◽  
Axiom System ◽  
Single Element ◽  
Effect Algebras ◽  
Algebraic Semantics ◽  
Implication Algebras ◽  
Lattice Effect ◽  
Ascending Chain Condition

Abstract Effect algebras form an algebraic formalization of the logic of quantum mechanics. For lattice effect algebras $${\mathbf {E}}$$ E , we investigate a natural implication and prove that the implication reduct of $${\mathbf {E}}$$ E is term equivalent to $${\mathbf {E}}$$ E . Then, we present a simple axiom system in Gentzen style in order to axiomatize the logic induced by lattice effect algebras. For effect algebras which need not be lattice-ordered, we introduce a certain kind of implication which is everywhere defined but whose result need not be a single element. Then, we study effect implication algebras and prove the correspondence between these algebras and effect algebras satisfying the ascending chain condition. We present an axiom system in Gentzen style also for not necessarily lattice-ordered effect algebras and prove that it is an algebraic semantics for the logic induced by finite effect algebras.

scholarly journals Weakly Ordered A-Commutative Partial Groups of Linear Operators Densely Defined on Hilbert Space

10.14311/1807 ◽  
2013 ◽  
Vol 53 (3) ◽  
Author(s):  
Jirí Janda
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Commutative Group ◽  
Linear Operators ◽  
Positive Linear Operators ◽  
Generalized Effect Algebra ◽  
Algebraic Description ◽  
Densely Defined

The notion of a generalized effect algebra is presented as a generalization of effect algebra for an algebraic description of the structure of the set of all positive linear operators densely defined on a Hilbert space with the usual sum of operators. The structure of the set of not only positive linear operators can be described with the notion of a weakly ordered partial commutative group (wop-group).Due to the non-constructive algebraic nature of the wop-group we introduce its stronger version called a weakly ordered partial a-commutative group (woa-group). We show that it also describes the structure of not only positive linear operators.

Abstract Hilbert Space

2018 ◽  
Author(s):  
John von Neumann
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
State Space ◽  
Function Space ◽  
Wave Functions ◽  
Mathematical Basis ◽  
Discrete Function ◽  
Capital Letter ◽  
Fundamental Vector

This chapter defines Hilbert space, which furnishes the mathematical basis for the treatment of quantum mechanics. This is done within the context of an equation introduced in the previous chapter, and which have accordingly the same meaning in the “discrete” function space FsubscriptZ of the sequences xsubscriptv (ν‎ = 1, 2, . . .) and in the “continuous” Fsubscript Greek Capital Letter Omega of the wave functions φ‎(q₁, . . . , qₖ) (q₁, . . . , qₖ run through the entire state space Ω‎). In order to define abstract Hilbert space, this chapter takes as a basis the fundamental vector operations af, f ± g, (f, g).

scholarly journals Generalized Jordan Semitriple Maps on Hilbert Space Effect Algebras

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Qing Yuan ◽  
Kan He
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Space Effect ◽  
Antiunitary Operator

Letℰ(H)be the Hilbert space effect algebra on a Hilbert spaceHwithdim⁡H≥3,α,βtwo positive numbers with2α+β≠1andΦ:ℰ(H)→ℰ(H)a bijective map. We show that ifΦ(AαBβAα)=Φ(A)αΦ(B)βΦ(A)αholds for allA,B∈ℰ(H), then there exists a unitary or an antiunitary operatorUonHsuch thatΦ(A)=UAU*for everyA∈ℰ(H).

scholarly journals Extensions of ordering sets of states from effect algebras onto their MacNeille completions

10.29007/lkdv ◽  
2018 ◽  
Author(s):  
Jiří Janda ◽  
Zdenka Riečanová
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Macneille Completion ◽  
Atomic Lattice ◽  
Space Effect ◽  
Lattice Effect

In [Riečanová Z, Zajac M.: Hilbert Space Effect-Representations of Effect Algebras] it was shown that an effect algebra E with an ordering set M of states can by embedded into a Hilbert space effect algebra E(l<sub>2</sub>(M)). We consider the problem when its effect algebraic MacNeille completion Ê can be also embedded into the same Hilbert space effect algebra E(l<sub>2</sub>(M)). That is when the ordering set M of states on E can be be extended to an ordering set of states on Ê. We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.

Nonlinear quantum mechanics in a q-deformed Hilbert space

2019 ◽  
Vol 383 (23) ◽  
pp. 2729-2738 ◽  
Author(s):  
Bruno G. da Costa ◽  
Ernesto P. Borges
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Nonlinear Quantum ◽  
Nonlinear Quantum Mechanics
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