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Journal of Physics A Mathematical and General

On state-dependent implication in quantum mechanics

Fedor Herbut

Published Nov 21, 1994• 21 citations
Publisher

Abstract

This paper studies a generalization of the a priori implication E<or=F, meaning of course EF=E, in the quantum logic P(H) of all projectors. The new definition can be applied also to new pairs of events, being based on the indistinguishability of some events when considered from the standpoint of a (fixed) state rho . The mathematical theory of factor Boolean algebras, on which this extension is based, is clarified.

DOI
10.1088/0305-4470/27/22/022

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The mixing structure of a given quantum state (statistical operator) is considered, that is, the pure states and associated weights of which it consists. Its influence on the state-dependent quantum logical implication determined by is also studied. It is found that it has no influence at all except through the null-projector. It is shown that the latter, when it determines the state-dependent implication, in a certain sense, carries with it all the Boolean mathematical structure of projectors in the given Boolean subalgebra of quantum logic , where is the `domain' of definition of the state-dependent implication.

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Tlic classical Kohnogorov's model of probability assumes that every pair of ev(Tits is siinult.aneouslyobservable.This principle is violated in several applications including quantum mechanics, artificial intelligence, psychology, sociology etc.In these areas iioneoinpatible events are encountered.These are events which can be observed separately, but not simultaneously, so they are not contained in a Boolean subalgebra i = classical subsystem) of the event structure describing the system in question.Various attempts have been made to generalize the probability theory to a more general structure admitting noneonipatibility.Among them, classes of subsets (more generally, concrete logics) wore studied for many years (see e.g.[15, 18]).Although some results were successfully generalized (see e.g.[4. 10, 19]).the theory proceeded slowly and with serious difficulties.Hero we introduce a more special-but still reasonably general-structure, a kernel logic.As it is described in terms of Boolean algebras using measure-theoretic notions, we believe that there is a greater chance to generalize classical results for Boolean algebras to kernel logics.Kernel logics seem to be interesting also from the algebraic point of view as a new const ruction technique for concrete logics.Its usefulness was proved by solutions of several quite nontrivial problems.Besides this, it scorns desirable to describe kernels of measures on Boolean algebras.

Czechoslovak Mathematical Journal•Jan 1, 1996•6 citations
This paper studies a generalization of the a priori implication E<or=F, meaning of course EF=E, in the quantum logic P(H) of all projectors. The new definition can be applied also to new pairs of events, being based on the indistinguishability of some events when considered from the standpoint of a (fixed) state rho . The mathematical theory of factor Boolean algebras, on which this extension is based, is clarified.
The mixing structure of a given quantum state (statistical operator) is considered, that is, the pure states and associated weights of which it consists. Its influence on the state-dependent quantum logical implication determined by is also studied. It is found that it has no influence at all except through the null-projector. It is shown that the latter, when it determines the state-dependent implication, in a certain sense, carries with it all the Boolean mathematical structure of projectors in the given Boolean subalgebra of quantum logic , where is the `domain' of definition of the state-dependent implication.
Ideal occurrence of an event (projector) leads to the known change of a state (density operator) into (the Lüders state). It is shown that two events and give the same Lüders state if and only if the equivalence relation is valid. This relation determines equivalence classes. The set of them and each class, are studied in detail. It is proved that the range projector of the Lüders state can be evaluated as , where denotes the greatest lower bound, and is the null projector of . State-dependent implication extends absolute implication (which, in turn, determines the entire structure of quantum logic). and are investigated in a closely related way to mutual benefit. Inherent in the preorder is the state-dependent equivalence , defining equivalence classes in a given Boolean subalgebra. The quotient set, in which the classes are the elements, has itself a partially ordered structure, and so has each class. In a complete Boolean subalgebra, both structures are complete lattices. Physical meanings are discussed.
Abstract Logic is an algebraic structure that defines a set of abstract rules which govern an area of interest. The abstraction property of the rules makes them reusable tools to model different problems and to reason with them. The proliferation of quantum theory brought attention to quantum logic which is a lattice of projectors and it is of importance to quantum computing. Unfortunately, basic tools like implication are not sufficiently studied in that logic, which prevents us from exploiting the power of quantum mechanics in reasoning. This note investigates the implication issue in quantum logic and defines a quantum implication operator for compatible events as well as for incompatible events. The suggested operator depends both on the angle between the vector sub-spaces of the involved events and the angles between the system state and the vector sub-spaces. It differentiates between three cases depending on the angle between the events’ sub-spaces. The article further shows through an example that some classical reasoning rules such as Modus Ponens and Modus Tollens hold given the suggested implication.
This paper begins with an overview of quantum mechanics, and then recounts a relatively recent algebraic extension of the Boolean algebra of probabilistic events to “conditional events” (order pairs of events). The main point is to show that a so-called “superposition” of two (or more) quantum events (usually with mutually inconsistent initial conditions) can be represented in this algebra of conditional events and assigned a consistent conditional probability. There is no need to imagine that a quantum particle can simultaneously straddle two inconsistent possibilities.
This paper shows that the non-Boolean logic of quantum measurements is more naturally represented by a relatively new 4-operation system of Boolean fractions—conditional events—than by the standard representation using Hilbert Space. After the requirements of quantum mechanics and the properties of conditional event algebra are introduced, the quantum concepts of orthogonality, completeness, simultaneous verifiability, logical operations, and deductions are expressed in terms of conditional events thereby demonstrating the adequacy and efficacy of this formulation. Since conditional event algebra is nearly Boolean and consists merely of ordered pairs of standard events or propositions, quantum events and the so-called “superpositions” of states need not be mysterious, and are here fully explicated. Conditional event algebra nicely explains these non-standard “superpositions” of quantum states as conjunctions or disjunctions of conditional events, Boolean fractions, but does not address the so-called “entanglement phenomena” of quantum mechanics, which remain physically mysterious. Nevertheless, separating the latter phenomena from superposition issues adds clarity to the interpretation of quantum entanglement, the phenomenon of influence propagated at faster than light speeds. With such treacherous possibilities present in all quantum situations, an observer has every reason to be completely explicit about the environmental–instrumental configuration, the conditions present when attempting quantum measurements. Conditional event algebra allows such explication without the physical and algebraic remoteness of Hilbert space.
Tlic classical Kohnogorov's model of probability assumes that every pair of ev(Tits is siinult.aneouslyobservable.This principle is violated in several applications including quantum mechanics, artificial intelligence, psychology, sociology etc.In these areas iioneoinpatible events are encountered.These are events which can be observed separately, but not simultaneously, so they are not contained in a Boolean subalgebra i = classical subsystem) of the event structure describing the system in question.Various attempts have been made to generalize the probability theory to a more general structure admitting noneonipatibility.Among them, classes of subsets (more generally, concrete logics) wore studied for many years (see e.g.[15, 18]).Although some results were successfully generalized (see e.g.[4. 10, 19]).the theory proceeded slowly and with serious difficulties.Hero we introduce a more special-but still reasonably general-structure, a kernel logic.As it is described in terms of Boolean algebras using measure-theoretic notions, we believe that there is a greater chance to generalize classical results for Boolean algebras to kernel logics.Kernel logics seem to be interesting also from the algebraic point of view as a new const ruction technique for concrete logics.Its usefulness was proved by solutions of several quite nontrivial problems.Besides this, it scorns desirable to describe kernels of measures on Boolean algebras.