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Journal of Physics A Mathematical and General

More on the state-dependent implication in quantum mechanics

Fedor Herbut

Published Jan 21, 1996• 5 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/29/2/023

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•Jul 30, 2020•0 citations
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.
Let be an arbitrary given composite-system state (statistical operator). It is shown that same-subsystem events imply each other state-dependently according to if and only if they act equally in the range of the corresponding subsystem state (reduced statistical operator). Opposite-subsystem events imply each other in the same way if and only if they are twin events, i.e. . If is a pure state, it is shown that the anti-unitary correlation operator also plays a decisive role in the latter implication.
Friedman and Putnam have argued (Friedman and Putnam 1978) that the quantum logical interpretation of quantum mechanics gives us an explanation of interference that the Copenhagen interpretation cannot supply without invoking an additional ad hoc principle, the projection postulate. I show that it is possible to define a notion of equivalence of experimental arrangements relative to a pure state φ, or (correspondingly) equivalence of Boolean subalgebras in the partial Boolean algebra of projection operators of a system, which plays a role in the Copenhagen explanation of interference analogous to the role played by the material equivalence, given φ, of certain propositions in the Friedman-Putnam quantum logical analysis. I also show that the quantum logical interpretation and the Copenhagen interpretation are equally capable of avoiding the paradoxical conclusion of the Einstein-Podolsky-Rosen argument (Einstein, Podolsky, and Rosen 1935). Thus, neither interference phenomena nor the correlations between separated systems provide a test case for distinguishing between the relative acceptability of the Copenhagen interpretation and the quantum logical interpretation as explanations of quantum effects.
This chapter contains a brief introduction to mathematical formalism and QM axiomatics. It is oriented to non-physicists. Since QM is a statistical theory it is natural to start with the classical probability model (Kolmogorov [ 191 ], 1933). Then we present the basics of the theory of Hilbert spaces and Hermitian operators and the representation of pure and mixed states by normalized vectors and density operators. This introduction is sufficient to formulate the axiomatics of QM in the form of five postulates. The projection postulate (the most questionable postulate of QM) is presented in a separate section. We sharply distinguish between the cases of quantum observables represented by Hermitian operators with nondegenerate and degenerate spectra, von Neumann’s and Ltiders’s forms of the projection postulate. The axiomatics is completed by a short section on the main interpretations of QM. The projection postulate (LUders’s form) plays a crucial role in the definition of quantum conditional (transition) probability. By operating with the latter we consider interference of probabilities for two incompatible observables as a modification of the formula of total probability by adding the interference term. This viewpoint to interference of probabilities was elaborated in a series of works by Khrennikov. Since classical probability theory is based on the 142 Boolean algebra of events, a violation of the law of total probability can be treated as the probabilistic sign of a violation of the laws of the Boolean logics. From this viewpoint, quantum theory can be considered as representing a new kind of logic, so-called quantum logic. The latter is also briefly presented in a separate section. We continue this review with a new portion of quantum mathematics, namely the notion of the tensor product of Hilbert spaces and the tensor product of operators. After the section on Dirac’s notation with ket and bra vector, we discuss briefly the notion of qubit and entanglement of a few qubits. This chapter concludes with a presentation of the detailed analysis of the probabilistic structure of the two-slit experiment in a bunch of different experimental contexts. This contextual structure leads to a violation of the law of total probability and non-Kolmogorovean probabilistic structure of this experiment.