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Hilbert Lattice Equations

Norman D. Megill

Published Aug 14, 2016• 4 citations
OpenAlex

Abstract

There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet's E_A, Godowski, Mayet-Godowski, and Mayet's E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet-Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations.

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There are five known classes of lattice equations that hold in every infinite dimensional Hilbert space underlying quantum systems: generalised orthoarguesian, Mayet's E_A, Godowski, Mayet-Godowski, and Mayet's E equations. We obtain a result which opens a possibility that the first two classes coincide. We devise new algorithms to generate Mayet-Godowski equations that allow us to prove that the fourth class properly includes the third. An open problem related to the last class is answered. Finally, we show some new results on the Godowski lattices characterising the third class of equations.
We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an n-variable generalized orthoarguesian equation which holds in any infinite dimensional Hilbert space. Then we strengthen Godowski's result by showing that in an ortholattice on which strong states are defined Godowski's equations as well as the orthomodularity hold. We also prove that all 6- and 4-variable orthoarguesian equations presented in the literature can be reduced to new 4- and 3-variable ones, respectively and that Mayet's examples follow from Godowski's equations. To make a breakthrough in testing these massive equations we designed several novel algorithms for generating Greechie diagrams with an arbitrary number of blocks and atoms (currently testing with up to 50) and for automated checking of equations on them. A way of obtaining complex infinite dimensional Hilbert space from the Hilbert lattice equipped with several additional conditions and without invoking the notion of state is presented. Possible repercussions of the results to quantum computing problems are discussed.
The Hilbert space of quantum mechanics has a dual representation in lattice theory, called the Hilbert lattice. In addition to offering the potential for new insights, the lattice-theoretical approach may be computationally efficient for certain kinds of quantum mechanics problems, particularly if, in the future, we are able to exploit what may be a “natural” fit with quantum computation. The equations that hold in the Hilbert space lattice representation are not completely known and are poorly understood, although much progress has been made in the last several years. This work contributes to the development of these equations, with special attention to the so-called generalized orthoarguesian equations. Many new results that do not appear in the literature are given, along with their detailed proofs. In addition, possible approaches for work towards answering some remaining open questions are discussed.
Abstract. We provide several new results on quantum state space, on lattice of subspaces of an infinite dimensional Hilbert space, and on infinite dimensional Hilbert space equations as well as on connections between them. In particular we obtain an n-variable generalized orthoarguesian equation which holds in any infinite dimensional Hilbert space. Then we strengthen Godowski’s result by showing that in an ortholattice on which strong states are defined Godowski’s equations as well as the orthomodularity hold. We also prove that all 6- and 4-variable orthoarguesian equations presented in the literature can be reduced to new 4- and 3-variable ones, respectively and that Mayet’s examples follow from Godowski’s equations. To make a breakthrough in testing these massive equations we designed several novel algorithms for generating Greechie diagrams with an arbitrary number of blocks and atoms (currently testing with up to 50) and for automated checking of equations on them. A way of obtaining complex infinite dimensional Hilbert space from the Hilbert lattice equipped with several additional conditions and without invoking the notion of state is presented. Possible repercussions of the results to quantum computing problems are discussed.
We use classes of Hilbert lattice equations for an alternative representation of Hilbert lattices and Hilbert spaces of arbitrary quantum systems that might enable a direct introduction of the states of the systems into quantum computers. More specifically, we look for a way to feed a quantum computer with algebraic equations of n-th order underlying an infinite dimensional Hilbert space description of quantum systems. A number of new results on states defined on Hilbert lattices are presented and discussed and a number of recently obtained results in the field of Hilbert space equations are reviewed.
We present a generalization of quantum error correction to infinite-dimensional Hilbert spaces. We find that, under relatively mild conditions, much of the structure known from systems in finite-dimensional Hilbert spaces carries straightforwardly over to infinite dimensions. We also find that, at least in principle, there exist qualitatively new classes of quantum error correcting codes that have no finite-dimensional counterparts. We begin with a shift of focus from states to algebras of observables. Standard subspace codes and subsystem codes are seen as the special case of algebras of observables given by finite-dimensional von Neumann factors of type I. The new classes of codes that arise in infinite dimensions are shown to be characterized by von Neumann algebras of types II and III, for which we give in-principle physical examples.