- Mathematisch-Geographische Fakultät, Katholische Universität Eichstätt-Ingolstadt, Eichstätt, Germany
Reasonable quantification of uncertainty is a major issue of cognitive infocommunications, and logic is a backbone for successful communication. Here, an axiomatic approach to quantum logic, which highlights similarity to and differences to classical logic, is presented. The axiomatic method ensures that applications are not restricted to quantum physics. Based on this, algorithms are developed that assign to an incoming signal a similarity measure to a pattern generated by a set of training signals.
1 Introduction
Reasonable quantification of uncertainty is a major issue of cognitive infocommunications, and logic is a backbone for successful communication. The motivation for writing this article came from some experience with fuzzy logic in industrial context, and from the observation that quantum logic appears to be superior to other “fuzzy” approaches; see, for instance, Schmitt and Nurnberger (2007).
This paper describes a mathematically rigorous pathway from classical logic to mathematical models that enable a quantification of uncertainty. Such models had been developed and studied especially in the context of quantum physics, where a link to probability is given by Born’s postulate. Hence, both classical logic with and, or, and negation, and the mathematics of quantum mechanics are in the focus of this article.
An axiomatic approach to quantum logic has two advantages: It highlights the relation between classical logic and quantum logic, and it shows that application of quantum logic is not restricted to quantum physics. The result is that a classical system of propositions can be represented as Boolean lattice, and a quantum system of propositions is represented by an atomistic orthomodular lattice. Quantum logic contains several variants of classical logic, as an atomistic orthomodular lattice has several Boolean sublattices.
In a human–machine communication process, the communicative acts often can be described by real functions defined on an interval [a, b]. Morover, a proposition often can be represented by a finitely generated subspace of L2 ([a, b]). The final section of this article describes how to use Gram-Schmidt processes for representing logical operations by dealing with generating families and constructing orthonormal families which span linear subspaces corresponding to logical expressions.
2 Quantum Logic: An Axiomatic Approach
In a historic perspective, quantum logic originated in an article by Birkhoff and von Neumann (1936) entitled The Logic of Quantum Mechanics, which appeared in the Annals of Mathematics. The authors discover logical structures in quantum mechanics and come to the conclusion that
“one can reasonably expect to find a calculus of propositions which is formally indistinguishable from the calculus of linear subspaces with respect to set products, linear sums, and orthogonal complements—and resembles the usual calculus of propositions with respect to and, or, and not.”
It was known at that time that the “usual calculus of propositions” means that the set of propositions carries the algebraic structure Boolean lattice with two binary operations called conjunction (logical and, ∧) and adjunction (logical or, ∨), and a unary operation called negation (logical not, ¬). Let us now go to the next major step towards the mathematics of quantum logic.
2.1 Piron’s Axiomatique Quantique
According to Piron (1964), the system
Axiom O: Implication is a partial order on
A requirement of this axiom is that implication is transitive, which reflects the classical Barbara syllogism.
The next axiom uses the notion indexed family of propositions, which is a map
Axiom T: For any family of propositions {aj: j ∈ J}, there is an infimum w.r.t. the partial order ⩽, i.e., a proposition
This proposition is denoted by u = inf{aj: j ∈ J} (Piron uses the notation ⋂Jaj).
This axiom requires that the logical conjunction of an arbitrary set of propositions is again a proposition.
The following lattice theoretic notation is used here:
Axiom C: There is on
Axiom C means that orthocomplementation is a mathematical model for logical negation.
Based on orthocomplement, the join of two propositions
The three axioms O, T and C ensure that the system of propositions
Axiom P: If a⩽b, then the sublattice generated by a and b is Boolean.
This axiom marks the difference between classical logic and quantum logic. It formulates that physical measurements that correspond to propositions satisfying the relation a⩽b are compatible.
In lattice theory, an element
Axiom A: 1) For any
2) If Q is point and a⩽x⩽a ⊔ Q, then x ∈ {a, a ⊔ Q}.
Axiom A requires that the set
Calling the join P ⊔ Q of two points P and Q a straight line, the points and straight lines make up an incidence geometry; see, for instance, Beutelspacher and Rosenbaum (2004). In addition, Piron (1964) proves that, in this structure, the following Veblen-Young-Axiom holds:
If A, B, C are points not all on the same line, and D and E (D ≠ E) are points such that B, C, D are on a line and C, A, E are on a line, there is a point F such that A, B, F are on a line and also D, E, F are on a line. (Veblen and Young, 1918, p. 2).
This connects quantum logic to projective geometry—a fact that will turn out to be crucial for our application.
2.2 Piron’s Theorem
The main result in Piron (1964) is that, provided the system
More precisely, he proves that there is a division ring
In this setting, if a proposition a corresponds to closed linear subspace
2.3 Solèr’s Condition and Hilbert Space
Given a division ring
Solèr’s theorem is based on a classical theorem of Frobenius, which states that there are exactly three different real division algebras: the real numbers
It is known from Piron’s theorem that any proposition corresponds to a closed subspace of
![www.frontiersin.org](https://www.frontiersin.org/files/Articles/662632/fcomp-03-662632-HTML/image_m/fcomp-03-662632-g001.jpg)
FIGURE 1. The relations between logic, projective geometry, linear algebra, orthogonal projections, and the computation of probabilities.
3 Quantification of Signal Similarity
3.1 Modeling With Hilbert Space
For modeling human–machine communicative situations, we restrict attention to real Hilbert spaces. In this case, the non-degenerate orthomodular Hermitian form reduces to a real scalar product. Specifically, we consider a fixed real interval [a, b] and the Hilbert space
Here, we use the intuitive ket-bra-notation introduced by Dirac (1939): An element of the Hilbert space is a ket vector
Note that in Dirac’s ket and bra notation, it appears more natural to consider the Hilbert space as a right vector space, i.e., the scalars come from the right-hand side. For calculating the projection probability pa(x) of a ket vector
In this calculation, it is used that an orthogonal projector is self-adjoint and idempotent.
Note that in Eq. 1, a certain “modeling freedom” is used. Here, it would be possible to introduce an appropriate weighting function, but this would only be reasonable if a good substantiation were provided.
3.2 Algorithms Used for Calculation
First assume for simplicity that all signals are sampled on an interval [a, b] with the same sampling rate. This means that there are time points t0 = a < t1 < … < tn = b such that, given a signal
Next, assume that training signals x1, … , xk are given, and that the question is whether or not an incoming signal x belongs to the pattern defined by the training signals. Or, more precisely, what is the probability p(x) that x belongs to the trained pattern? For answering this question, I propose the following steps:
1) Employ a Gram-Schmidt process for constructing an orthonormal base |v1⟩, …, |vℓ⟩ for the subspace generated by |x1⟩, …, |xk⟩—note that, as the generating set is finite, this subspace is closed, and that its dimension is ℓ⩽k.
2) Normalize the incoming signal by setting
3) Evaluate the formula
A noteworthy special case is when just one training signal x1 is given. Then,
3.3 Application to Logic
By Piron’s theorem, a logical proposition does not correspond to training signals, but rather to the subspace generated by them. The atoms of the lattice of propositions are the one-dimensional subspaces given
What about representing logical operations and, or, and negation? For a concise notation, assume that two propositions a and b correspond to two sets of training signals spanning closed subspaces
1) Logical “or”: Running a Gram-Schmidt process on the sequence
leads to an orthonormal family generating the closed linear subspace
2) Relative negation: Here, the aim is to construct an orthonormal family generating the subspace
which corresponds to the proposition (a ⊔ b) ⊓ a′. To this end, run Gram-Schmidt on the sequence 3, and remove the first part that belongs to
3) Logical “and”: Apply de Morgan’s law to relative orthocomplements in
This reduces the logical “and” to logical “or” and relative negation. Along these lines, algorithms described above can be combined to construct an orthonormal base for the intersection of finitely generated subspaces.
4 Application to Signal Comparison
Quantum-inspired uncertainty quantification can be applied to signal comparison tasks. The procedure is as follows.
1) Ensure that all training signals have equal length. This can be done by cutting appropriately and/or by using linear transformations. The result should be a set of training signals x1, … , xk, which are defined on the same interval.
2) For handling the problem of possible different sampling rates, collect all sampling points and fill the missing data by linear interpolation. This is reasonable as we are only interested in the calculation of scalar products.
3) For an incoming signal, use cutting and/or rescaling to ensure that it is defined on the same interval as the training data.
4) Compute the projection probability p(x) of the incoming signal x to these training patterns using the procedure described in section 3.2.
Then, p(x) provides a measure of similarity between an incoming signal and the set of training signals.
Data Availability Statement
The original contributions presented in the study are included in the article/Supplementary Material. Further inquiries can be directed to the corresponding author.
Author Contributions
The author confirms being the sole contributor of this work and has approved it for publication.
Conflict of Interest
The author declares that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
Publisher’s Note
All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.
Acknowledgments
The idea to this logically founded quantification of uncertainty arose slowly during a long-term collaboration with Matthias Wolff and Ingo Schmitt, BTU Cottbus-Senftenberg. I am grateful to the Katholische Universität Eichstätt-Ingolstadt, where I hold a part-time teaching position, for supporting this work in many respects, including financially.
References
Beutelspacher, A., and Rosenbaum, U. (2004). Projektive Geometrie : von den Grundlagen bis zu den Anwendungen (Wiesbaden: Vieweg). edn, 2.
Birkhoff, G., and Neumann, J. V. (1936). The Logic of Quantum Mechanics. Ann. Mathematics 37, 823–843. doi:10.2307/1968621
Boole, G. (1847). The Mathematical Analysis of Logic : Being an Essay towards a Calculus of Deductive Reasoning. Cambridge: Macmillan.
Dirac, P. A. M. (1939). A New Notation for Quantum Mechanics. Math. Proc. Camb. Phil. Soc. 35, 416–418. doi:10.1017/s0305004100021162
Schmitt, I., and Nurnberger, A. (2007). “Image Database Search Using Fuzzy and Quantum Logic,” in 2007 IEEE International Fuzzy Systems Conference, London, UK, 23-26 July 2007, 1–6. doi:10.1109/FUZZY.2007.4295682
Solèr, M. P. (1994). Charakterisierung von Hilberträumen als spezielle orthomodulare Räume. Dissertation Universität Zürich, 22 Seiten.
Keywords: axiomatic quantum logic, atomistic orthomodular lattice, Hilbert space, orthogonal projection, Born’s postulate, uncertainty quantification, trapezoidal rule, Gram-Schmidt process
Citation: Wirsching G (2022) Quantum-Inspired Uncertainty Quantification. Front. Comput. Sci. 3:662632. doi: 10.3389/fcomp.2021.662632
Received: 01 February 2021; Accepted: 08 November 2021;
Published: 11 January 2022.
Edited by:
Gennaro Cordasco, University of Campania Luigi Vanvitelli, ItalyReviewed by:
Adam Csapo, Széchenyi István University, HungaryAnna Esposito, University of Campania Luigi Vanvitelli, Italy
Copyright © 2022 Wirsching. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
*Correspondence: Günther Wirsching, guenther.wirsching@ku.de