We begin by discussing the history of quantum logic, dividing it into three eras or lives. The first life has to do with Birkhoff and von Neumann's algebraic approach in the 1930's. The second life has to do with attempt to understand quantum logic as logic that began in the late 1950's and blossomed in the 1970's. And the third life has to do with recent developments in quantum logic coming from its connections to quantum computation. We discuss our own work connecting quantum logic to quantum computation (viewing quantum logic as the logic of quantum registers storing qubits), and make some speculations about mathematics based on quantum principles.
We begin by discussing the history of quantum logic, dividing it into three eras or lives. The first life has to do with Birkhoff and von Neumann's algebraic approach in the 1930's. The second life has to do with attempt to understand quantum logic as logic that began in the late 1950's and blossomed in the 1970's. And the third life has to do with recent developments in quantum logic coming from its connections to quantum computation. We discuss our own work connecting quantum logic to quantum computation (viewing quantum logic as the logic of quantum registers storing qubits), and make some speculations about mathematics based on quantum principles.
§1. Introduction. Our understanding of Nature comes in layers, so should the development of logic. Classic logic is an indispensable part of our knowledge, and its interactions with computer science have recently dramatically changed our life. A new layer of logic has been developing ever since the discovery of quantum mechanics. G. D. Birkhoff and von Neumann introduced quantum logic in a seminal paper in 1936 [1]. But the definition of quantum logic varies among authors (see [2]). How to capture the logic structure inherent in quantum mechanics is very interesting and challenging. Given the close connection between classical logic and theoretical computer science as exemplified by the coincidence of computable functions through Turing machines, recursive function theory, and λ-calculus, we are interested in how to gain some insights about quantum logic from quantum computing. In this note we make some observations about quantum logic as motivated by quantum computing (see [5]) and hope more people will explore this connection. The quantum logic as envisioned by Birkhoff and von Neumann is based on the lattice of closed subspaces of a Hilbert space, usually an infinite dimensional one. The quantum logic of a fixed Hilbert space ℍ in this note is the variety of all the true equations with finitely many variables using the connectives meet, join and negation. Quantum computing is theoretically based on quantum systems with finite dimensional Hilbert spaces, especially the states space of a qubit ℂ2. (Actually the qubit is merely a convenience.
. We put forward a new take on the logic of quantum mechanics, following Schrödinger's point of view that it is composition which makes quantum theory what it is, rather than its particular propositional structure due to the existence of superpositions, as proposed by Birkhoff and von Neumann. This gives rise to an intrinsically quantitative kind of logic, which truly deserves the name ‘logic’ in that it also models meaning in natural language, the latter being the origin of logic, that it supports automation, the most prominent practical use of logic, and that it supports probabilistic inference. The physics and the logic of quantum-ish logic. In 1932 John von Neumann formalized Quantum Mechanics in his book “Mathematische Grundlagen der Quantenmechanik”. This was effectively the official birth of the quantum mechanical formalism which until now, some 75 years later, has remained the same. Quantum theory underpins so many things in our daily lives including chemical industry, energy production and information technology, which arguably makes it the most technologically successful theory of physics ever. However, in 1935, merely three years after the birth of his brainchild, von Neumann wrote in a letter to American mathematician Garrett Birkhoff: “I would like to make a confession which may seem immoral: I do not believe absolutely in Hilbert space no more.” (sic)—for more details see [73]. Soon thereafter they published a paper entitled “The Logic of Quantum Mechanics” [13]. Their ‘quantum logic’ was cast in order-theoretic terms, very much in the spirit of the then reigning algebraic view of logic, with the distributive law being replaced with a weaker (ortho)modular law. This resulted in a research community of quantum logicians [68, 71, 47, 30]. However, despite von Neumann's reputation, and the large body of research that has been produced in the area, one does not find a trace of this activity neither in the mainstream physics, mathematics, nor logic literature. Hence, 75 years later one may want to conclude that this activity was a failure. What went wrong? The mathematics of it. Let us consider the raison d'être for the Hilbert space formalism.
By quoting extensively from unpublished letters written by John von Neumann to Garret Birkhoff during the preparatory phase (in 1935) of their ground-breaking 1936 paper that established quantum logic, the main steps in the thought process leading to the 1936 Birkhoff–von Neumann paper are reconstructed. The reconstruction makes it clear why Birkhoff and von Neumann rejected the notion of quantum logic as the projection lattice of an infinite dimensional complex Hilbert space and why they postulated in their 1936 paper that the quantum propositional system should be isomorphic to an abstract projective geometry. Looking at the paper now I see, that I forgot to say this, which should be said somewhere in the first §: That while common logics did apply to quantum mechanics, if the notion of simultaneous measurability is introduced as an auxiliary notion, we wished to construct a logical system, which applies directly to quantum mechanics – without any extraneous secondary notions like simultaneous measurability. And in order to have such a consequent, one-piece system of logics, we must change the classical class calculus of logics. (J. von Neumann to G. Birkhoff, November 21, 1935)
Quantum computer programming is emerging as a new subject domain from multidisciplinary research in quantum computing, computer science, mathematics (especially quantum logic, lambda calculi, and linear logic), and engineering attempts to build the first non-trivial quantum computer. This paper briefly surveys the history, methods, and proposed tools for programming quantum computers circa late 2007. It is intended to provide an extensive but non-exhaustive look at work leading up to the current state-of-the-art in quantum computer programming. Further, it is an attempt to analyze the needed programming tools for quantum programmers, to use this analysis to predict the direction in which the field is moving, and to make recommendations for further development of quantum programming language tools.
Quantum logic has been introduced by Birkhoff and von Neumann as an attempt to base the logical primitives, the propositions and the relations and operations among them, on quantum theoretical entities, and thus on the related empirical evidence of the quantum world. We give a brief outline of quantum logic, and some of its algebraic properties, such as nondistributivity, whereby emphasis is given to concrete experimental setups related to quantum logical entities. A probability theory based on quantum logic is fundamentally and sometimes even spectacularly different from probabilities based on classical Boolean logic. We give a brief outline of its nonclassical aspects; in particular violations of Boole-Bell type consistency constraints on joint probabilities, as well as the Kochen-Specker theorem, demonstrating in a constructive, finite way the scarcity and even nonexistence of two-valued states interpretable as classical truth assignments. A more complete introduction of the author can be found in the book "Quantum Logic" (Springer, 1998)