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Communications in computer and information science

Formal Concept Analysis and Structures Underlying Quantum Logics

Ondrej Krídlo, Manuel Ojeda‐Aciego

Published Jan 1, 2018• 7 citations
Publisher
DOI
10.1007/978-3-319-91473-2_49

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In a earlier work [1, 2, 3, 4] we have put forward the hypothesis that two layers of thought can be identified in the human thought process, which we have called the ‘classical logical ’ layer and the ‘quantum conceptual ’ layer. In the classical logical layer thought processes are given form by an underlying Boolean logical conceptual process, its non deterministic ‘lack of knowledge ’ version modeled by Kolmogorovian probability and in an idealized form by a stochastic process. In the quantum conceptual layer, the thought process gets form under the influence of the totality of the surrounding conceptual landscape. The key event within this layer is that different concepts figure as individual entities, even when they are combinations of other concepts. A phenomenon of conceptual emergence is taking place in the quantum conceptual layer mathematically described by superposition within the quantum formalism used to model this layer. The quantum conceptual thought process is indeterministic in essence; i.e. there is not necessarily an underlying deterministic process. It is however an equally well structured layer than the classical logical one, and we belief that it is mainly the lack of insight about the structure of this quantum conceptual layer that often leads to the idea that the thought processes described in it are mainly unstructured and correspond to intuitive or irrational thought. In the present paper we analyze more in detail the cognitive and mathematical structures of these two layers of thought. More specifically we put forward a concrete
Why did it take us 50 years since the birth of the quantum mechanical formalism to discover that unknown quantum states cannot be cloned? Yet, the proof of the ‘no-cloning theorem’ is easy, and its consequences and potential for applications are immense. Similarly, why did it take us 60 years to discover the conceptually intriguing and easily derivable physical phenomenon of ‘quantum teleportation’? We claim that the quantum mechanical formalism doesn't support our intuition, nor does it elucidate the key concepts that govern the behaviour of the entities that are subject to the laws of quantum physics. The arrays of complex numbers are kin to the arrays of 0s and 1s of the early days of computer programming practice. Using a technical term from computer science, the quantum mechanical formalism is ‘low-level’. In this review we present steps towards a diagrammatic ‘high-level’ alternative for the Hilbert space formalism, one which appeals to our intuition. The diagrammatic language as it currently stands allows for intuitive reasoning about interacting quantum systems, and trivialises many otherwise involved and tedious computations. It clearly exposes limitations such as the no-cloning theorem, and phenomena such as quantum teleportation. As a logic, it supports ‘automation’: it enables a (classical) computer to reason about interacting quantum systems, prove theorems, and design protocols. It allows for a wider variety of underlying theories, and can be easily modified, having the potential to provide the required step-stone towards a deeper conceptual understanding of quantum theory, as well as its unification with other physical theories. Specific applications discussed here are purely diagrammatic proofs of several quantum computational schemes, as well as an analysis of the structural origin of quantum non-locality. The underlying mathematical foundation of this high-level diagrammatic formalism relies on so-called monoidal categories, a product of a fairly recent development in mathematics. Its logical underpinning is linear logic, an even more recent product of research in logic and computer science. These monoidal categories do not only provide a natural foundation for physical theories, but also for proof theory, logic, programming languages, biology, cooking, … So the challenge is to discover the necessary additional pieces of structure that allow us to predict genuine quantum phenomena. These additional pieces of structure represent the capabilities nature has provided us with to manipulate entities subject to the laws of quantum theory.
Abstract Traditionally by ‘quantum logic’, one understands the study of the lattice of the subspace of Hilbert space. Here, ‘quantum logic’ is taken to be the underlying logic of quantum physics. Following Schrödinger’s idea that the standard concept of identity cannot be applied to quantum objects, a kind of ‘non-reflexive’ logic called Schrödinger logic is presented, in which the reflexive law of identity is taken not to apply. First order Schrödinger logics are presented, and extended to higher order logics and also to an intensional logic. Classical semantics is presented, and a weak completeness theorem is sketched. The last part of the chapter covers a logic of sortal predicates for which a quasi-set semantics is delineated. This completes the formal framework capable of accommodating quantum non-individuality and thus the initial understanding of Born, Heisenberg, and Schrödinger, as well as the second horn of the above metaphysical underdetermination, can be formally represented.