Reasoning About Quantum Systems (original) (raw)
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Reasoning about Quantum Information: An Overview of Quantum Dynamic Logic
Applied Sciences
This paper provides an overview of quantum dynamic logics, showing how they have been designed and illustrating how these logics can be applied to verify the correctness of quantum protocols. Similar to the advantages of using dynamic logics to reason about the flow of classical information, the quantum analogues of these logics are tailored to the task of reasoning about the flow of quantum information. We present our logical systems in a modular way, starting with the qualitative logic of quantum measurements and unitary evolutions in single quantum systems, which can already express non-classical effects, e.g., the state-changing interference induced by quantum tests, their non-commutativity, etc. We then move on to logics for compound quantum systems that can capture the non-local features of quantum information: separability, entanglement, correlated measurements, Bell states, etc. We then briefly summarize the logic of quantum probabilities and sketch some applications to quan...
Extending Classical Logic for Reasoning About Quantum Systems
Handbook of Quantum Logic and Quantum Structures, 2009
A logic extending classical reasoning and supporting quantum reasoning is presented. The quantum logic is obtained by applying the exogenous semantics approach to propositional logic. The design is guided by the postulates of quantum mechanics and inspired by applications in quantum computation and information. The models of the quantum logic are superpositions of classical valuations. In order to achieve a recursive axiomatization, the superpositions are taken in inner product spaces over algebraic closures of arbitrary real closed fields.
2010
Current research in Logic is no longer confined to the traditional study of logical consequence or valid inference. As can be witnessed by the range of topics covered in this special issue, the subject matter of logic encompasses several kinds of informational processes ranging from proofs and inferences to dialogues, observations, measurements, communication and computation. What interests us here is its application to quantum physics: how does logic handle informational processes such as observations and measurements of quantum systems? What are the basic logical principles fit to handle and reason about quantum physical processes? These are the central questions in this paper. It is my aim to provide the reader with some food for thought and to give some pointers to the literature that provide an easy access to this field of research. In the next section I give a brief historical sketch of the origin of the quantum logic project. Next I will explain the theory of orthomodular lattices in section 2. Section 3 covers the syntax and semantics of traditional quantum logic. In section 4, I focus on the limits of quantum logic, dealing in particular with the implication problem. This paves the way to section 5 on modal quantum logic. I end with section 6 on dynamic quantum logic, giving the reader a taste of one of the latest new developments in the field.
Reasoning about Quantum Actions: A Logician's Perspective
2013
In this paper I give an overview of how the work on quantum dynamic logic for single systems (as developed in [2]) builds on the concepts of (dynamic) modal logic and incorporates the methodology of logical dynamics and action based reasoning into its setting. I show in particular how one can start by modeling quantum actions (i.e. measurements and unitary evolutions) in a dynamic logic framework and obtain a setting that improves on the known theorems in traditional quantum logic (stated in the context of orthomodular lattices).
A proposal for a new approach to Quantum Logic
Article CITATIONS 0 READS 17 2 authors, including: Some of the authors of this publication are also working on these related projects: interpretation of analytical mechanics through the two dichotomies. Search of a new formualtion of quantum mechanics relying on the alternative choices of the Dirac-von Neumann's one A new View project Antonino Drago University of Naples Federico II 70 PUBLICATIONS 88 CITATIONS SEE PROFILE All content following this page was uploaded by Antonino Drago on 14 January 2015.
LQP: the dynamic logic of quantum information
Mathematical Structures in Computer Science, 2006
The main contribution of this paper is the introduction of a dynamic logic formalism for reasoning about information flow in composite quantum systems. This builds on our previous work on a complete quantum dynamic logic for single systems. We extend that work here to a sound (but not necessarily complete) logic for composite systems, which brings together ideas from the Quantum Logic tradition with concepts from (dynamic) Modal Logic and from Quantum Computation. This logic of Quantum Programs (LQP) is capable of expressing important features of quantum measurements and unitary evolutions of multi-partite states, as well as giving logical characterizations to various forms of entanglement (e.g. the Bell states, the GHZ states etc.). We present a finitary syntax, a relational semantics and a sound proof system for this logic. As applications, we use our system to give formal correctness proofs for the Teleportation protocol and for a standard Quantum Secret Sharing protocol; a while range of other quantum circuits and programs, including other known protocols (e.g. Superdense Coding, Entanglement Swapping, Logic-Gate Teleportation etc.), can be similarly verified using our logic.
Quantum Logic as a Dynamic Logic
2011
We address the old question whether a logical understanding of Quantum Mechanics requires abandoning some of the principles of classical logic. Against Putnam and others 1 , our answer is a clear " no ". Philosophically, our argument is based on combining a formal semantic approach, in the spirit of E. W. Beth's proposal of applying Tarski's semantical methods to the analysis of physical theories, with an empirical-experimental approach to Logic, as advocated by both Beth and Put-nam, but understood by us in the view of the operational-realistic tradition of Jauch and Piron, i.e. as an investigation of " the logic of yes-no experiments " (or " questions "). Technically, we use the recently-developed setting of Quantum Dynamic Logic [4, 6] to make explicit the operational meaning of quantum-mechanical concepts in our formal semantics. Based on our recent results [4], we show that the correct interpretation of quantum-logical connectives is dynamical, rather than purely propositional. We conclude that there is no contradiction between classical logic and (our dynamic reinterpretation of) quantum logic. Moreover, we argue that the Dynamic-Logical perspective leads to a better and deeper understanding of the " non-classicality " of quantum behavior than any perspective based on static Propositional Logic.
Probabilistic logic of quantum observations
Logic Journal of the IGPL, 2018
A probabilistic propositional logic, endowed with an epistemic component for asserting (non-)compatibility of diagonizable and bounded observables, is presented and illustrated for reasoning about the random results of projective measurements made on a given quantum state. Simultaneous measurements are assumed to imply that the underlying observables are compatible. A sound and weakly complete axiomatization is provided relying on the decidable first-order theory of real closed ordered fields. The proposed logic is proved to be a conservative extension of classical propositional logic.
The logic of quantum mechanics-Take II
2012
Abstract: We put forward a new take on the logic of quantum mechanics, following Schroedinger'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.
Arxiv preprint quant-ph/0101028, 2001
We investigate some forms of quantum logic arising from the standard and the unsharp approach.