A substructural logic for quantum measurements (original) (raw)
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Connections among quantum logics. Part 1. Quantum propositional logics
International Journal of Theoretical Physics, 1985
In this paper, we propose a theory of quantum logics which is general enough to enable us to reexamine previous work on quantum logics in the context of this theory. It is then easy to assess the differences between the different systems studied. The quantum logical systems which we incorporate are divided into two groups which we call "quantum propositional logics" and "quantum event logics." We include the work of Kochen and Specker (partial Boolean algebras), Greechie and Gudder (orthomodular partially ordered sets), Domotar (quantum mechanical systems), and Foulis and Randall (operational logics) in quantum propositional logics; and Abbott (semi-Boolean algebras) and Foulis and Randall (manuals) in quantum event logics. In this part of the paper, we develop an axiom system for quantum propositional logics and examine the above structures in the context of this system.
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...
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.
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.
07 02 02 3 v 1 2 F eb 2 00 7 Contextual logic for quantum systems
2005
In this work we build a quantum logic that allows us to refer to physical magnitudes pertaining to different contexts from a fixed one without the contradictions with quantum mechanics expressed in no-go theorems. This logic arises from considering a sheaf over a topological space associated to the Boolean sublattices of the ortholattice of closed subspaces of the Hilbert space of the physical system. Differently to standard quantum logics, the contextual logic maintains a distributive lattice structure and a good definition of implication as a residue of the conjunction. PACS numbers: 03.65.Ta, 02.10.-v ∗Fellow of the Consejo Nacional de Investigaciones Cient́ıficas y Técnicas (CONICET)
Algebras of measurements: the logical structure of quantum mechanics
International Journal of …, 2006
In Quantum Physics, a measurement is represented by a projection on some closed subspace of a Hilbert space. We study algebras of operators that abstract from the algebra of projections on closed subspaces of a Hilbert space. The properties of such operators are justified on epistemological grounds. Commutation of measurements is a central topic of interest. Classical logical systems may be viewed as measurement algebras in which all measurements commute.
Contextual logic for quantum systems
Journal of Mathematical Physics, 2005
In this work we build a quantum logic that allows us to refer to physical magnitudes pertaining to different contexts from a fixed one without the contradictions with quantum mechanics expressed in no-go theorems. This logic arises from considering a sheaf over a topological space associated to the Boolean sublattices of the ortholattice of closed subspaces of the Hilbert space of the physical system. Differently to standard quantum logics, the contextual logic maintains a distributive lattice structure and a good definition of implication as a residue of the conjunction.
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.
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.
A theory of computation based on quantum logic (I)
2005
The (meta)logic underlying classical theory of computation is Boolean (twovalued) logic. Quantum logic was proposed by Birkhoff and von Neumann as a logic of quantum mechanics more than sixty years ago. It is currently understood as a logic whose truth values are taken from an orthomodular lattice. The major difference between Boolean logic and quantum logic is that the latter does not enjoy distributivity in general. The rapid development of quantum computation in recent years stimulates us to establish a theory of computation based on quantum logic. The present paper is the first step toward such a new theory and it focuses on the simplest models of computation, namely finite automata. We introduce the notion of orthomodular lattice-valued (quantum) automaton. Various properties of automata are carefully reexamined in the framework of quantum logic by employing an approach of semantic analysis. We define the class of regular languages accepted by orthomodular lattice-valued automata. The acceptance abilities of orthomodular lattice-valued nondeterministic automata and their various modifications (such as deterministic automata and automata with ε−moves) are compared. The closure properties of orthomodular lattice-valued regular languages are derived. The Kleene theorem about equivalence of regular expressions and finite automata is generalized into quantum logic. We also present a pumping lemma for orthomodular lattice-valued regular languages. It is found that the universal validity of many properties (for example, the Kleene theorem, the equivalence of deterministic and nondeterministic automata) of automata depend heavily upon the distributivity of the underlying logic. This indicates that these properties does not universally hold in the realm of quantum logic. On the other hand, we show that a local validity of them can be recovered by imposing a certain commutativity to the (atomic) statements about the automata under consideration. This reveals an essential difference between the classical theory of computation and the computation theory based on quantum logic.