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Trends in Logic 48
Maria Luisa Dalla ChiaraRoberto GiuntiniRoberto LeporiniGiuseppe Sergioli
Quantum Computation and LogicHow Quantum Computers Have Inspired Logical Investigations
Editorial AssistantAndrea Robitzsch, University of Osnabrück, Osnabrück, Germany
Editorial BoardArnon Avron, University of Tel Aviv, Tel Aviv, Israel
Katalin Bimbó, University of Alberta, Edmonton, CanadaGiovanna Corsi, University of Bologna, Bologna, ItalyJanusz Czelakowski, University of Opole, Opole, PolandRoberto Giuntini, University of Cagliari, Cagliari, Italy
Rajeev Goré, Australian National University, Canberra, AustraliaAndreas Herzig, University of Toulouse, Toulouse, France
Wesley Holliday, UC Berkeley, Berkeley, USAAndrzej Indrzejczak, University of Łodz, Łodz, PolandDaniele Mundici, University of Florence, Florence, Italy
Sergei Odintsov, Sobolev Institute of Mathematics, Novosibirsk, RussiaEwa Orłowska, Institute of Telecommunications, Warsaw, Poland
Peter Schroeder-Heister, University of Tübingen, Tübingen, GermanyYde Venema, University of Amsterdam, Amsterdam, The Netherlands
Andreas Weiermann, University of Ghent, Ghent, BelgiumFrank Wolter, University of Liverpool, Liverpool, UK
Ming Xu, Wuhan University, Wuhan, PR ChinaJacek Malinowski, Polish Academy of Sciences, Warszawa, Poland
Founding editorRyszard Wójcicki, Polish Academy of Sciences, Warsaw, Poland
SCOPE OF THE SERIES
TRENDS IN LOGICStudia Logica Library
Editor-in-ChiefHeinrich Wansing, Ruhr-University Bochum, Bochum, Germany
VOLUME 48
The book series Trends in Logic covers essentially the same areas as the journal StudiaLogica, that is, contemporary formal logic and its applications and relations to other dis-ciplines. The series aims at publishing monographs and thematically coherent volumesdealing with important developments in logic and presenting significant contributions tological research.
The series is open to contributions devoted to topics ranging from algebraic logic, modeltheory, proof theory, philosophical logic, non-classical logic, and logic in computer scienceto mathematical linguistics and formal epistemology. However, this list is not exhaustive,moreover, the range of applications, comparisons and sources of inspiration is open andevolves over time.
More information about this series at http://www.springer.com/series/6645
Maria Luisa Dalla ChiaraRoberto Giuntini • Roberto LeporiniGiuseppe Sergioli
Quantum Computationand LogicHow Quantum Computers Have InspiredLogical Investigations
123
Maria Luisa Dalla ChiaraDepartment of PhilosophyUniversity of FlorenceFlorence, Italy
Roberto GiuntiniDepartment of PhilosophyUniversity of CagliariCagliari, Italy
Roberto LeporiniDepartment of Management, Informationand Production EngineeringUniversity of BergamoDalmine, Bergamo, Italy
Giuseppe SergioliDepartment of PhilosophyUniversity of CagliariCagliari, Italy
ISSN 1572-6126 ISSN 2212-7313 (electronic)Trends in LogicISBN 978-3-030-04470-1 ISBN 978-3-030-04471-8 (eBook)https://doi.org/10.1007/978-3-030-04471-8
Library of Congress Control Number: 2018961691
© Springer Nature Switzerland AG 2018This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or partof the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations,recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmissionor information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilarmethodology now known or hereafter developed.The use of general descriptive names, registered names, trademarks, service marks, etc. in thispublication does not imply, even in the absence of a specific statement, that such names are exempt fromthe relevant protective laws and regulations and therefore free for general use.The publisher, the authors and the editors are safe to assume that the advice and information in thisbook are believed to be true and accurate at the date of publication. Neither the publisher nor theauthors or the editors give a warranty, express or implied, with respect to the material contained herein orfor any errors or omissions that may have been made. The publisher remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.
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Dedicated to the memory of David Foulis,Peter Mittelstaedt and Giuliano Toraldo diFrancia
Preface
After the publication of Richard Feynman’s pioneering contributions (in the 80sof the last century), investigations in the field of quantum computation theory havebecome more and more intense. In spite of some initial skepticism, importantachievements have recently been obtained in the technological realizations ofquantum computers, which nowadays cannot be any longer regarded as mere“thought experiments”. These researches have naturally inspired new theoreticalideas, stimulating also a new interest for foundational and philosophical debatesabout quantum theory.
As is well known, classical computers have a perfect abstract model representedby the concept of Turing machine. Due to the intuitive strength of this concept andthe high stability of the notion of Turing computability (which has turned out to beequivalent to many alternative definitions of computability) for a long time theChurch-Turing thesis (according to which a number-theoretic function f is com-putable from an intuitive point of view iff f is Turing-computable) has been regardedas a deeply reasonable conjecture. This hypothesis seems to be also confirmed by anumber of studies about alternative concepts of computing machine that at firstsight may appear “more liberal”. A significant example is represented by the notionof non-deterministic (or probabilistic) Turing machine. Interestingly enough, onehas proved that non-deterministic Turing-machines do not go beyond the “limitsand the power” of deterministic Turing machines; for, any probabilistic Turingmachine can be simulated by a deterministic one.
To what extent have quantum computers “perturbed” such clear and wellestablished approaches to computation-problems? After Feynman’s contributions,the abstract mathematical model for quantum computers has often been representedin terms of the notion of quantum Turing machine, the quantum counterpart of theclassical notion of Turing machine. But what exactly are quantum Turing machi-nes? So far, the literature has not provided a rigorous “institutional” concept ofquantum Turing machine. Some definitions seem to be based on a kind of “imi-tation” of the classical definition of Turing machine, by referring to a tape (wherethe symbols are written) and to a moving head (which changes its position on thetape). These concepts, however, seem to be hardly applicable to physical quantum
vii
computers. We need only think of the intriguing situations determined by quantumuncertainties that, in principle, should also concern the behavior of moving heads.In this book, we will consider a more general concept, represented by the notion ofabstract quantum computing machine, which neglects both tapes and movingheads. Do abstract quantum computing machines go beyond the computationallimits of classical Turing machines? In other words, does quantum computationtheory lead us to a refutation of the Church-Turing thesis? In spite of some inter-esting examples discussed in the literature, this hard problem seems to be stillundecided.
Quantum computation theories have naturally inspired new ideas in the field oflogic, bringing about some important conceptual changes in the quantum-logicalinvestigations. The interaction between quantum theory and logic has a long historythat started in 1936 with the publication of Birkhoff and von Neumann’s celebratedarticle “The logic of quantum mechanics”. At the very beginning, this article did notraise any great interest either in the physical or in the logical community. Strangelyenough, logicians did not immediately recognize the most “revolutionary logicalidea” of quantum logic: the possible divergence between the concepts of maximalinformation and logically complete information.
As is well known, the pure states of a classical physical system S (a gas-molecule, a table, a planet, etc.) represent pieces of information that are at the sametime maximal and logically complete. The information provided by a pure state ofS cannot be consistently extended to a richer knowledge; at the same time, suchinformation decides all possible events that may occur to S. For this reason, thenotion of pure state of a classical physical object seems to be very close to the ideaof complete concept, investigated by Leibniz: although many properties of anindividual object (say, the Moon) may be unknown to human minds, God knowsthe complete concept of any object (living either in the actual or in some possibleworld), and this concept represents a maximal and logically complete informationabout the object in question.
Due to the celebrated uncertainty-principles (discovered by Heisenberg), com-plete concepts (in Leibniz’ sense) cannot exist for quantum objects. Consider aquantum system S (say, an electron) in a pure state that assigns an exact value to itsvelocity in the x-direction. In such a case, the position of S (with respect to thex-direction) will be completely indeterminate: the object S turns out to benon-localized. Quantum objects, are in a sense, “poor”; and their “poverty” con-cerns the number of physical properties that can be satisfied at the same time.Furthermore, quantum properties seem to behave in a contextual way: propertiesthat are completely indeterminate in a given context may become actual anddeterminate in a different context (for instance, after the performance of anappropriate measurement). Hence, the system of properties that are determinate fora given quantum object turns out to be context-dependent. Quantum pure statesrepresent pieces of information that are at the same time maximal (since they cannotbe consistently extended to a richer knowledge) and logically incomplete (sincethey cannot decide all the relevant properties of the objects under investigation).This divergence between the concepts of maximal knowledge and logically
viii Preface
complete knowledge represents a characteristic logical aspect of the quantum worldthat may appear prima facie strange, since it is in contrast with a basic theorem ofclassical logic (and of many alternative logics): Lindenbaum’s theorem, accordingto which any non-contradictory set of sentences T can be extended to a set ofsentences T′ that is at the same time non-contradictory (no contradiction can bederived from T′), logically complete (for any sentence a of the language either a orits negation ¬a belongs to T′), maximal (all proper extensions of T′, formalized inthe same language of T′, are contradictory).
Apparently, quantum undecidabilities turn out to be much stronger than thesyntactical undecidabilities discovered by Gödel’s incompleteness theorems. In thequantum world, undecidability is not only due to the limited proof-theoreticcapacities of finite minds: against “Leibniz’ dream” even an infinite omniscientmind should be bound to quantum uncertainties.
Birkhoff and von Neumann’s quantum logic (as well as its further developments)represent, in a sense, static logics. The basic aim of these logics is the descriptionof the abstract structure of all possible quantum events that may occur to a givenquantum system and of the relationships between events and states. In thisframework, the logical connectives are interpreted as (generally irreversible)operations, which do not reflect any time-evolution either of the physical system orof the observer.
Quantum computation theory has inspired a completely different approach toquantum logic, giving rise to new forms of logics that have been called quantumcomputational logics. The basic objects of these logics are pieces of quantuminformation: possible states of quantum systems that can store and transmit theinformation in question, evolving in time. Accordingly, any formula of a quantumcomputational language can be regarded as a synthetic logical description of aquantum logical circuit. In this way, linguistic expressions acquire a characteristicdynamic meaning, representing possible computational actions.
The most natural semantics for quantum computational logics is a form ofholistic semantics, where some puzzling features of quantum entanglement (oftendescribed as mysterious and potentially paradoxical) are used as a positive semanticresource. Against the compositionality principle (a basic assumption of classicallogic and of many other logics), the meaning of a compound expression of aquantum computational language cannot be generally represented as a functionof the meanings of its well-formed parts. The procedure goes from the whole to theparts, and not the other way around. Furthermore, meanings are essentiallycontext-dependent. In this way, quantum computational logics turn out to be anatural abstract tool that allows us to model semantic situations (even far frommicrophysics), where holism, contextuality, vagueness and ambiguity play anessential role, as happens in the case of natural languages and in the languages ofarts (say, poetry or music).
The aim of this book is providing a general survey of the main concepts,questions and results that have been studied in the framework of the recent inter-actions between quantum information, quantum computation and logic.
Preface ix
Chapter 1 is an introduction to the basic concepts of the quantum-theoreticformalism used in quantum information. It is stressed how the characteristicuncertainties of the quantum world have brought about some deep logical inno-vations, due to the divergence between the concepts of maximal information andlogically complete information. It is explained how Birkhoff and von Neumann'squantum logic and the more recent forms of unsharp (or fuzzy) quantum logics havenaturally emerged from the mathematical environment of quantum theory.
Chapter 2 gives a synthetic presentation of the main “mathematical characters”of the quantum computational game: qubits, quregisters, mixtures of quregisters,and quantum logical gates. The basic idea of quantum computer theory is thatcomputations can be performed by some quantum systems that evolve in time.Accordingly, by applying Schrödinger’s equation, it is natural to assume thatquantum information is processed by special examples of unitary operators (calledquantum logical gates), which transform in a reversible way the pure states of thequantum systems that store the information in question. The last section of thechapter illustrates possible physical implementations of some quantum logicalcircuits by means of special variants of the Mach-Zehnder interferometer, anapparatus that has played an important role in the philosophical debates aboutquantum theory.
Chapter 3 investigates the puzzling entanglement-phenomena. The Einstein-Podolsky-Rosen paradox (EPR) is logically analyzed and it is shown howEPR-situations have later on been transformed into powerful resources, even from atechnological point of view. As a significant example, teleportation-experiments arebriefly illustrated.
Chapter 4 introduces the reader to quantum computational logics, new forms ofquantum logic inspired by the theory of quantum circuits. The basic idea of theselogics is that sentences denote pieces of quantum information, while logical con-nectives are interpreted as special examples of quantum logical gates. The mostnatural quantum computational semantics is a holistic and contextual theory ofmeanings, where quantum entanglement can be used as a logical resource. Theconcept of logical consequence, defined in this semantics, characterizes a weakform of quantum logic (called holistic quantum computational logic), where manyimportant logical arguments (which are valid either in classical logic or in Birkhoffand von Neumann’s quantum logic) are possibly violated.
Chapter 5 develops a quantum computational semantics for a language that canexpress sentences like “Alice knows that everybody knows that she is pretty”. Thebasic question is: to what extent is it possible to interpret quantifiers and epistemicoperators as special examples of Hilbert-space operations? It is shown how theselogical operators have a similar logical behavior, giving rise to a “reversibility-breaking”. Unlike logical connectives, quantifiers and epistemic operators can berepresented as particular quantum maps that are generally irreversible. An inter-esting feature of the epistemic quantum semantics is the failure of the unrealisticphenomenon of logical omniscience: Alice might know a given sentence withoutknowing all its logical consequences.
x Preface
Chapter 6 is devoted to a “many-valued generalization” of the classical part ofquantum computational logics, which only deals with bits, registers and gates thatare reversible versions of Boolean functions (in the framework of a two-valuedsemantics). One can generalize this approach, by assuming a many-valued classicalbasis for quantum computation. In this way, qubits are replaced by qudits: quantumsuperpositions living in a Hilbert space whose dimension may be greater than two.The qudit-semantics gives rise to some interesting physical applications.
Chapter 7 introduces the concept of abstract quantum computing machine,which represents an adequate mathematical model for the description of concretequantum computers. To what extent can abstract quantum computing machines besimulated by classical Turing machines? Does quantum computation give rise topossible violations of the Church-Turing thesis? These hard questions did not find,so far, a definite answer.
Chapter 8 describes some possible applications of the holistic quantumsemantics to fields (far apart from microphysics), where ambiguity, vagueness,allusions and metaphors play an essential role. Some characteristic examples thatarise in the framework of musical languages are illustrated.
Chapter 9 discusses some recent debates about foundational and philosophicalquestions of quantum theory, which have been stimulated by researches in the fieldof quantum information and quantum computation. “Information interpretations”according to which quantum theory should be mainly regarded as a “revolutionaryinformation theory” have been opposed to more traditional “realistic views”,according to which the pure states of the quantum-theoretic formalism shall always“mirror” objective properties of physical systems that exist (or may exist) in the realworld.
Chapter 10 contains a survey of the definitions of the main mathematical con-cepts used in the book.
Florence, Italy Maria Luisa Dalla ChiaraCagliari, Italy Roberto GiuntiniBergamo, Italy Roberto LeporiniCagliari, Italy Giuseppe Sergioli
Preface xi
Acknowledgements
Many ideas discussed in this book have been inspired by long collaborations withother scholars. Some of them have been co-authors of previous contributions ofours. We are especially grateful to Enrico Beltrametti, Gianpiero Cattaneo, AnatolijDvurečenskij, Hector Freytes, Richard Greechie, Stanley Gudder, Antonio Ledda,Eleonora Negri, Francesco Paoli, Sylvia Pulmannová and Miklos Rédei. Our workhas also been greatly stimulated by interactions with scientists and philosopherswho have represented for us an important scientific and human point of reference:Dirk Aerts, Alexandru Baltag, Sergio Bernini, Jeremy Butterfield, MicheleCamerota, Andrea Cantini, Roberto Casalbuoni, Ettore Casari, Gianni Cassinelli,Elena Castellani, Domenico Costantini, Marcello D’Agostino, Francesco DeMartini, Christian de Ronde, Antonio Di Nola, Dov Gabbay, Claudio Garola, AmitHagar, Décio Krause, Pekka Lahti, Federico Laudisa, Pierluigi Minari, MioaraMugur Schächter, Daniele Mundici, Mirko Navara, Giulio Peruzzi, Pavel Pták,Jaroslaw Pykacz, Zdenka Riečanová, Giovanni Sambin, Frank Schroeck, SonjaSmets, Sandro Sozzo, Karl Svozil, Silvano Tagliagambe, Bas van Fraassen andPaola Verrucchi. We would like to mention also the periodical meetings of theInternational Quantum Structures Association (IQSA), splendid occasions forscientific collaboration.
Some great scholars and friends are no longer with us. We fondly rememberSławomir Bugajski, Paul Busch, Roberto Cignoli, David Finkelstein, David Foulis,GianCarlo Ghirardi, Peter Mittelstaedt, Franco Montagna, Belo Riečan, CharlesRandall, Alberto Rimini, Gottfried Rüttimann, Patrick Suppes and Giuliano Toraldodi Francia.
Finally, we warmly thank Heinrich Wansing (editor-in-chief of Trends in Logic),who has encouraged our work and patiently waited for the final version of thisbook. We are also deeply grateful to the anonymous referee who has proposed to usmany interesting and useful suggestions.
This work has been partially supported by Regione Autonoma della Sardegna inthe framework of the project “Time-logical evolution of correlated microscopicsystems” (CRP 55, L.R. 7/2007, 2015).
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Contents
1 The Mathematical Environment of Quantum Information . . . . . . . 11.1 Physics and Logic in Classical Information Theory . . . . . . . . . . 11.2 From the Classical to the Quantum-Theoretic Formalism . . . . . . 31.3 The Mathematics of Quantum Theory . . . . . . . . . . . . . . . . . . . 51.4 Composite Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161.5 Time Evolution and Quantum Measurements . . . . . . . . . . . . . . 191.6 The Unsharp Approaches to Quantum Theory . . . . . . . . . . . . . 241.7 Quantum Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Pieces of Quantum Information and Quantum Logical Gates . . . . . 312.1 Qubits, Quregisters and Mixtures of Quregisters . . . . . . . . . . . . 312.2 Quantum Logical Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.3 Quantum Logical Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452.4 Physical Implementations by Optical Devices . . . . . . . . . . . . . . 47References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
3 Entanglement: Mystery and Resource . . . . . . . . . . . . . . . . . . . . . . . 533.1 Entangled Pure States . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 The Einstein-Podolsky-Rosen Paradox . . . . . . . . . . . . . . . . . . . 543.3 Quantum Teleportation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
4 From Quantum Circuits to Quantum Computational Logics . . . . . 654.1 A New Approach to Quantum Logic: Quantum
Computational Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.2 A Sentential Quantum Computational Language . . . . . . . . . . . . 664.3 A Holistic Computational Semantics . . . . . . . . . . . . . . . . . . . . 694.4 Quantum Computational Logical Arguments . . . . . . . . . . . . . . 78References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
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5 Individuals, Quantifiers and Epistemic Operators . . . . . . . . . . . . . . 855.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Truth-Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 865.3 A First-Order Epistemic Quantum Computational Language . . . 885.4 A Holistic Quantum Computational Semantics for
a Fragment of the Language L1 . . . . . . . . . . . . . . . . . . . . . . . 915.5 An Epistemic Quantum Computational Semantics . . . . . . . . . . . 955.6 Physical Examples and Relativistic-Like Epistemic Effects . . . . 1055.7 Quantifiers as Quantum Maps . . . . . . . . . . . . . . . . . . . . . . . . . 112References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6 From Qubits to Qudits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1176.2 Qudit-Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1186.3 Quantum Logical Gates in Qudit-Spaces . . . . . . . . . . . . . . . . . 1206.4 Łukasiewicz-Quantum Computational Logics . . . . . . . . . . . . . . 124References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126
7 What Exactly Are Quantum Computations? Classicaland Quantum Turing Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1277.2 Classical Deterministic and Probabilistic Machines . . . . . . . . . . 1287.3 Quantum State Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1317.4 Abstract Quantum Computing Machines . . . . . . . . . . . . . . . . . 135References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
8 Ambiguity in Natural and Artistic Languages: A QuantumSemantic Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1398.2 Vague Possible Worlds and Metaphors . . . . . . . . . . . . . . . . . . 1408.3 A Quantum Semantics for Musical Languages . . . . . . . . . . . . . 142References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
9 Quantum Information in the Foundational and PhilosophicalDebates About Quantum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 1519.1 Information Interpretations of Quantum Theory . . . . . . . . . . . . 1519.2 The Collapse-Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1529.3 Determinism, Indeterminism, Realism . . . . . . . . . . . . . . . . . . . 1539.4 A “Quantum Logical Thinking” in Different Fields . . . . . . . . . . 159References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
10 Mathematical Survey . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16310.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16310.2 Algebraic Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16310.3 Hilbert-Space Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
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Index
AAbelian group, 166Abstract quantum computing machine, 134Abstract quantum logic, 29, 65Adjoint operator, 10, 171Agatha Christie, 105Aharonov, D., 44, 137Amplitude, 7, 157Ancilla, 39, 67, 124Approximately universal gate-system, 45Aristotle, 141Atomic structure, 89, 114
BBaltag, A., 86Belief operation, 99Bell state, 17, 40, 54, 59, 73, 141Birkhoff and vonNeumann’s quantum logic,
27, 28, 53, 65, 83, 160Birkhoff, G., 8, 155Bit, 1Bit-flip knowledge operation, 109Bit flip quantum operation, 107Bit-phase-flip quantum operation, 107Bloch sphere, 105Bohr, N., 58, 151, 154Boolean algebra, 2, 39Bounded lattice, 164Bounded operator, 14, 171Brahms, J., 145
CCanonical basis, 7, 32, 86, 118, 131Canonical truth-perspective, 86, 108Cantor’s theorem, 140
Cauchy sequence, 5, 169Church-Turing thesis, 127Clara’s theme, 145Classical deterministic state machine, 129Classical observable, 3Classical probabilistic state machine, 129Clock, 136Closed subspace, 8, 118, 170Collapse of the wave function, 22, 61, 136,
152Complete Boolean algebra, 2Complete lattice, 164Composite system, 2, 18, 22, 31, 50, 55, 59,
152Compositionality-principle, 66, 117Computation of a deterministic state
machine, 129Computation of a probabilistic state
machine, 130Computation of a quantum state machine,
132Computation-path, 130Computing machine, 127Configuration, 131, 136Consistent-histories approaches to quantum
mechanics, 158Contraposition, 28, 165Control bit, 36, 50Convex combination, 3Copenhagen interpretation, 151
DDensely defined operator, 171Density operator, 14, 173Depolarizing quantum operation, 108Depth, 47, 68, 132
© Springer Nature Switzerland AG 2018M. L. Dalla Chiara et al., Quantum Computation and Logic,Trends in Logic 48, https://doi.org/10.1007/978-3-030-04471-8
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176 Index
Deterministic completion of quantum the-ory, 155
Deterministic physical theory, 154Deutsch, D., 157Dictum de omni-principle, 113Dirac, P., 5, 168Distributive lattice, 164Distributivity, 9Division ring, 5, 167Double negation, 28, 165Duns Scotus-principle, 28
EEffect, 24, 119Effect algebra, 165Effect-valued measure, 27Eigenvalue, 170Eigenvector, 13, 170Einstein, A., 53, 56, 154, 156Element of reality, 56Entanglement, 16Epistemic agent, 85, 95, 99Epistemic domain, 85, 95, 109Epistemic quantum computational seman-
tics, 95Epistemic realization, 101Epistemic situation, 59, 100EPR-paradox, 54Excluded middle-principle, 4, 8, 28, 104,
149, 165Extensional meaning, 78, 117Extra-musical meaning, 144
FFactorizable state, 17Factorized state, 23, 72, 152Falsity, 1, 66, 86, 118Falsity-property, 32, 119Feynman, R., 127Field, 167First-order epistemic realization, 114First-order quantum computational seman-
tics, 86Frege, G., 65, 117
GGestalt, 139Ghirardi, G., 158Gödel’s incompleteness theorem, 5Goethe, G.W., 144Group, 166
HHadamard-gate, 41, 48, 66, 111, 122, 124Halting state, 128, 136Heine, H., 144, 147Hilbert lattice, 29Hilbert space, 5, 169Holistic behavior, 40, 73, 143Holistic quantum computational logic, 77Holistic semantics, 73
IIncompatible physical quantities, 27Infimum, 8, 156, 164Information interpretation of quantum the-
ory, 151Initial state, 20, 23, 128, 152Inner product, 5, 20, 168Intensional meaning, 117, 142Internal state, 128Isolated system, 19
KKant, E., 151Kerr-effect, 50Kerr-Mach-Zehnder interferometer, 50Knowledge operation, 95, 109Kochen, S., 155Kraus operator, 22, 107Kripke-semantics, 85
LLattice, 9, 29, 65, 163Lieder, 145, 147Lindenbaum-property, 156Linear combination, 6, 169Linear operator, 170Locality-principle, 56Logical omniscience, 86, 105Łukasiewicz-conjunction, 120Łukasiewicz’ logics, 120Łukasiewicz-quantumcomputational logics,
124
MMach–Zehnder circuit, 132Mach-Zehnder interferometer, 48Maximal information, 2, 154Measurable set, 2Metric, 5, 168Metrically complete pre-Hilbert space, 169
Index 177
Min-conjunction, 120Mittelstaedt, P., 159Mixed state, 3, 13, 18Mixture, 3, 16, 32Musical phrase, 142Musical theme, 143Musical thought, 142, 145
NNegation-gate, 34, 45, 47, 121No-go theorem, 155Non-contradiction, 26, 28, 120Norm, 5, 168
OOrthoarguesian law, 29Orthocomplement, 7, 26, 165Orthogonality-relation, 6, 26, 166Ortholattice, 164Orthomodularity, 9Orthomodular lattice, 9, 26, 29, 65, 165Orthonormal basis, 6, 85, 169
PPartial Boolean algebra, 156Partial classical logic, 155Partial order, 9, 14, 25, 164Partial trace, 18Pauli matrix, 61Phase-flip quantum operation, 108Phase space, 2, 16Physical completeness-principle, 56Physical event, 2, 142, 154Podolsky, B., 53, 56Positive operator, 14, 171Pre-Hilbert space, 168Pre-order relation, 33, 77, 87Probabilistic equivalence, 33Probabilistic identity, 96Program-path, 130Projection, 9, 15, 24, 32, 156, 172Projection theorem, 8Projection-valued measure, 13Pure state, 2, 17, 22, 32, 53, 56, 131, 149,
151, 155
QQuantifier, 85, 88, 91, 112Quantifier-configuration, 114Quantum computational logics, 29, 47, 65
Quantum event, 5, 11, 27, 56, 65, 83Quantum logical circuit, 45, 49, 65Quantum logical gate, 34Quantum logics, 27Quantum-measurement problem, 24Quantum observable, 12, 27Quantum state machine, 131Quantum Turing machine, 127Qubit, 32Qudit, 118Qudit-space, 118Quregister, 32
RRealism, 153, 154Reality-principle, 56Reduced state, 16, 18, 31, 108, 141Relative conditional phase shifter, 50Relative phase shifter, 48Relativistic-like epistemic effect, 101Reversible conjunction, 39Reversible negative conjunction, 39Rimini, A., 158Ring, 166Ring with unity, 167Rosen, N., 53, 56
SSchiller, F., 144Schrödinger’s axiom, 20Schrödinger’s equation, 24, 55, 152Schrödinger, E., 53Schubert, 144, 145Schumann, 145Score, 142Self-adjoint operator, 10, 171Shi, Y., 44, 137σ -complete Boolean algebra, 2σ -complete lattice, 164Skolem-paradox, 140Smets, S., 86Specker, E.P., 155Spin, 54, 66, 123Square root of identity, 41, 66Square root of negation-gate, 41, 66, 123State machine, 128Superposition, 7, 32, 41, 122, 131, 142, 157,
169Supremum, 8, 156, 164Swap-gate, 34, 36, 38, 46
178 Index
TTarget bit, 35, 50Tarski, A., 153Teleportation, 59Tensor product, 17, 174Tertium non datur-principle, 28, 117, 154Theme and Variations, 145Toffoli-conjunction, 122Toffoli-gate, 36, 45, 50, 66, 73, 88, 122Toffoli–Łukasiewicz conjunction, 122Toffoli–Łukasiewicz gate, 121Trace, 12, 173Trace-class operator, 173Truth, 1, 28, 66, 86Truth-perspective, 85, 95Truth-property, 32, 119Turing machine, 127
UUncertainty-principles, 4, 155Understanding operation, 99Unitary operation, 20, 34, 87, 123Unitary operator, 19, 172Unsharp approaches to quantum theory, 24Unsharp (fuzzy) quantum logics, 29
VVague possible world, 139Vector space, 167Von Neumann, J., 8, 152Von Neumann-Lüders’ axiom, 21, 55
WWeak distributivity, 28, 83Weber, T., 158Weight, 3, 14Width, 47, 68, 132Wieck Schumann, C., 147Wire, 1, 45Word, 128Word-input, 128, 133Word-output, 132
XXOR-gate, 35, 66, 73
ZZarlino, 143Zeilinger, A., 62Zermelo–Fraenkel theory, 140