UNITEXT for Physics Guido Fano S.M. Blinder Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers Mathematical Methods and Conceptual Foundations UNITEXT for Physics Series editors Michele Cini, Roma, Italy Attilio Ferrari, Torino, Italy Stefano Forte, Milano, Italy Guido Montagna, Pavia, Italy Oreste Nicrosini, Pavia, Italy Luca Peliti, Napoli, Italy Alberto Rotondi, Pavia, Italy Paolo Biscari, Milano, Italy Nicola Manini, Milano, Italy Morten Hjorth-Jensen, East Lansing, USA UNITEXTforPhysicsseries,formerly UNITEXT CollanadiFisicae Astronomia, publishestextbooksandmonographsinPhysicsandAstronomy,mainlyinEnglish language, characterized of a didactic style and comprehensiveness. The books published in UNITEXT for Physics series are addressed to graduate and advanced graduate students, but also to scientists and researchers as important resources for their education, knowledge and teaching. More information about this series at http://www.springer.com/series/13351 Guido Fano S.M. Blinder (cid:129) Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers Mathematical Methods and Conceptual Foundations 123 GuidoFano S.M.Blinder Dipartimento di Fisica eAstronomia University of Michigan Universitàdi Bologna AnnArbor, MI Bologna USA Italy ISSN 2198-7882 ISSN 2198-7890 (electronic) UNITEXTfor Physics ISBN978-3-319-58731-8 ISBN978-3-319-58732-5 (eBook) DOI 10.1007/978-3-319-58732-5 LibraryofCongressControlNumber:2017940230 ©SpringerInternationalPublishingAG2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of 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 transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Foreword Manyyearsago,mymathteacherassignedmethetasktogiveapresentationtothe restoftheclass.Timetoprepare,2weeks;topic,grouptheoryanditsapplicationto quantummechanics.IpolitelyobservedthatIhadnotyethadacourseonquantum theory; I knew nothing about it. Teacher’s reply was: “Well, you study it.” His name was Guido Fano. ThisishowGuidoFanohadmelearningquantumtheory.Athome,alone,with abunchofbooksonmydesk,Dirac’soneontopofall.Ihavebeenforevergrateful tohim,forthisandmuchmore.IfIgotintotheoreticalphysics,it’sthankstohim, the trust that I found, with surprise, from him. Today, when I realize that there is something that I should know and do not know (which of course happen contin- uously), I hear his voice in my mind saying “Well, you study it.” But the second lesson I learned from Guido Fano, both about life and quantum theory,wasevenmorecrucial.Aftermypresentationtotheclass,Ithankedhimand said: “I have now learned quantum mechanics.” His reply was “No you haven’t.” How true. Andnotjustbecauseacoupleofweekscouldn’tsufficetodigestthetheory,but becausenearlyacenturyfromHeisenbergandDirachasn’tsufficedthecommunity of us all to come to terms with this theory. In the months after my presentation, I kept discovering, first from Fano himself, and then over and over for the rest of mylife,howmultifacetedisthistheory,andhowslipperyouractualunderstanding toit.ThemoreIhavelearnedaboutthetheory,thelessclearithasbecome.When comesthemomentofsayingwhatwereallyhavelearnedabouttheworldinfinding thatquantummechanicsworkssoeffectively—well,wedisagreeamongourselves, andtherearenearlyasmanyopinionsasphysicists(notcountingthephilosophers). Longtimelater,withthisnewbookonquantumtheory,onceagainGuidoFano, togetherwithhisoverseascolleagueS.M.Blinder,opensupnewsidesofquantum mechanics for me. The beauty of this book, in my opinion, is that it merges very differenttraditionsofthinkingaboutthetheoryandteachingit.Foratheorystillso perplexing, this is what we need; in order to understand it better, to learn it better, but also to learn how to better use it. v vi Foreword Ifhadtostartagainlearningquantumtheoryfromscratch,asIdid40yearsago, I would do as I did: at home, alone, with many texts on my desk. But I would certainly have, next to Dirac’s one, this book. Marseille, France Carlo Rovelli February 2017 Preface Thismonographistheresultofmanyyearsofexperienceandcontemplationbytwo octogenarian mathematical physicists, on opposite sides of the Atlantic, who, after all these years, are still endlessly fascinated by the marvelous intellectual edifice that has become quantum mechanics. And which, in the twenty-first century, continues its proliferation into entirely new avenues of human accomplishment, including experiment-based answers to metaphysical questions and the limitless potential of quantum computation. The main purpose of the book is to make accessible to nonspecialists the still evolving fundamental concepts of QM and the terminology in which these are expressed. Hence our title: “Twenty-First Century Quantum Mechanics: Hilbert Space to Quantum Computers.” Among the concepts which we emphasize are the following: (cid:129) The wavefunction of a particle: associated with a “cloud” of probability, such thatthedensityofthecloudisgreaterinregionswhichhaveahigherprobability of containing the particle. (cid:129) The Heisenberg uncertainty principle: conjugate pairs of observables, such as thepositionandthevelocityofaparticle,cannotbothbepreciselydeterminedat the same time. (cid:129) Isotropic vectors: vectors with complex components, which are orthogonal to the rotation axis and thus remain invariant under rotation; these are used to construct spinors. (cid:129) The spin of elementary particles: the quantum counterpart of rotations of clas- sical objects, described by spinors. (cid:129) The question of whether the fundamental laws of physics violate local realism. Locality means that the influence of one particle on another cannot exceed the speedoflight.Realismmeansthatquantumstateshavewell-definedproperties, independent of our knowledge of them. (cid:129) The possible existence of hidden variables: something analogous to the enor- mous number of microscopic details of molecular motions, which exhibit vii viii Preface themselves in the determinate macroscopic properties of matter, such as temperature, pressure, etc. (cid:129) Conceptual problems associated with measurement, superposition and deco- herence in quantum systems: collapse of the wavefunction, Schrödinger’s cat, and quantum entanglement. (cid:129) Quantum computers: if they can be made practicable, enormous enhancements in computing power, artificial intelligence and secure communication will follow. Needless to say, the quantum theory raises very profound metaphysical and epistemological questions on the description about the “objective world.” An important precept of Felix Klein’s famous Erlangen program was that “A geometryisthestudyofinvariantsunderagroupoftransformations.”Theseideas, subsequentlyelaboratedinthecontributionsofEinstein,Dirac,andthephilosopher Nozick,haveledto,atleast,aprovisionalunderstandingofwhatconstitutesreality. The need for an observer to “search for invariants” is of such generality that it must apply even in the lives of animals. Imagine a gazelle cautiously eying two lions.Supposethatsheglancesatthefirstlion,andthenthesecond.Indoingso,she mustturnherhead.Buttheremustbesomeprocessinherbrainthatenablesherto realize that,evenaftersheturnsherhead(andthusregistersadifferent image),the first lion is still there, surely an instance of “invariance.” If we wanted to build a robot capable of distinguishing objects, then, when the robot’s eyes move, its programming must include the capability of performing mathematical transforma- tions among images viewed at varying angles at different times. Nature probably doesnotworkinpreciselythesameway,butthefundamentalconceptualfeatures: (1) variability of images and (2) recognition of invariants, or common elements, must still be applicable. These ideas certainly pertain to the classical view of Nature; what are their manifestations in quantum mechanics? The analog of the rotation of a classical observer is the evolution of the wavefunction. However there are two distinctly differentmodesofevolutioninquantummechanics.Oneisacontinuousevolution, following the time-dependent Schrödinger equation; the second, called collapse of the wave function, is a random and instantaneous event brought about by a measurementorperturbationofthequantumsystem.Thiswas,atleast,thepointof viewofthefoundingfathersofquantummechanics,mainlyBohr,Heisenberg,and Dirac. The most eminent critic of such ideas (a probabilistic interpretation of QM) was none other than Albert Einstein, as epitomized in his famous pronouncement: “God does not play dice with the Universe!” A major aspect of the epistemological problem has been resolved by actual experiments (by Alain Aspect and others), motivated by the deep insights of John StewartBell.Thishasrevealedamajorincompatibilitybetweentheworldviewsof classicalandquantumphysics.Bell’stheoremstatesthatitisimpossibletoexplain Preface ix theresultsofquantumphysicsusingthecausalityofclassicalphysics,thusnegating the possible existence of local hidden variables. Quantum mechanics differs fun- damentally from classical mechanics in that the underlying microscopic behavior is not determinate. Measurement and decoherence: according to the traditional (“Copenhagen”) interpretation of quantum mechanics, wave function collapse occurs when a mea- surement is performed. However there remains the problem of when the collapse actuallyoccurs?Inthepast,somephysiciststhoughtthatcollapseisbroughtabout whenaconsciousobserver“takesnote”ofthenewstateofasystem;butthispoint of view is now in the minority, since it is more reasonable to think that any inanimateapparatuscanalsomakeameasurementandproducea“quantumjump.” Amorerealisticapproachtothisproblemistoconsiderthemicroscopicsystem,the measuring apparatus and the environment as a single composite system. The wavefunction of the complete system must change during an exceedingly short interval of time from a “superposition of states,” to just one of these states. This phenomenon is called decoherence; it can be proved mathematically rigorously in some models, although there is still much work to be done. The superposition of two wavefunctions for a macroscopic object is also con- sideredintheinfamousSchrödinger’scatGedankenexperiment.Acatisconfinedto a closed box with a Geiger counter, which detects randomly-occurring radioactive decaysinasampleofradium.TheGeigercounterisconnectedtoavialofcyanide, whichisbrokenwhenadecayparticleisdetected,killingtheunfortunatecat.Until the box is opened, its state can only be described as a “superposition,” of a “live cat”anda“deadcat.”AccordingtotheCopenhagenversionofquantummechanics, the cat “becomes” dead or alive only after an observer opens the box. As para- doxical as it seems, superposition of quantum states of macroscopic objects has now been achieved, for example, in a SQUID (superconducting quantum inter- ference device). Quantum computing proposes to apply uniquely quantum-mechanical phe- nomena, such as superposition and entanglement to operate on quantum units of information, called qubits. In contrast to classical bits, which can represent a variablewithjusttwovalues,say0and1,qubitscan,inconcept,containaninfinite continuum of information, in terms of superpositions of two basis qubits, such as jWi¼aj0iþbj1i. A quantum computer could, in principal, be capable of solving problems in a matter of seconds, which might take a classical computer several centuries to accomplish. Several hypothetical quantum algorithms have already been proposed for large-integer factorization and other applications. Current real- izations ofquantum computers arestill very far from having such capabilities. But apartfrompotentialpracticalapplications,quantumcomputingremainsaprofound subjectoffundamentalinterestfor bothcomputerscienceandquantummechanics.