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SPRINGER BRIEFS IN MATHEMATICAL PHYSICS 41 Blake C. Stacey A First Course in the Sporadic SICs 123 SpringerBriefs in Mathematical Physics Volume 41 SeriesEditors NathanaëlBerestycki,UniversityofVienna,Vienna,Austria MihalisDafermos,MathematicsDepartment,PrincetonUniversity,Princeton,NJ, USA AtsuoKuniba,InstituteofPhysics,TheUniversityofTokyo,Tokyo,Japan MatildeMarcolli,DepartmentofMathematics,UniversityofToronto,Toronto, Canada BrunoNachtergaele,DepartmentofMathematics,Davis,CA,USA HalTasaki,DepartmentofPhysics,GakushuinUniversity,Tokyo,Japan SpringerBriefs are characterized in general by their size (50–125 pages) and fast productiontime(2–3monthscomparedto6monthsforamonograph). Briefsareavailableinprintbutareintendedasaprimarilyelectronicpublicationto beincludedinSpringer’se-bookpackage. 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Stacey A First Course in the Sporadic SICs BlakeC.Stacey UniversityofMassachusettsBoston Boston,MA,USA ISSN2197-1757 ISSN2197-1765 (electronic) SpringerBriefsinMathematicalPhysics ISBN978-3-030-76103-5 ISBN978-3-030-76104-2 (eBook) https://doi.org/10.1007/978-3-030-76104-2 ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2021 Thisworkissubjecttocopyright.AllrightsaresolelyandexclusivelylicensedbythePublisher,whether thewholeorpartofthematerialisconcerned,specificallytherightsoftranslation,reprinting,reuse ofillustrations,recitation,broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,and transmissionorinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilar ordissimilarmethodologynowknownorhereafterdeveloped. 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ThisSpringerimprintispublishedbytheregisteredcompanySpringerNatureSwitzerlandAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland Preface Physicshassocialandaestheticaspectsthatmanyofitsmostvocalboostersareloath toadmit.Evenwhenwephysicistsletdownourhairinmemoirsandinterviewsand othersemi-informalvenuesandadmitthatthebeautyofoursubjectmovesus,we arereluctanttoconfessthatthe“beauty”weinvokeisanintenselypersonalquality. A matter of taste, to be frank about it. To me, SICs offer the enigmatic appeal of patternshalfwayseen,likeaforestinapredawnfog. Thisbookisnotaboutcompletedpatterns.WedonotknowhowmanySICsthere are,andevenifwenarrowourattentiontothesporadicSICs,theprimaryfocusof thesechapters,thereareplentyofunresolvedquestionsontheedgesofwhatwewill coverhere.ItisevenpossiblethattheclassificationofsporadicSICsisincomplete, andastructurefoundtomorrowwillhavetobeaddedtothelist.Ihavetriedtopoint outthesequestionswhereIcould.Manyoftheexercisesarenotmorethanastepor twofromtheresearchfrontier. It is my hope that you have at least one reaction to this book that I did not intend.Acertainamountofvaguenessisnecessary,Ithinksometimes,forfruitful scientific communication. For if my statement creates precisely the consequences Iplanned,matchingmyintentionineverydetail,thentherewouldbenoroomfor spontaneity,forthegenerationofleapsImyselfwasunabletomake.Thereisnopoint inyourthinkingonlywhatIhavealreadythought!Thechallengeliesinmakingthose moments of imprecise fantasy, of ideas-for-an-idea, fall in the appropriate places amid the direct and declarative surroundings. “To construct a pleasingly thrilling contrapuntalstructure,”tosoundlikeapersonwhothinkstheyknowmusictheory. TheconceptualbackgroundtomyworkwithSICsistheresearchprogramknown asQBism,anongoingprojectinquantumfoundationswhosegoalistoidentifythe characteristicsofnaturewhichmakequantumtheorysuchagoodtoolfornavigating initreferences[77–80].Forthemostpart,thisbookwillskimoverthesurfaceof thosephilosophicalwaters,astheyrunquitedeepandmypagecounthereisquite finite.Ihavealsomostlyelidedthetopicofhowtoimplementinthelaboratorythe measurementsdefinedbySICs.ThisisfortheverygoodreasonthatIamatheorist andshouldbekeptasfarawayfromlaboratoriesaspossible.Iamnotthebestone toexplainhowtobuildmachinesthatgoclick,butIknowthatexperimentalistsare v vi Preface cleverfolk.Tosuggestthathavingfourdifferentkindsofclickinsteadoftwoisan impossibleextravagancewouldinsulttheircapabilities. Iwouldliketothankmyfamilyforkeepingintouchatasafedistance,andmy housemates for putting up with me in close proximity. My collaborators deserve more gratitude than a preface can contain. I hesitate to list names in the certainty thatIwillomitanimportantone,butIknowIcannotletthismomentpasswithout expressingmyappreciationforMarcusAppleby,GabrielaBarretoLemos,JohnB. DeBrota,ChristopherA.Fuchs,JacquesL.Pienaar,andHuangjunZhu.Backinthe days when we had conferences, I had wonderful learning experiences about SICs atWorcesterPolytechnicInstitute,theOhioStateUniversity,twoMarchMeetings of the American Physical Society, and at MIT. More recently, we have had video chats, upon occasion managing to bridge four continents. As these meetings have been a highlight of the past year, I owe a kind word to the participants, including IngemarBengtsson,IrinaDumitru,MaryFries,AmandaGefter,SachinGupta,Bob Henderson, Kim Reece, Kathryn Schaffer, Juan Varela, and Matthew Weiss. John BaezandKarolZ˙yczkowskigavemeopportunitiestoexpoundontheseideasinblog postsandvideoconference,respectively.Someearlydraftsofwhatwouldbecome thisbookwerewrittenwhileIhadthesupportoftheJohnTempletonFoundation;all oftheopinionsexpressedherearemyownandnotthoseoftheJTF.DavidHarden andEricDownessawthisbookinmanuscriptandprovidedcorrections.Ifanyerrors remain, Eric knows that I will transfer all the responsibilities to him; we met in college,andheiswellawarethatIhavenoshame. I found my way to SICs many years ago, during a time when life had left me emotionally adrift. They turned out to be the research problem I needed. I do not knowwhethertheycanprovideanyoneelseamomentofrespite,butthisbookisa goodopportunitytofindout. NearBoston,MA,USA BlakeC.Stacey Contents 1 EquiangularLines .............................................. 1 1.1 Introduction ............................................... 1 1.2 RealLines ................................................. 2 1.3 ComplexLines ............................................. 6 References ..................................................... 10 2 OptimalQuantumMeasurements ................................ 13 2.1 Introduction ............................................... 13 2.2 SICRepresentationsofQuantumStates ........................ 15 2.3 ConstructingSICsUsingGroups ............................. 22 References ..................................................... 24 3 GeometryandInformationTheoryforQubitsandQutrits ......... 27 3.1 Qubits .................................................... 27 3.2 Qutrits .................................................... 28 3.3 Coherence ................................................. 31 References ..................................................... 36 4 SICsandBellInequalities ....................................... 39 4.1 Mermin’sThree-QubitBellInequality ......................... 40 4.2 TheHoggarSIC ............................................ 41 4.3 QubitPairsandTwinnedTetrahedralSICs ..................... 45 4.4 FailureofHiddenVariablesforQutrits ........................ 50 4.5 QuantumTheoryfromNonclassicalProbabilityMeshing ........ 52 References ..................................................... 53 5 TheHoggar-TypeSICs ......................................... 57 5.1 Introduction ............................................... 57 5.2 SimplifyingtheQBicEquation ............................... 59 5.3 TripleProductsandCombinatorialDesigns .................... 60 5.4 TheTwinoftheHoggarSIC ................................. 67 5.5 CombinatorialDesignsfromtheTwinHoggarSIC .............. 69 5.6 Quantum-StateCompatibility ................................ 72 5.7 FromPauliOperatorstoRealEquiangularLines ................ 78 vii viii Contents 5.8 ConcludingRemarks ........................................ 80 References ..................................................... 81 6 SporadicSICsandtheExceptionalLieAlgebras .................. 83 6.1 RootSystemsandLieAlgebras ............................... 83 6.2 E ........................................................ 86 6 6.3 E ........................................................ 88 8 6.4 E ........................................................ 90 7 6.5 TheRegularIcosahedronandReal-Vector-SpaceQuantum Theory .................................................... 92 6.6 OpenPuzzlesConcerningExceptionalObjects ................. 95 References ..................................................... 100 7 Exercises ...................................................... 103 References ..................................................... 111 Index ............................................................. 113 Chapter 1 Equiangular Lines Iintroducetheproblemoffindingmaximalsetsofequiangular lines,inbothitsrealandcomplexversions,attemptingtowrite thetreatmentthatIwouldhavewantedwhenIfirstencountered thesubject.Equiangularlinesintersectintheoverlapregionof quantuminformationtheory,theoctonionsandHilbert’stwelfth problem.Thequestionofhowmanyequiangularlinescanfit intoaspaceofagivendimensioniseasytopose—ahigh-school studentcangraspit—yetitishardtoanswer,beingasyet unresolved.Thiscontrastofeaseanddifficultygivestheproblem aclassiccharm. 1.1 Introduction Tomotivatethedefinition,wecanstartwiththemostelementaryexample:thediag- onalsofaregularhexagon.Anytwoofthemcrossandcreatewhattheschoolbooks callsupplementaryverticalangles.Withoutlossofinformation,wecantake“the” angle defined by the pair of lines tobe the smaller of these twovalues. Moreover, thisvalueisthesameforallthreepossiblepairsoflines:Foranytwodiagonals,their angle of intersection willbe π/3. We can statethis in a way amenable to general- izationifwelayaunitvectoralongeachofthethreediagonals.Whicheverwaywe choosetoorientthevectors,theirinnerproductswillsatisfy (cid:2) 1, j =k; |(cid:2)v ,v (cid:3)|= (1.1) j k α, j (cid:4)=k. When a set of vectors {v : j =1...,N} enjoys this property, it yields a set of j equiangular lines. An orthonormal basis is equiangular, with α =0. The question becomesmoreintriguingwhenwepushthesize N ofthesetbeyondthedimension d.Forexample,ifwestepfromR2 uptoR3,itisalreadyhardtoguesshowmany ©TheAuthor(s),underexclusivelicensetoSpringerNatureSwitzerlandAG2021 1 B.C.Stacey,AFirstCourseintheSporadicSICs, SpringerBriefsinMathematicalPhysics41, https://doi.org/10.1007/978-3-030-76104-2_1