2023年7月11日发(作者：)

Threeplayer,TwoStrategy,MaximallyEntangledQuantumGamesAdenAhmed∗SteveBleiler†FaisalShahKhan‡August10,2008AbstractWedevelopanoctonionicrepresentationofthepayofffunctionforathreeplayer,twostrategy,maximallyentangledquantumgame.1AFormalismforQuantizationUpuntilnow,toquantizegamesmostauthorshave,likeMeyer,focusedtheireffortsonthequantizationoftheplayers’strategyspaces,essentiallythedomainofthepayofffunctionthatdefinesthegametobequantized(seeforexample[7,8,11,12]).Theprincipaltechniqueistoidentifythesestrategyspaceswithanorthogonalbasisofsomequantumsystem,inorderthatplayersmaynowtakesuperpositionsorevenmixedsuperpos-tion,playersmaynowcorrelatethently,mereaccesstothehigherran-domizationofsuperposition(asopposedtorealprobabilisticcombination)orthecorrelationofstrategicchoicesviaentanglementallowspayoffstothepl[3]amathematicalformalismforgamequantizationthatfocusesonthequantizationofthepayoutsoftheoriginalgameGtobequantized,andex-pressesthequantizedversionasa(proper)extensionoftheoriginalpayout∗PhDcandidate,DepartmentofMathematicsandStatistics,PortlandStateUniversity,Portland,:ahmedao@.†ProfessorofMathematics,DepartmentofMathematicsandStatistics,:bleilers@.‡PhDcandidate,DepartmentofMathematicsandStatistics,tFaculty,UniversityofPortland,PortlandOregon,:faisal@.1arXiv:0808.1391v1

[quant-ph]

10

Aug

2008functionintheset-theoreticsense(justasintheclassicalcase)cally,duringextensions,ishtopasstoamoregeneralnotionofrandomization,henwithaHpurposehereletusassumethatHisfinitedimensional,andthatwehaveafinitntumsuperpositionofXwithrespecttothebasisBwemeanacomplexprojectivelinearcombinationofelementsofX;thatis,arepre-sentativeofanequivalenceclassofcomplexlinearcombinationswheretheequiecontextisclearastothebasistowhichthesetXisidentified,denotethesetofquantumsuper-positionsforXasQS(X).Ofcourse,itisalsopossibletodefinequantumsuperpositionsforinfinitesets,butforthepurposehere,llowscanbeeasilygeneralizedtotheinfinderlyingspaceofcomplexlinearcombinationsisaHilbertspace,wecanassignalengthtoeachlinearcombinationand,uptophase,alwaysrepresentaphquantumsuperpositionofXwecanobtainaprobabilitydis-tributionoverXbyassigningtoeachcomponenttheratioofthesquareofthelengthofitscoeffimple,theprobabilitydistributionproducedfromαx+βyisjust|α|2|β|222x+22y|α|+|β||α|+|β|CallthisfunctionQS(X)→∆(X)aquantummeasurementwithrespecttoX,andnotethatgeometricallyquantummeasurementisdefinedbypro-jectinganormalizedquantumsuperpositionontothevariouselementsofmeas,thisfunctionbyqXclearfromthecontext,enafiniten-playergameG,supposewehavea