Quaternionic Computing

equivalenttotheidentityoperationandbrasaresimplytransposedkets.Similarlythematrixdaggeroperator( )canbereplacedwiththematrixtransposeoperator(t).

Inthiscase,wemustreplaceunitarytransformationswithorthonormaltransformations,asthesearetheonlyinner-productpreservingoperationsonthisinner-productspace.Onecouldconceiveamodelinwhichthestatevectorsalwayshaverealamplitudes,butinwhicharbitraryunitarytransformations(onthecomplexHilbertspace)areallowed,aslongastheendresultisstillarealamplitudevector.Itiselementarytoshowthatorthonormaltransformationsaretheonlyonesthathavethisproperty,andhencethismodelisasgeneralascanbe,giventhefactthatweinsistthattheamplitudesbereal.

RebitsandStates

Inquantumcomputingandquantuminformationtheory,wede nethequbitasthemostelementaryinformation-containingsystem.Abstractly,thestateofaqubitcanbedescribedbya2-dimensionalstatevector

|Φ =α|0 +β|1 ,s.t. Φ 2=

a2+b2=1(4)

Inthiscase,thearbitraryphasefactorcanonlybe+1or 1,andtherebitequivalencerelationwhichreplacesEquation3is

Φ≡Φ′ |Φ =eiθ|Φ′ ,whereθ∈{0,π} |Φ =±|Φ′ (5)(6)

Similarlyasforqubits,singlerebitstatesdohaveanicegeometricalinterpretation:theyareisomorphictothecircumference,having|0 and|1 atoppositeextremes.OnewaytoseethisistoconsiderthelocusofpointsintheBlochsphereforwhicheiθ=1,orinotherwords,thosewithnocircularpolarisation.Unfortunately,thereisnosuchnicegeometricrepresentationofanarbitraryn-qubitstate,andwebelievethesameistrueforn-rebitstates.

Thecomputationalbasisvectorsforarebitarestill|0 and|1 ,andforarbitraryn-rebitsystemstheycanalsoberepresentedasn-bitstrings.Themeasurementruleinde ningtheprobabilitiesofobtainingthecorrespondingbitstringasaresultisessentiallythesameasEquation1,

Pr(|Φ →“b”)=| Φ|b |2= Φ|b 2

(7)