Unitary invariants of qubit systems

We give an algorithm allowing to construct bases of local unitary invariants of pure k-qubit states from the knowledge of polynomial covariants of the group of invertible local filtering operations. The simplest invariants obtained in this way are explicit

UnderconsiderationforpublicationinMath.Struct.inComp.Science

Unitaryinvariantsofqubitsystems

arXiv:quant-ph/0604202v1 27 Apr 2006

Jean-GabrielLuque1,Jean-YvesThibon1andFr´ed´ericToumazet2

InstitutGaspardMonge,Universit´edeMarne-la-Vall´ee,F-77454Marne-la-Vall´eecedex,France.email:{luque,jyt}@univ-mlv.fr2

Laboratoired’InformatiquedeParisNord,InstitutGalil´ee-Universit´eParis13/ParisNord,99av.J.-B.Clement93430Villetaneuse,France.email:ft@lipn.univ-paris13.frReceived1February2008

1

Wegiveanalgorithmallowingtoconstructbasesoflocalunitaryinvariantsofpurek-qubitstatesfromtheknowledgeofpolynomialcovariantsofthegroupofinvertiblelocal http://www.mianfeiwendang.completesetsofgeneratorsareobtainedforuptofourqubits,andthestructureoftheinvariantalgebrasisdiscussedindetail.

1.Introduction

Fromamathematicalpointofview,QuantumInformationTheorydealswith nitedi-mensionalHilbertspaces,thestatespacesof nitek-partitesystems,whichhavethespecialform

H=V1 V2 ··· Vk,

(1)

whereViisthe nitedimensionalstatespaceoftheithpart(orparticle)ofthesystem,mostofthetimeassumedtobeaqubit,whichmeansthatdimVi=2.

Theinterestingnon-classicalbehaviorsonwhichthetheoryisbasedalreadyoccurfortwo-qubitsystems,fortheso-calledentangledstates,thoseψ∈V1 V2whichcannotbewrittenintheformv1 v2.ThepropertiesofsuchstatesarethebasisoftheEPRparadox(Einsteinetal.andsinceitsdiscovery,theentanglementphenomenonhasbeenthoroughlyinvestigatedbyphysicists,cf.(Bell1966;Clauseretal.Aspectetal.1982;

BennettandWiesnerandmorerecentlybymathematicians,e.g.,andBrylinsky2002;Klyachko2002;Thereis,however,nogeneralagreementonthede nitionofentanglementforsystems

withmorethantwoparts.Klyachkohasproposed(Klyachko2002;KlyachkoandShumovskytoregardasentangledthestateswhicharesemi-stablefortheactionofthegroupofin-

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