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Minimal multipartite entanglement detecion

Minimal multipartite entanglement detecion. QIPA 2.08.2013. Dr Marcin Wieśniak, Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gdański. Marcin Wieśniak, Koji Maruyama , Physical Review A 85 , 062313 (2011). General motivation. Entanglement is useful im many various protocols .

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Minimal multipartite entanglement detecion

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  1. Minimalmultipartiteentanglementdetecion QIPA 2.08.2013 Dr Marcin Wieśniak, Instytut Fizyki Teoretycznej i Astrofizyki, Uniwersytet Gdański Marcin Wieśniak, KojiMaruyama, PhysicalReview A 85, 062313 (2011)

  2. General motivation • Entanglementisuseful im manyvariousprotocols. • Evenifitcannot be useddirectly, itcanalsoimprove the situation (distilation and oscillations) • Usuallyitsdetectionrequires a big experimentaleffort.

  3. Derivation of Bell Inequalities with subcorrelations Problem: some N-qubitstates do not possesanycorrelationsbetweenallqubits, yetare N-partiteentangled (DK, AS(D), U(S), AW(?)) Genesis (theory): we act with the NOT gate on allqubits, and equally mix the result with the originalstate. This map changes the sign of allthreenontrivial Pauli Matrices, so the correlationsbetweenalloddnumbers of parties. NOT (timeinversion) isanantiunitarytransformation, but we act with itonly on purestates, soitdoesn’tgenerateanynonphisicality. Generation (experiment): we producespecific (N+1)-qubitstates and trace out one qubit. As anexample, we give the 5-qubit singletstate \ (M. Radmark, MW, M. Żukowski, M. Bourennane) i orDickestatemixtures (N. Kiesel et al., R. Predevel et al.)

  4. Werner-Wolf-Wienfurter-Żukowski-BruknerInequality Only one expression in pair Has modulo 1 for LHVT, the otherisvanishing. When we add a pair of suchevents, we can be left with only A1, or A2 in case of subtraction. In bothcases we get mirror tightinequalities. So in this place, somethingnew, for example a trivialmeasurementcan be substitued

  5. Poperties: • Weakerviolation: for five qubits of order of 2, ratherthan 2,82, as in case of standard Bell Inequalities. 1 iseasier to be simulated by bylocalrealism. • The inequalty from the lastslideis not violated. • We havefound the necessarycondition for the violation • Theseinequalitesarerelated to the problem of pairmatching on a hypercube. The matchiscounted as a way to connecttwovertices by anedge in such a way, thateachvertexbelong to at most one group. We don’tknow the number of suchmatchings, for three, therearetwosuch (imperfect) matchings • It ispossible to violatethisinequality with N-partiteentangledstates (evenwithout • N-partitecorrelations)

  6. QuadraticEntanglementCriteria Conditions for violating WWWŻB (and WNŻ) inequalities: Sum of observables: Difference of obsevables: Clauser-Horne-Shimony-Holtineq. In thislanguage: We nowapplythe Cauchy –Schwartz inequality and getthat the inequalitycan be violatediftherearesuchlocalcoordinate system that

  7. Non-negativity of th e probabilities Non-negavity of the state Ifoperatorsrelated to correlation tensor elementsa,b,canticomommute with one another, then Sketch of a proof For operators, whichmutuallycommute, choose one of theircommoneigenstates to maximizethe sum of squares of eigenvalues. Cut-anticommutativity: Operators A1xB1 and A2xB2 anticommute with respect to cutif we {A1,A2}=0 or {B1,B2}=0. In such a case, the sum of the meanvalues for statesseparable with respect to thiscutdoes not exceed 1.

  8. Convexity The function we want to maximizetakes form We shouldworryonlyaboutpurestates

  9. Trees and graphs of anticommutation Constructinganticommutativitygraphs Choose a set of correlation tensor elements. Representeach element by a vertex. Putanedgeif the twooperatorsanticommute. Assignvalues 0 or 1 to vertices. Two 1’s cannot be connected by the Sum of squaredmeanvaluesisequal to the independecenumber, i.e. the thelargestnumber we candistributeover the graph Repeatprocedure for allcuts.

  10. Case 1: CHSH 1 xx xx xy xy yy yy yx yx 1 1 • The methodis: • General-youcanconstructmanydifferentcriteria • Elastic-youcanfocus on specifickind of entanglement • Constructive-everycriteriondefines a set of statesthatmaximizeit. • Economical-we do not need to optimizeoverallstates. The complexityishidden in constructing and analyzinggraphs. • …

  11. … and (sometimes) minimal Case 3: clusterstate Case 2: GHZ

  12. Experiment In collaboration with Max-Planck Institute in Garching: C. Schwemmer, L. Knips, H. Weinfurter 2 photons-4 qubits (2 path, 2 polarization) Analyzer Analyzer PBS PBS f Type-II

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