χ型纠缠相干态的产生及贝尔型不等式的研究

χ型纠缠相干态的产生及贝尔型不等式的研

究

宋思谕1,2,王书浩1,2,龙桂鲁1,2

1清华大学物理学院低维量子物理国家重点实验室，北京　100084

2清华大学清华信息科学技术国家实验室，北京　100084

摘要：我们提出了一种新的纠缠相干态-χ型纠缠相干态的产生方案。这个过程是通过克尔介质和线性光学器件来实现的。基于光子对称性测量和光子阈值测量，我们研究了χ型纠缠相干态的量子非局域性。对于贝尔型不等式，它显示出强的违背。

关键词：纠缠相干态，态产生，贝尔型不等式

中图分类号：O43

Generationoftheχ-typeentangledcoherentstateandviolationsofBell-typeinequality

SiYuSong1,2,ShuhaoWang1,2,GuiLuLong1,2

1StateKeyLaboratoryofLow-DimensionalQuantumPhysicsandDepartmentofPhysics,

TsinghuaUniversity,Beijing,100084

2TsinghuaNationalLaboratoryForInformationScienceandTechnology,Tsinghua

University,Beijing,100084

Abstract:Weproposeaprotocoltogenerateanewtypeofstate,calledtheχ-typeentangledcoherentstate.Theprocessisrealizedbyusingoneπ-cross-Kerrmediumandlinearopticalcomponents.Thequantumnon-localityoftheχ-typeentangledcoherentstateisinvestigatedbasedonphoton-parityandphoton-thresholdmeasurements.WeshowthatstrongerviolationsofBell-typeinequalityappearintheχ-typeentangledstate.Keywords:entangledcoherentstates,stategeneration,Bell-typeinequality0INTRODUCTION

Entanglementisakeyfeatureinquantummechanics,whichcontributestothenon-localandnon-classicalbehaviorofquantumsystems.Entanglementhasgreatpotentialinquantuminformationprocessing,includingquantumcryptography[1,2,3,4],superdosecoding[5,6,7],基金项目：NationalNaturalScienceFoundation(11175094)，NationalBasicResearchProgram(2009CB929402),Na-tionalBasicResearchProgram(2011CB9216002),theSRFDPunderGrant([1**********]007)

作者简介：SongSiYu(1986-)，female，PHD，majorresearchdirection：quantuminformation.Correspondenceauthor：LongGuiLu(1962-)，male，professor，majorresearchdirection：quantuminformation.

quantumteleportation[8,9,10,11,12,18,19,20,17,22,13,14,21,16,15],andquantumcomputation[28,23,24,25,26,27].TheBellinequalitywasfirstproposedin1964[29],andviolationsofthisinequalitycanberegardasevidenceofentanglementintwo-qubitsystems

[30].TheBell-Mermaninequality[31]isthemulti-qubitversionoftheBellinequalityinRef.

[29].Thisinequalityisbasedontheassumptionthateverypartyisallowedtochoosebetweentwomeasurementsettings.TheBell-typeinequalityforcontinuousvariableentangledstatesoftwo-modeelectromagneticfieldsaregivenbyBanaszekandWódkiewicz,whoshowthatsuchinequalitycanbeobtainedthroughaWignerdistributionfunction[32].Themulti-modecontinuousvariablestateshavebeenthoroughlyinvestigatedthereafter[33,26].

Entangledphotonsareimportantinformationcarriers.Thediscretedegreeoffreedomsofphotons,forinstance,thepolarization,aretypicallyusedforinformationprocessing.Suchentangledphotonscanbegeneratedthroughparametricdownconversion.However,anotherrealizationofinformationcarriersisfacilitatedbycoherentsuperposedstates(CSSs)

|CSS⟩Even/Odd=N(|α⟩±|−α⟩),(1)

wherethesubscriptEven/OddcorrespondstotheevenandoddCSSs,andNisanormalizationconstantequaltoN=[2(1+e−2|α|)]−.Theencodingmethodofentangledcoherentstates(ECSs)[23]usingCSSsasqubitsissimilartothatofentangleddiscretephotonsystem.Themethodhastheadvantageofbeingrobustagainstdecoherence[34],thusthestatehasbeenwidelyused[18,19,20,22,21,23,24,25,26].

Thecoherentstatesaretheeigenstatesoftheannihilationoperatoraˆ,i.e.,aˆ|α⟩=α|α⟩,whereαistheeigenvaluecorrespondingto|α⟩,whichiscomplex.Thecoherentstatescanbeexpressedas∞∑αnˆ(α)|0⟩|n⟩=D(2)|α⟩=en=0√ˆ(α)=withthenumberstateis|n⟩=[(ˆa†)n/|0⟩,andthedisplacementoperatorisD−|α|221exp(αaˆ†−α∗aˆ).Thecoherentstates|α⟩and|−α⟩areencodedas0and1,respectively,whereweassumethatαisreal.Giventhat|⟨0|1⟩|2=|⟨α|−α⟩|2=exp(−4|α|2),thecoherentstatesareorthogonalwhenαissufficientlylarge.

Ifthemultiplesofannihilationoperatorsareinvolvedinthecoherentstate,suchstateiscalledamulti-modecoherentstate.Multi-modeECSshaveseveraltypes:theGHZ-typeECS

[19,20,26,21],theW-typeECS[22,26,35,36],thecluster-typeECS[37],etc.TheseECSscanbeconstructedbyusingbeamsplitters(BSs),phaseshifters(PSs),andnonlinearKerrmedia[26,37],whichhavebeenusedinnumerousproposedschemes.