òô &OUBOHMFNFOU EFUFDUJPO JO OBOPTDBMF TZTUFNT
FMM UFTU GPS PSCJUBM FOUBOHMFNFOU JO B RVBOUVN EPU TZTUFN
8F OPX EFNPOTUSBUF IPX UP FYUSBDU UIF [FSPGSFRVFODZ DVSSFOU DSPTTDPSSFMBUJPOT GSPN UIF RVBOUVN EPU TZTUFN EJTDVTTFE JO 1BQFS * BOE VTF UIFN UP GPSNVMBUF B #FMM JOFRVBMJUZ 5P UIJT FOE XF mSTU JOUSPEVDF DPVOUJOH mFMETχγ UIBU DPVQMF UP UIF OFU QBSUJDMF DVSSFOU UP FBDI MFBEγ ZJFMEJOH UIF DPVOUJOHmFME SFTPMWFE -JOECMBE FRVBUJPO <DG &R òñõ>
Eˆρ(U) EU = −J [ ˆ)FĊ, ˆρ(U)] + γ Γγ GγD−χγ[ˆE†
γ, ˆρ(U)] + ( − Gγ)Dχγ[ˆEγ, ˆρ(U)]
+Γϕ
D[ˆE†γEγ, ˆρ(U)]
, òóð
XIFSFDχ[ˆ-, ˆρ] = FJχˆ-ˆρˆ-†−
{ˆ-†ˆ-, ˆρ} JT UIF DPVOUJOHmFME SFTPMWFE EJTTJQBUPS
6TJOH &R ñóó XF PCUBJO UIF DVNVMBOU HFOFSBUJOH GVODUJPO PG UIF MPOHUJNF TUBUJTUJDT GSPN &R òóð CZ FYUSBDUJOH UIF FJHFOWBMVF UIBU HPFT UP [FSP XIFO UIF [FSP DPVOUJOH mFMET HP UP [FSP
'χ=
αβ
(FJ(χα+χβ−χ−χ)− )1αβ òóñ
UP MFBEJOH PSEFS JO UIF UVOOFMJOH BNQMJUVEFT )FSFα = "+, "− BOE β = #+, #− BT
CFGPSF ɨF USBOTGFS SBUFT BSF HJWFO CZ
1αβ = |Uβα|(Γα+ Γβ+ Γϕ)
(Γα+ Γβ+ Γϕ)+ ε
αβ
, òóò
XJUI UIF FOFSHZ EJĊFSFODFεαβ ≡ εα+εβ+6αβ−(ε+ε+6) GSPN UXPQBSUJDMF SFT POBODF ɨF FYQSFTTJPO JO &R òóñ DPSSFTQPOET UP GPVS EJĊFSFOU DPUVOOFMJOH QSPDFTTFT XIFSF UIF UXP FMFDUSPOT JO UIF FOUBOHMFS EPUT BSF USBOTGFSSFE UP UIF GPVS EJĊFSFOU DPNCJO BUJPOT PG EFUFDUPS EPUT &BDI USBOTGFS QSPDFTT JT 1PJTTPOJBO BOE UBLFT QMBDF BU B SBUF TFU CZ
1αβ 1αβNBZ CF JOUFSQSFUFE BT UIF VOOPSNBMJ[FE KPJOU EFUFDUJPO QSPCBCJMJUJFT PG IBWJOH POF FMFDUSPO USBOTGFSSFE UP EPUα BOE BOPUIFS UP EPU β
.PTU JNQPSUBOUMZ UIFTF RVBOUJUJFT NBZ CF FYUSBDUFE GSPN [FSPGSFRVFODZ DVSSFOU DSPTT DPSSFMBUJPOT ɨJT DBO CF TFFO CZ DBMDVMBUJOH UIF DSPTTDPSSFMBUJPOT BT EFSJWBUJWFT PG òóñ XJUI SFTQFDU UP EJĊFSFOU DPVOUJOH mFMET ZJFMEJOH
4αβ = F1αβ = F ∂'χ ∂(Jχα)∂(Jχβ)
χα=χβ=. òóó 'SPN UIJT FYQSFTTJPO XF DBO BMTP PCUBJO UIF DPSSFMBUJPO GVODUJPO JO &R òóï XIFSF XF TFUζ" = ζ# = TJODF BMM DIBSHF USBOTGFS FWFOUT BSF BTTVNFE UP CF QFSGFDUMZ EFUFDUFE GPS UIF RVBOUVN EPU TFUVQ
8F SFDBMM UIBU UIF EFUFDUPS TFUUJOHT BSF EFUFSNJOFE CZ UIF UVOOFMJOH DPVQMJOHT CFUXFFO UIF FOUBOHMFS BOE UIF EFUFDUPS EPUT #Z UVOJOH UIFN XJUI FMFDUSPTUBUJD TJEF HBUFT POF NBZ DIPPTF EJĊFSFOU TFUUJOHT UP QFSGPSN B #FMM UFTU )PXFWFS UIJT UVOJOH NVTU CF NBEF JO TVDI B XBZ UIBU UIF FNJUUFE TUBUF <TFF &R òñô> JT OPU BMUFSFE 5P UIJT FOE XF BTTVNF UIBU UIF UVOJOH PG UIF UVOOFMJOH NBZ CF QBSBNFUSJ[FE XJUI BO BOHMFθ"BT
U"+= U"cos θ", U"− = U"cos θ"
U"−= U"sin θ", U"+= −U"sin θ", òóô XIFSF U
"= U
"++U
"−JT UIF UPUBM QSPCBCJMJUZ GPS BO FMFDUSPO JO UIF VQQFS FOUBOHMFS EPU UP UVOOFM UP EFUFDUPS TZTUFN " BOE U" = U
"++ U
"−JT UIF DPSSFTQPOEJOH QSPCBCJMJUZ GPS BO FMFDUSPO JO UIF MPXFS FOUBOHMFS EPU " TJNJMBS QBSBNFUSJ[BUJPO JT BTTVNFE UP IPME BU EFUFDUPS TZTUFN # 6OEFS UIJT SPUBUJPO PG UIF UVOOFMJOH SBUFT GPS XIJDI|U"+/U"−| = |U"−/U"+|
BOE|U#+/U#−| = |U#−/U#+| UIF FNJUUFE TUBUF JT UIF TBNF .PSF TQFDJmDBMMZ UIF DPFG
mDJFOUT EFmOJOH UIF FNJUUFE TUBUF JO &R òñô BSF HJWFO CZ D ∝ U"U#BOE D ∝ U"U#
<TFF &R òñô>
*OTFSUJOH UIFTF QBSBNFUSJ[BUJPOT JO &R òóò BOE VTJOH &R òòø XF PCUBJO UIF NBYJNBM WBMVF PG UIF #FMM QBSBNFUFS UIBU POF DBO PCUBJO GPS UIF FNJUUFE TUBUF
BNBY =
+ sinθ, òóõ
XIFSFθ = arctan(D/D) −Ż ≤ θ ≤ Ż ɨJT WBMVF PG UIF #FMM QBSBNFUFS JT DMFBSMZ MBSHFS UIBO VOMFTT D PS D JT [FSP ɨJT EFNPOTUSBUFT UIBU UIF JOFRVBMJUZ JO &R òòø DBO CF WJPMBUFE o BU MFBTU JO QSJODJQMF o XIFOFWFS UIF FNJUUFE TUBUF JT FOUBOHMFE 8F IBWF UIVT TIPXFE IPX UIF FNJUUFE TUBUF DBO CF EFUFDUFE VTJOH [FSPGSFRVFODZ DVSSFOU DSPTT DPSSFMBUJPOT ɨBU TBJE JU JT TUJMM FYQFSJNFOUBMMZ IJHIMZ DIBMMFOHJOH UP QFSGPSN B #FMM UFTU ɨF NBJO SFBTPO JT UIBU UIF UVOJOH PG UIF UVOOFMJOH DPVQMJOHT IBWF UP CF NBEF JO B WFSZ DPOUSPMMBCMF XBZ *O SFBMJUZ UIF DPOUSPM PG UIF UVOOFMJOH DPVQMJOHT NBZ CF WFSZ MJNJUFE BOE JU DBO CF DIBMMFOHJOH UP UVOF FH POF PG UIF DPVQMJOHT XJUIPVU BĊFDUJOH UIF PUIFST 0OF XPVME UIFSFGPSF JOTUFBE MJLF UP VTF B EFUFDUJPO TDIFNF CBTFE PO B TNBMMFS TFU PG TFUUJOHT BOE NFBTVSFNFOUT TVDI BT FOUBOHMFNFOU XJUOFTTFT
òôñ &OUBOHMFNFOU XJUOFTTFT
"MUIPVHI UIF BCPWF DBMDVMBUJPOT TIPX UIBU B #FMM UFTU NBZ o BU MFBTU JO QSJODJQMF o CF QFS GPSNFE JO PVS RVBOUVN EPU TZTUFN SFBMJ[JOH TVDI B UFTU TUJMM SFNBJOT B CJH IVSEMF UP PWFS DPNF "T BMSFBEZ QPJOUFE PVU B #FMM UFTU SFRVJSFT ðõ EJĊFSFOU DSPTTDPSSFMBUJPO NFBTVSF NFOUT JO GPVS EJĊFSFOU CBTFT #PUI GPS UIF UVOOFMJOH DPVQMJOHT VTFE JO UIF RVBOUVN EPU TZTUFN BT XFMM BT GPS PUIFS OBOPTDBMF EFUFDUJPO TDIFNFT TVDI BT UIPTF CBTFE PO GFSSPNBH OFUJD MFBET GPS TQJO FOUBOHMFNFOU EFUFDUJPO JU JT DIBMMFOHJOH UP GVMMZ DPOUSPM UIF EFUFDUPS TFUUJOHT ɨJT NBLFT JU EJċDVMU JO QSBDUJDF UP SFBMJ[F BMM UIF SJHIU TFUUJOHT UIBU BSF OFFEFE GPS B #FMM UFTU *U JT UIFSFGPSF EFTJSBCMF UP VTF BMUFSOBUJWF EFUFDUJPO TDIFNFT XIFSF UIF SFRVJSFE OVNCFS PG NFBTVSFNFOUT JO QBSUJDVMBS UIF OVNCFS PG TFUUJOHT DBO CF SFEVDFE
0OF TVDI LJOE PG EFUFDUJPO TDIFNFT JT CBTFE PO FOUBOHMFNFOU XJUOFTTFT <øô ðñô> XIJDI BSF UIF NBJO PCKFDUT PG 1BQFS ** ɨF JEFB CFIJOE XJUOFTTFT JT UP VTF BOZ PCTFSWBCMF XIJDI JT DBQBCMF PG EJTUJOHVJTIJOH FOUBOHMFE TUBUFT GSPN UIF TFQBSBCMF POFT UP EFUFDU FOUBOHMFNFOU .PSF QSFDJTFMZ JG BO PCTFSWBCMF GPS BU MFBTU POF FOUBOHMFE TUBUF QSPEVDFT BO FYQFDUBUJPO WBMVF UIBU OP TFQBSBCMF TUBUF DBO ZJFME JU DBO CF VTFE BT B XJUOFTT GPS UIBU FOUBOHMFE TUBUF .BUIFNBUJDBMMZ XF EFmOF B XJUOFTT PQFSBUPSW BT BOZ PQFSBUPS GVMmMMJOH
W = US{W ˆρ} > max
ˆρT
US{W ˆρT} PS W = US{W ˆρ} < min
ˆρT
US{W ˆρT} òóö GPS BU MFBTU POF FOUBOHMFE TUBUF ˆρ XIFSF UIF NJOJNJ[BUJPONBYJNJ[BUJPO JT QFSGPSNFE PWFS
BMM TFQBSBCMF TUBUFT ˆρT ɨF mSTU DPOEJUJPO XF DBMM UIF wVQQFS MJNJU DPOEJUJPOw BOE UIF TFDPOE DPOEJUJPO UIF wMPXFS MJNJU DPOEJUJPOw *O UIF MJUFSBUVSF UIFTF DPOEJUJPOT BSF PGUFO SFOPSNBMJ[FE UP
US{W ˆρ} < BOE US{W ˆρT} ≥ ∀ˆρT. òó÷ )PXFWFS IFSF XF XJMM GPDVT PO UIF BDUVBM FYQFSJNFOUBMMZ NFBTVSFE RVBOUJUJFT BOE UIVT TUJDL UP UIF DPOEJUJPOT JO &R òóö
W W A B Separable states Entangled states Detectable states
'JHVSF 5IF TFU PG BMM TUBUFT EFOTJUZ NBUSJDFT JT DPOWFY 5IJT TFU JT NBEF VQ PG UIF DPOWFY TVCTFU PG TFQBSBCMF TUBUFT ZFMMPX BOE UIF OPODPOWFY TVCTFU PG FOUBOHMFE TUBUFT HSFFO "O FOUBOHMFNFOU XJUOFTT NBZ CF SFQSFTFOUFE BT B IZQFSQMBOF TFQBSBUJOH UIF FOUBOHMFE TUBUFT UIBU BSF EFUFDUBCMF GSPN UIF TFQBSBCMF TUBUFT )FSF 8"JT B XJUOFTT TVQFSJPS UP UIF OPOPQUJNBM 8#
8F OPUF UIBU UIF #FMM QBSBNFUFS JO &R òòø JUTFMG NBZ CF TFFO BT B XJUOFTT *U EFUFDUT BMM FOUBOHMFE TUBUFT UIBU ZJFME B MBSHFS WBMVF UIBOB = &OUBOHMFNFOU XJUOFTTFT BSF UIVT B
NPSF HFOFSBM DMBTT PG EFUFDUJPO TDIFNFT UIBO UIF #FMM UFTUT "OPUIFS FYBNQMF PG B XJUOFTT JTW = σY⊗ σY+ σZ ⊗ σZ ɨF HSFBU CFOFmU PG FOUBOHMFNFOU XJUOFTTFT DPNQBSFE UP #FMM UFTUT JT JOEFFE UIFJS HFOFSBMJUZ 'JSTU JO DPOUSBTU UP #FMM JOFRVBMJUJFT UIFSF JT B XJUOFTT GPS FWFSZ FOUBOHMFE TUBUF <øô> UIBU DBO EFUFDU JU ɨJT GBDU JT JMMVTUSBUFE JO 'JH òô GPS FWFSZ FOUBOHMFE TUBUF UIFSF JT B XJUOFTT SFQSFTFOUFE CZ B IZQFSQMBOF UIBU TFQBSBUFT JU GSPN UIF DPOWFY TFU PG TFQBSBCMF TUBUFT 4FDPOE XJUOFTTFT BMMPXT GPS EFUFDUJPO TDIFNFT UIBU SFRVJSF NVDI GFXFS NFBTVSFNFOUT <ðñõoðñ÷> UIBO DPOWFOUJPOBM #FMM UFTUT ɨJT IPMET USVF FWFO GPS NVMUJQBSUJUF FOUBOHMFNFOU <ðñø>
.JOJNBM FOUBOHMFNFOU XJUOFTTFT CBTFE PO [FSPGSFRVFODZ DSPTTDPSSFMBUJPOT
8F OPX BEESFTT IPX FOUBOHMFNFOU XJUOFTTFT DBO CF VTFE UP NJOJNJ[F UIF OVNCFS PG [FSP GSFRVFODZ DVSSFOU DSPTTDPSSFMBUJPOT OFFEFE UP EFUFDU FOUBOHMFNFOU UIF UPQJD PG 1BQFS ** *O QSFWJPVT XPSLT FOUBOHMFNFOU XJUOFTTFT IBWF CFFO GPSNVMBUFE JO UFSNT PG [FSPGSFRVFODZ DSPTTDPSSFMBUJPOT XJUI UIF BJN PG EFDSFBTJOH UIF OVNCFS PG NFBTVSFNFOUT OFFEFE UP XJU OFTT FOUBOHMFNFOU JO TPMJETUBUF TZTUFNT <òðoòó> )FSF XF XJMM QSFTFOU UIF JEFB CFIJOE 1BQFS ** UIBU JT IPX UP mOE UIF NJOJNBM OVNCFS PG NFBTVSFNFOUT OFFEFE UP EFUFDU FOUBOHMF NFOU
#BTFE PO &R òòô XF TUBSU CZ XSJUJOH EPXO BO FYQSFTTJPO GPS BO PCTFSWBCMF DPSSFTQPOEJOH UP BO BSCJUSBSZ OVNCFS / DVSSFOU DSPTTDPSSFMBUJPOT VQ UP VOJONQPSUBOU QSFGBDUPS PGFΓ CFUXFFO MFBE "+ BOE MFBE #+
W =
/
J=
( + ζ"aJ· σ) ⊗ ( + ζ#bJ· σ). òóø )FSF aJ BOE bJ EFOPUF UIF QPMBSJ[BUJPO WFDUPST VTFE BT UIF JUI NFBTVSFNFOU TFUUJOH BU EFUFDUPS " BOE # SFTQFDUJWFMZ 8F BSF BTTVNJOH UIBU UIF EFUFDUPS FċDJFODJFTζ" BOEζ#
BSF UIF TBNF UISPVHIPVU POF BOE UIF TBNF FYQFSJNFOU ɨF XPSL PG 1BQFS ** GPDVTFT PO
JOWFTUJHBUJOH XIBU UIF TNBMMFTU QPTTJCMF OVNCFS / JT UIBU NBLFTW B XJUOFTT BOE XIBU
FOUBOHMFE TUBUFT BSF EFUFDUBCMF GPS UIBU /