ɨF TJOHMFU TUBUF JO &R òð JT KVTU POF PG NBOZ FYBNQMFT PG FOUBOHMFE TUBUFT TIBSJOH UIF TUBSUMJOH GFBUVSFT JMMVTUSBUFE JO UIF &13 QBSBEPY 8F OPX HJWF B QSPQFS EFmOJUJPO PG FO UBOHMFE TUBUFT BOE UIF DPNQMFNFOUBSZ TFU PG TUBUFT UIF TFQBSBCMF TUBUFT 8F BMTP QSFTFOU TPNF PG UIF NFUIPET UP EFUFSNJOF NBUIFNBUJDBMMZ XIFUIFS B DFSUBJO TUBUF WFDUPS PS EFOTJUZ NBUSJY JT FOUBOHMFE PS TFQBSBCMF
5P EFmOF FOUBOHMFNFOU BOE TFQBSBCJMJUZ XF mSTU OFFE UP TQFDJGZ PWFS XIBU QBSUJUJPO T
(a) (b) A B A B C A B C
'JHVSF B " CJQBSUJUF TZTUFN DPOTJTUJOH PG UXP QBSUJDMFT ZFMMPX EJTDT XJUI B QBSUJUJPO TFQBSBUJOH UIF UXP TVCTZTUFNT C 'PS B NVMUJQBSUJUF TZTUFN DPOTJTUJOH PG NPSF UIBO UXP QBSUJDMFT IFSF UISFF UIF TUBUFT NBZ CF FOUBOHMFE FJUIFS XJUI SFTQFDU UP POMZ POF QBSUJUJPO MFGU mHVSF PS TFWFSBMBMM QBSUJUJPOT SJHIU mHVSF *O UIJT UIFTJT XF NBJOMZ GPDVT PO CJQBSUJUF FOUBOHMFNFOU
B TUBUF JT FOUBOHMFETFQBSBCMF TFF 'JH òð " TUBUF NBZ CF FOUBOHMFE XJUI SFTQFDU UP POF QBSUJUJPO CVU OPU UP BOPUIFS 'PDVTJOH PO CJQBSUJUF FOUBOHMFNFOU 'JH òð B XF DPOTJEFS B TZTUFN QBSUJUJPOFE JOUP UXP TVCTZTUFNT " BOE # XJUI )JMCFSU TQBDFTH"BOEH# 8F TUSFTT UIBU BMM EFmOJUJPOT CFMPX DBO CF FBTJMZ HFOFSBMJ[FE UP NVMUJQBSUJUF FOUBOHMFNFOU XIFSF NPSF UIBO POF QBSUJUJPO JT JOWPMWFE TFF 'JH òð C
òñð 1VSF TUBUFT
*O UIF DBTF PG QVSF TUBUFT SFQSFTFOUFE CZ TUBUF WFDUPST|Ψ ∈ H"⊗ H# B TUBUF JT TFQBSBCMF JG BOE POMZ JG JU NBZ CF XSJUUFO JO B QSPEVDU GPSN
|Ψ = |Φ"⊗ |lj#, òõ
XIFSF|Φ" ∈ H"BOE|lj# ∈ H# BSF TUBUF WFDUPST PG TVCTZTUFNT " BOE # SFTQFDUJWFMZ "OZ QVSF TUBUF XIJDI JT OPU TFQBSBCMF BOE UIVT DBOOPU CF XSJUUFO BT JO &R òõ JT BO
FOUBOHMFE TUBUF
$POTJEFSJOH B NFBTVSFNFOU PG BO PCTFSWBCMF ˆ"a PO TVCTZTUFN " BOE BOPUIFS PCTFSWBCMF ˆ#bPO TVCTZTUFN # UIF FYQFDUBUJPO WBMVF GPS B DPNCJOFE NFBTVSFNFOU GBDUPSJ[FT
&(a, b) = Ψ|ˆ"a⊗ ˆ#b|Ψ = Φ|ˆ"a|Φ"lj|ˆ#b|lj#, òö GPS TFQBSBCMF TUBUFT JF MPDBM NFBTVSFNFOU PVUDPNFT BSF JOEFQFOEFOU .PTU JNQPSUBOUMZ UIJT JT OPU UIF DBTF GPS FOUBOHMFE TUBUFT GPS XIJDI DPSSFMBUJPOT o RVBOUVN DPSSFMBUJPOT o CFUXFFO UIF TVCTZTUFNT EP FYJTU
4DINJEU EFDPNQPTJUJPO
5P EFUFSNJOF JG B CJQBSUJUF TUBUF NBZ CF FYQSFTTFE JO B QSPEVDU GPSN PS OPU POF NBZ FNQMPZ UIF 4DINJEU EFDPNQPTJUJPO <ñ> PG B TUBUF WFDUPS
|Ψ = E J= αJ|ΦJ"⊗ |ljJ#, E J= α J = , ò÷ XIFSF{|ΦJ"}E" J=BOE{|ljJ#}E#
J=GPSN CBTFT GPSH"BOEH# XJUI E = min(E", E#) )FSF UIF 4DINJEU DPFċDJFOUTαJ ≥ BSF QPTJUJWF OVNCFST UIBU GVMMZ DIBSBDUFSJ[F UIF TFQBSBCJM
JUZFOUBOHMFNFOU PG B CJQBSUJUF TZTUFN *O QSBDUJDF UIF 4DINJEU EFDPNQPTJUJPO JT PCUBJOFE CZ XSJUJOH UIF TUBUF WFDUPS JO B NBUSJY GPSN BOE UIFO BQQMZJOH B TJOHMF WBMVF EFDPNQPTJUJPO ɨF OVNCFS PG OPO[FSP 4DINJEU DPFċDJFOUT JT DBMMFE UIF 4DINJEU SBOL PS UIF 4DINJEU OVNCFS 8IFOFWFS UIF 4DINJEU SBOL JT MBSHFS UIBO POF UIF TUBUF JT FOUBOHMFE *OUFSFTU JOHMZ UIF 4DINJEU SBOL JT SFMBUFE UP UIF QVSJUZ PG UIF SFEVDFE EFOTJUZ NBUSJY PG POF PG UIF TVCTZTUFNT Q(ˆρ") = US" (US#ρ)ˆ = Q(ˆρ#) = US# (US"ρ)ˆ = E J= α J ≤ . òø &RVBMJUZ IPMET JG BOE POMZ JG UIF 4DINJEU SBOL JT POF UIJT NFBOT UIBU B CJQBSUJUF QVSF TUBUF JT FOUBOHMFE JG BOE POMZ JG JUT SFEVDFE TUBUFT BSF NJYFE
#FMM TUBUFT
'PS B UXP RVCJU TZTUFN UIF GPVS #FMM TUBUFT BSF EFmOFE BT UIF FOUBOHMFE TUBUFT
|Ψ± = √
(| + − ± | − +) , |Φ± = √
(| + + ± | − −) , òðï PG XIJDI UIF TJOHMFU TUBUF JO &R òð DPSSFTQPOET UP|Ψ− ɨFJS SFEVDFE TUBUFT XIFO JO
UFHSBUJOH PVU UIF EFHSFFT PG GSFFEPN PG POF PG UIF TVCTZTUFNT BSF NBYJNBMMZ NJYFE ɨFTF TUBUFT BSF LOPXO BT NBYJNBMMZ FOUBOHMFE TUBUFT ɨFZ GPSN B LFZ JOHSFEJFOU JO TVQFSEFOTF DPEJOH <ö> RVBOUVN UFMFQPSUBUJPO <÷> BOE RVBOUVN DSZQUPHSBQIZ <ø> ɨVT NBOZ FOUBO HMFNFOU HFOFSBUJPO TDIFNFT BJN BU DSFBUJOH UIJT DMBTT PG TUBUFT
òññ .JYFE TUBUFT
ɨF EFmOJUJPO PG FOUBOHMFE QVSF TUBUFT JO UFSNT PG DPSSFMBUJPOT NBZ CF FYUFOEFE UP BMM NJYFE TUBUFT )PXFWFS TPNF DBSF IBT UP CF UBLFO TJODF NJYFE TUBUFT NBZ BMTP DPOUBJO DMBT TJDBM DPSSFMBUJPOT XIJDI BSF OPU PXJOH UP FOUBOHMFNFOU 8F UIFSFGPSF IBT UP EJTUJOHVJTI
CFUXFFO DMBTTJDBM DPSSFMBUJPOT BOE RVBOUVN DPSSFMBUJPOT $MBTTJDBM DPSSFMBUJPOT BSF DPSSFMB UJPOT XIJDI DBO CF QSPEVDFE GSPN MPDBM PQFSBUJPOT BOE DMBTTJDBM DPNNVOJDBUJPO -0$$ ɨJT JODMVEFT BMM PQFSBUJPOT UIBU NBZ CF QFSGPSNFE MPDBMMZ PO FBDI TVCTZTUFN FJUIFS JOEF QFOEFOUMZ PS DPSSFMBUFE WJB DMBTTJDBM DPNNVOJDBUJPO XJUI UIF PUIFS TVCTZTUFN ɨFSF JT OP TJNQMF FYQSFTTJPO GPS QBSBNFUSJ[JOH -0$$ PQFSBUJPOT JO HFOFSBM CVU UIFZ NBZ BMM CF XSJUUFO BT B QSPEVDU PG MPDBM PQFSBUJPOT
ˆ7 = ˆ7"⊗ ˆ7#, òðð
XIFSF ˆ7"BOE ˆ7#BDU POMZ PO TVCTZTUFN " BOE # SFTQFDUJWFMZ 4VSQSJTJOHMZ UIF DPOWFSTF JT OPU USVF OPU BMM QSPEVDUT PG MPDBM PQFSBUJPOT DBO CF JNQMFNFOUFE BT -0$$ PQFSBUJPOT <øñ> "O JNQPSUBOU DMBTT PG -0$$ PQFSBUJPOT BSF UIF MPDBM VOJUBSZ PQFSBUJPOT
ˆ
6 = ˆ6"⊗ ˆ6#, òðñ
XIFSF ˆ6"BOE ˆ6#BSF CPUI VOJUBSZ PQFSBUPST BDUJOH PO FBDI TVCTZTUFN ɨFTF DPSSFTQPOE UP B DIBOHF PG CBTJT BU FBDI TVCTZTUFN XIJDI EP OPU DIBOHF UIF BNPVOU PG FOUBOHMFNFOU " CJQBSUJUF NJYFE TUBUF XIJDI POMZ DPOUBJOT DMBTTJDBM DPSSFMBUJPOT NBZ CF XSJUUFO BT
ˆ ρ = J QJρˆ(J)" ⊗ ˆρ(J)# , J QJ = . òðò
ɨJT DPSSFTQPOET UP BO JODPIFSFOU NJYUVSF XIFSF XF XJUI QSPCBCJMJUZ QJmOE TVCTZTUFN " JO UIF TUBUF ˆρ(J)" BOE TVCTZTUFN # JO UIF TUBUF ˆρ(J)# 0OMZ JG QJ = GPS BMM J FYDFQU POF UIJT TUBUF SFEVDFT UP B QSPEVDU TUBUF GPS XIJDI OP DPSSFMBUJPOT CFUXFFO UIF TVCTZTUFNT FYJTU BU BMM *U GPMMPXT GSPN &R òðò UIBU UIF TFU PG TFQBSBCMF TUBUFT JT DPOWFY B QSPQFSUZ UIBU XJMM CF QBSUJDVMBSMZ JNQPSUBOU XIFO XF EJTDVTT FOUBOHMFNFOU XJUOFTTFT
8F OPUF UIBU UIF FYQFDUBUJPO WBMVF PG UXP MPDBM NFBTVSFNFOUT GPS B TFQBSBCMF TUBUF JT HJWFO CZ &(a, b) = Ψ|ˆ"a⊗ ˆ#b|Ψ = J QJUS ˆ"aρˆ(J)" US ˆ#bρˆ(J)# . òðó ɨJT FYQFDUBUJPO WBMVF DBO BMXBZT CF NJNJDLFE CZ POF PCUBJOFE GSPN B MPDBM IJEEFO WBSJBCMF UIFPSZ <TFF &R òò>
8F EFmOF BOZ TUBUF XIJDI DBOOPU CF XSJUUFO JO UIF GPSN HJWFO CZ &R òðò UIVT DPOUBJO JOH RVBOUVN DPSSFMBUJPOT BT FOUBOHMFE 2VJUF TVSQSJTJOHMZ OPU BMM FOUBOHMFE NJYFE TUBUFT WJPMBUF UIF QSJODJQMF PG MPDBM SFBMJTN 0OF FYBNQMF JT UIF 8FSOFS TUBUFT <øò>
ˆ
ρ8 = λˆ*/ + ( − λ)|Ψ−Ψ−|, òðô XIJDI NBZ NPEFM B TJOHMFU TUBUF UIBU JT FYQPTFE UP OPJTF ɨFZ BSF FOUBOHMFE GPS ≤ λ <
/ CVU WJPMBUFT MPDBM SFBMJTN POMZ GPS ≤ λ < /√
1FSFTo)PSPEFDLJ DSJUFSJPO
%FUFSNJOJOH XIFUIFS B NJYFE TUBUF JT TFQBSBCMF PS FOUBOHMFE JT BO JODSFBTJOHMZ DIBMMFOHJOH UBTL BT UIF EJNFOTJPOT PG B )JMCFSU TQBDF JODSFBTFT *O HFOFSBM UIFSF JT OP LOPXO FċDJFOU NFUIPE UP EFUFSNJOF JG B TUBUF JT FOUBOHMFE )PXFWFS GPS UXPRVCJU TZTUFNT XIJDI BSF UIF SFMFWBOU TZTUFNT GPS 1BQFST * BOE ** UIFSF JT B TJNQMF BOE FċDJFOU NFUIPE LOPXO BT UIF 1FSFTo)PSPEFDLJ DSJUFSJPO <øó> ɨJT DSJUFSJPO QSFTDSJCFT UIBU B TUBUF JT TFQBSBCMF JG BOE POMZ JUT QBSUJBM USBOTQPTF
ˆ ρ5" = ⎛ ⎝ JKLM DLMJK|J"|L#M|#K|" ⎞ ⎠ 5" = JKLM DLMKJ|J"|K#L|#M|" òðõ JT BMTP B QIZTJDBM TUBUF JF B QPTJUJWF PQFSBUPS ɨJT TUBUFNFOU JT DMFBS GSPN &R òðò ɨVT BOZ TUBUF QSPEVDJOH BO VOQIZTJDBM TUBUF VOEFS UIF QBSUJBM USBOTQPTF NVTU CF FOUBOHMFE 'PS TZTUFNT XJUI EJNFOTJPOT Y PS Y UIF 1FSFTo)PSPEFDLJ DSJUFSJPO JT B OFDFTTBSZ BOE TVċDJFOU DPOEJUJPO GPS FOUBOHMFNFOU )PXFWFS GPS MBSHFS EJNFOTJPOT UIF DSJUFSJPO JT POMZ B TVċDJFOU DPOEJUJPO JO UIBU DBTF UIFSF BSF FOUBOHMFE TUBUFT XIJDI NBZ ZJFME QIZTJDBM TUBUFT VOEFS UIF QBSUJBM USBOTQPTF <øô>