CS286.2 Lecture 14: Quantum de Finetti Theorems II

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1 CS286.2 Lecure 14: Quanum de Fne Theorems II Scrbe: Mara Okounkova 1 Saemen of he heorem Recall he las saemen of he quanum de Fne heorem from he prevous lecure. Theorem 1 Quanum de Fne). Le ρ Dens C 2 ) n), k n and n k. Then here exss an m such ha f for any [n] k, j [n] m and x [d 8 ] m, ρ j,x s he pos-measuremen sae on quds... k obaned afer measurng j... j m usng Λ and obanng oucomes x 1... x m, we have E E j,x ρ j,x ρ j,x... ρ j,x k 1 4 lnd)18d)k k 2. 1) Ths saemen s dfferen from he prevous saemen of he quanum de Fne heorem as ρ s no assumed o be permuaon nvaran hs s replaced by he fac ha he ndces j ha are measured, and he ndces ha are kep, are dsrbued unformly n [n] subjec o beng dsnc ndces), he expecaon over x s aken ousde f we move nsde, we oban a convex combnaon of producs as n he prevous de Fne, bu akng he expecaon value ousde makes he saemen sronger), and because he bound has a larger dependence on he dmenson d k nsead of he prevous d). Ths las pon s he man drawback of he heorem and s drecly due o he use of he nformaonally complee measuremen Λ s an open problem wheher hs can be avoded). Theorem 1 has a classcal analogue, wh he same defnons of n, m, k, and : Theorem 2 Classcal Analogue). Le P be a dsrbuon on S n for some se S of sze d. Le = 1... k, j = j1... j m, and x = x 1... x m be sampled accordng o he margnal P j. Le P j,x be he margnal on condoned on he varables ndexed by j beng equal o he values specfed by x. Then E E j,x P j,x 2 Classcal = Quanum P j,x... P j,x k 1 2k2 ln d. 2) Our pah now s o prove ha he quanum saemen follows from he classcal saemen, and hen prove he classcal saemen, and fnally o apply hs o QPCP. Proof. Consder he dsrbuon P obaned from measurng all quds n ρ usng Λ we assume ha Λ s nformaonally complee wh dsoron 18d) k/2, as before, and s measured as a ensor produc across all qus). By he classcal saemen of he heorem, we have, 1

2 E E j,x P j,x P j,x... P j,x k 1 2k2 lnd 8 ), snce P s a dsrbuon on he se [d 8 ] n of all possble oucomes of Λ. Now observe ha by defnon P j,x = Λ k ρ j,x ) and P j,x l = Λρ j,x l ) for l = 1,..., k. Hence from he above we ge E E j,x Λ k ρ j,x ) Λρ j,x )... Λρ j,x k ) 1 2k2 lnd 8 ) Usng he fac ha Λ k s nformaonally complee wh dsorson 18d) k/2 yelds he quanum heorem. 3 Proof of Classcal Saemen Now our am s o prove he classcal analogue of he heorem, whch we wll do usng an nformaon heorec proof. Frs, we wll defne varous nformaon heorec quanes boh classcal and quanum) o help us n he effor. Le ρ AB DensC d C d ) be a densy marx, wh reduced saes ρ A and ρ B. Defnon 3. The enropy of a sae ρ A s defned as Sρ A ) = Trρ A ln ρ A ) = Sλρ A )), 3) where λρ A ) are he egenvalues of ρ A hese form a dsrbuon snce ρ A s a densy marx). Noe ha f ρ AB = ψ ψ s pure hen he egenvalues of ρ A are he squares of Schmd coeffcens n he decomposon of ψ across he A : B paron. Defnon 4. The muual nformaon IA : B) ρ s defned as IA : B) ρ = Sρ A ) + Sρ B ) Sρ AB ). 4) We wll also use condonal muual nformaon. If ρ ABX s a qqc sae,.e. ρ ABX = x p x ρ x AB x x for some dsrbuon p x and densy marces ρ x AB, hen IA : B X) ρ ABX = x p x IA : B) ρ x AB = SA X) + SB X) SAB X) where SA X) = SAX) SX). Example 5. If ρ = ρ A ρ B, hen Sρ A ρ B ) = Sρ A ) + Sρ B ) and so IA : B) ρ = 0. In oher words, n a produc sae, neher subsysem carres nformaon abou he oher. Example 6. On he oher hand, f ρ = ψ ψ s pure, hen Sρ) = 0 and IA : B) = 2Sρ A ). In parcular, s ψ = 1 d s he maxmally enangled sae, hen IA : B) = 2 ln d. Clam 7. In fac hese wo exreme examples saurae he followng nequales, whch always hold: 0 IA : B) 2 mnln d, ln d ) 5) One key propery of muual nformaon s gven by Pnsker s nequaly. 2

3 Clam 8. For any bpare sae ρ AB, IA : B) ρ 1 2 ρ AB ρ A ρ B ) Noe ha, f IA : B) = 0, hen Pnsker s nequales mples ha he sae mus be a ensor produc. Muual nformaon has several useful properes. Proposon 9. Muual nformaon sasfes he chan rule and monooncy under measuremens M A on A, as well as racng ou: IA : BX) = IA : X) + IA : B X) 7) IA : B) MA Id B ρ AB ) IA : B) ρab 8) IA : B) TrC ρ ABC ) IAC : B) ρ ABC. 9) Monooncy s nuve: gven a sysem A, we canno ncrease he nformaon ha A conans abou B by performng an operaon on A alone. In he classcal case s easy o prove, bu n he quanum case requres much more work! We wll ake hese properes for graned here. We are ready o prove he classcal saemen of he de Fne heorem. For smplcy we wll do he proof for k = 2, bu he proof for general k s vrually he same. We show he followng: Clam 10. Under he same noaon as Theorem 2, E 0 m< E =1, 2 ), j=j1...j m ) IX 1 : X 2 X j1... X jm ) 1 2 E IX : X ) 10) where X... X n ) are random varables wh dsrbuon P and s shorhand for [n]\{}. Frs le s show ha he clam leads o he classcal heorem. Proof of Theorem 2. Snce X ranges over a se of sze d we have IX : X ) 2 ln d. Applyng Pnsker s nequaly, So we oban as desred. P jx, 2 P jx P jx IX : X 1 X j1... X jm ). E m E j E x P jx, 2 P jx P jx ln d, Now le s prove he clam. Proof of Clam 10. By monooncy, and hen by he chan rule, we have 1 E IX : X ) 1 E,j 1,...j IX : X j1... X j ) where he las lne s a smple relabelng of coordnaes. = 1 1 E,j 1,...j IX : X jm+1 X j1... X jm ) m=0 = E 0 m< E =1, 2 ), j=j1...j m ) IX 1 : X 2 X j1... X jm ), 3

4 4 Applcaon o QPCP Now we apply hs saemen of he quanum de Fne heorem o he quanum PCP conjecure o show he followng. Theorem 11. Le H be a 2-local Hamlonan on quds such ha he neracon graph has degree exacly D. Then here exss a produc sae ψ = ψ 1... ψ n such ha d ψ H ψ λ 0 H) + 2 ) 1/3 ln d O m, 11) D where m = nd 2 s he number of local erms n H. Ths heorem can be seen as a srong no-go for QPCP. Indeed, shows ha QPCP s false on D- regular graphs for any value of he gap parameer γ = Ωd 2 ln d/d) 1/3 assumng QMA =NP). Indeed, he heorem shows ha, for hese values of γ here s always a produc sae ha has energy less han b = λ 0 H) + γ and can serve as a classcal wness for hs fac. Snce QPCP assers hardness for consan γ, we see ha he conjecure can only be rue on graphs of consan degree D unless he local dmenson d s allowed o grow faser han D). In an earler lecure we also saw as an exercse) ha QPCP requred graphs of hgh dmenson n order o be rue. Togeher hese wo saemens narrow down he knd of local Hamlonans for whch he conjecure may be rue: her neracon degree should have hgh dmenson canno be embedded on any lace of dmenson Olog n)), bu consan degree. Proof. Le ψ be he ground sae of H, and he Λ be he IC measuremen, defned as before. Le ρ = ψ ψ. j,x, defne σ j,x = j ρ j,x 1 d m I j 1...j m 12) In oher words, hs sae s ρ j,x f j and Id /d oherwse. By he quanum de Fne heorem appled o ρ wh k = 2 and o be deermned laer, we have E 1, 2 E j,x ρ j,x ρ j,x ρ j,x d 2 1 = 2 ) O ln d 13) For he sae we defned, we have E, j,x ρ j,x σ j,x σ j,x d ) O ln d + 2 n 14) where he las quany comes from he rare cases when j. Resrcng he expecaon on = 1, 2 ) o he Dn/2 pars ha correspond o edges of he consran graph of he local Hamlonan and applyng Jensen s nequaly we oban E =1, 2 ) E E j,x ρ j,x σ j,x σ j,x 1 2 = O d 2 ln d + ) n 15) n D 4

5 By defnon of he race norm and usng ha for any j he reduced densy of Ex ρ j,x on he n quds ha were no measured s he same as ha of ρ, we oban E j ψ H ψ ψ E x σ j,x ψ O d 2 ln d + ) n nd + D, 16) n D where here he mulplcave nd comes from swchng he expecaon over edges o a sum over edges snce H s gven as a sum of local erms), and he addonal D corresponds o mposng a maxmum energy penaly of 1 for each edge nvolvng one of he measured qubs. Choosng he ha mnmzes he rgh-hand sde, nd 2/3 ln d 1/3 /D 1/3, we oban he desred resul: here wll exs a leas one choce of j such ha he separable sae Ex σ j,x has low energy; usng lneary of he Hamlonan here s a leas one pure produc sae n he suppor of E x σ j,x ha has a mos he same energy. 5