New bounds on classical and quantum one-way communication complexity

New b ounds on classical and quan tum one-w a y comm unication complexit y Rahul Jain 1 ⋆ and Shengyu Zhang 2 ⋆⋆ 1 Universit y of W aterloo rjain@cs.u waterloo.ca 2 Califo rnia Institute of T ec hn o logy shengyu@caltec h.edu Abstract. In this pap er we provide n e w b o un ds on classical and quantum distributional comm un ication complexity in the t wo-part y , one-wa y mo del of comm un i cation. In the classical one-w ay mod el , our b ound exten ds t h e well kn o wn upp er boun d of K remer, Nisan and R on [KNR95] t o include non - pro duct distributions. Let ǫ ∈ (0 , 1 / 2) b e a constan t. W e show th at for a bo olean function f : X × Y → { 0 , 1 } and a non-pro duct distribution µ on X × Y , D 1 ,µ ǫ ( f ) = O (( I ( X : Y ) + 1) · VC ( f ) ) , where D 1 ,µ ǫ ( f ) represen ts the one-w a y distributional comm unication complexity of f with error at most ǫ under µ ; V C ( f ) represen ts t h e V apnik-Chervonenkis dimension of f and I ( X : Y ) represents the mutual information, und er µ , b etw een the rand om inputs of the tw o parties. F or a non-b o olean function f : X × Y → { 1 , . . . , k } ( k ≥ 2 an in teger), w e show a simi lar up p er b ound on D 1 ,µ ǫ ( f ) in terms of k , I ( X : Y ) and the pseudo-dimension of f ′ def = f k , a generalization of the VC -dimension for non-b o olean funct ions. In the quantum one-wa y model w e pro vide a lo we r bound on the distributional comm unication complexity , u nder product distributions, of a function f , in terms t h e w ell studied complexity measure of f referred to as the r e ctangle b ound or the c orruption b ound of f . W e sho w for a non-b o olean total fun ction f : X × Y → Z and a produ ct distribution µ on X × Y , Q 1 ,µ ǫ 3 / 8 ( f ) = Ω ( rec 1 ,µ ǫ ( f )) , where Q 1 ,µ ǫ 3 / 8 ( f ) rep resen ts the quantum one-w ay distributional comm unication complexity of f with error at most ǫ 3 / 8 under µ and rec 1 ,µ ǫ ( f ) represents the one-w ay rectangle b ound of f with error at most ǫ under µ . Similarly for a non- b oolean partia l function f : X × Y → Z ∪ { ∗} and a p ro du ct distribution µ on X × Y , w e show, Q 1 ,µ ǫ 6 / (2 · 15 4 ) ( f ) = Ω ( rec 1 ,µ ǫ ( f )) . ⋆ School of Computer Science, and I nstitute for Quantum Co mpu ting, Universit y of W aterlo o, 200 Un iversi ty Ave. W., W aterloo, ON N2L 3G1, Canada. R esearch supp orted in part by ARO/NSA USA. ⋆⋆ Computer Science D ep artment and Institute for Quan tum Computing, California Institute of T echnology , 1200 E California Bl, IQI, MC 107-81, Pasadena, CA 91125, USA . This w ork wa s supported by the National Science F oundation under grant PHY-045672 0 and the Army Research Office under gran t W911NF-05-1- 0294 through Institut e for Quantum In formation at California I nstitute of T echnology . 1 In tro duction Communication co mplexity studies the minimum a mount o f comm unication that t wo or more parties need to compute a giv en function or a relation of their inputs. Since its inception in the seminal paper by Y ao [Y ao7 9], communication complexity has been an imp ortant and widely studied r esearch area. This is th e case both because o f the in teresting and intriguing mathematics in volv ed in its s tudy , and also be c a use of the fundamen tal co nnections it bea rs with many o ther areas in theoretical computer science, such as data structures, streaming algorithms, circuit lower b ounds, decision tree complexity , VLSI designs, etc. Different models of c ommunication have been prop osed and studied. In the basic and standar d two-p arty inter active mo del, tw o pa rties sa y Alice and Bob , each receive an input say x ∈ X and y ∈ Y , resp ectively . They in teract with ea ch other p oss ibly comm unicating several messages in order to jointly compute, say a given function f ( x, y ) of their inputs. If only one message is allow ed, say from Alice to Bob , and Bo b outputs f ( x, y ) without any further interaction with A lic e , then the model is said to be one-way . Though seemingly simple, this mo del has n umerous nontrivial q uestions as w ell as applications to other a r eas suc h as low er bo unds of strea ming algorithms, see for example [Mut05]. Other mo dels like the Simu ltane ous message p assing ( SMP ) mo del, and mu lti-p arty mo dels are also studied. W e refer r eaders to the textb o ok [KN97] for a co mprehensive in tro duction to the field of clas sical communication complexity . In 1993, Y ao [Y ao 9 3] introduced qu antum communication complexity and since then it has a lso b ecome a very active and vibrant a rea o f resea rch. In the quantum communication models, the parties are allow ed t o use quantum co mputers to process their inputs and to use quantum channels to send messag es. In this pap er we a r e primarily concerned with the one-w ay mo del and we a ssume that the single mes sage is a lwa ys, say fro m Alice to Bob . Let us first briefly discuss a few classical mo dels. In the deterministic o ne-wa y model, the pa rties act in a deterministic fashion, and compute f correctly on a ll input pairs ( x, y ). The minim um communication req uired for ac complishing this is called as the deterministic c omplexity of f and is denoted by D 1 ( f ). Allowing the parties to use ra ndomness and allo wing them to err on their inputs with a small non-ze r o probability , often results in considera ble savings in comm unication. The communication of the b est public-coin one-wa y proto col that has er ror at most ǫ o n all inputs, is re fer red to a s the one-way public-c oin r andomize d communication complexity of f and is deno ted b y R 1 , pub ǫ ( f ). Similarly w e can define the one-way private-c oin r andomize d communication complexit y of f , denoted b y R 1 ǫ ( f ) and in the quant um mo del, the one-way quantum communication complexit y of f , denoted b y Q 1 ǫ ( f ). Please refer to Sec. 2.2 for explicit definitions. When the subscript is omitted, ǫ is a s sumed to b e 1 / 3. Sometimes the require ment on co mm unicatio n pro to cols is less stringent and it is only r equired that the av erage error, under a given distribution µ on the inputs, is small. The communication of the bes t one-way clas s ical proto col that has average er ror at most ǫ under µ , is referred to as t he o ne-way distributional communication complexity of f and is denoted by D 1 ,µ ǫ ( f ). W e can define the one-way distributional quan tum c ommunication complexit y Q 1 ,µ ǫ ( f ) in a s imila r wa y . A useful connec tio n betw een the public-coin randomized and distributiona l communication complex ities via the Y ao’s Principal [Y ao 77] states that for a given ǫ ∈ (0 , 1 / 2 ), R 1 , pub ǫ ( f ) = max µ D 1 ,µ ǫ ( f ). A distribution µ , that achiev es the ma x imu m in Y ao’s Principa l, that is for which R 1 , pub ǫ ( f ) = D 1 ,µ ǫ ( f ), is referred to as a har d distribution for f . This principal also holds in many other mo dels and a llows for a go o d handle on the public-coin r andomized complexity in scenar ios wher e the distributional co mplex it y is m uch easier to understand. Often, the distributional complexity when the inputs of Alice and Bob are drawn indep endently from a pro duct distribution, is eas ier to unders tand. Nonetheless, often as is the case with sev era l important functions lik e Set Disjoint ness ( DISJ ) a nd Inner Pro duct ( IP ), the maximum in Y ao’s Pr incipal, in the one-wa y mo del, o ccur s for a pro duct distribution, and hence it pav es the wa y for understanding the public-coin ra ndomized complex it y . Let us now discuss our first main r e sult which is in the classical one-wa y mo del. W e ask the reader to refer to Sec. 2 for the definitions of v arious quan tities inv olved in the discussion below. 1.1 Classical upp er b o und F or a b o olean function f : X × Y → { 0 , 1 } , its V apnik-Chervonenkis (V C) dimension , deno ted by V C ( f ), is an important complexit y measure, widely studied sp ecially in the cont exts of c omputational le arning the ory . Kr emer, Nisan a nd Ron [KNR95, Thm. 3.2] found a be a utiful connec tio n b etw een the distributional complexit y of f under pr o duct dis tributions on X × Y , and V C ( f ), as follows. Theorem 1 ([KNR95]). L et f : X × Y → { 0 , 1 } b e a b o ole an function and let ǫ ∈ (0 , 1 / 2) b e a c onstant. L et µ b e a pro duct distribution on X × Y . Ther e is a universal c onstant κ such that, D 1 ,µ ǫ ( f ) ≤ κ · 1 ǫ log 1 ǫ · VC ( f ) . (1) Note that such a relation ca nnot hold for non- pro duct distr ibutions µ since otherwise it would translate, via the Y ao’s Pr inc ipa l, into R 1 , pub ǫ ( f ) = O ( VC ( f )), for all b o olean f . This is not tr ue as is exhibited by sev era l functions for exa mple the Greater Than ( GT n ) function, in which Alice and Bob need to determine which of their n - bit inputs is big ger. F or this function, R 1 , pub ǫ ( GT n ) = Θ ( n ) but VC ( GT n ) = 1. Nonetheless for these functions, any hard dis tribution µ , is highly corr elated betw een X and Y . Therefore it is conceiv able that such a relationship, as in Eq. 1, could still hold, po ssibly after taking into account the a mo unt of cor relation in a given non-pro duct distribution. This q uestion, although probably never explicitly asked in any previo us work, app ear s to b e quite fundamen tal. W e answer it in the pos itive by the following. Theorem 2. L et f : X × Y → { 0 , 1 } b e a b o ole an function and let ǫ ∈ (0 , 1 / 2 ) b e a c onstant. L et µ b e a distribution (p ossibly non-pr o duct) on X × Y . L et X Y b e joint r andom variables d istribute d ac c or ding to µ . Ther e is a u niversal c onstant κ such that, D 1 ,µ ǫ ( f ) ≤ κ · 1 ǫ log 1 ǫ ·  1 ǫ · I ( X : Y ) + 1  · VC ( f ) In p articular, for c onstant ǫ , D 1 ,µ ǫ ( f ) = O (( I ( X : Y ) + 1) · V C ( f )) Ab ove I ( X : Y ) re pr esents the mutual information b etwe en c orr elate d r andom variables X and Y , distribute d ac c or ding to µ . Let us disc us s be low a few asp ects of this result and its r elationship with what is previously k nown. Note that in combination with Y ao’s Principal, Thm. 2 gives us the follo wing (where the mutual information is now considered under a har d dis tribution for f ). R 1 , pub ( f ) = O (( I ( X : Y ) + 1) · VC ( f )) . (2) 1. It is easily observed using Sauer’s Lemma (Lem. 2, Sec. 2.) that the deterministic complexity of f has D 1 ( f ) = O ( VC ( f ) · log |Y | ) . (3) This is because Alice can simply tell the na me of f x in O ( VC ( f ) · log |Y | ) bits since |F | ≤ |Y | V C ( f ) . Now our result (2) is on one hand stro nger than (3) in the s ense I ( X : Y ) ≤ log |Y | a lwa ys, and I ( X : Y ) could be m uch smaller than log |Y | dep ending on µ . An example of such a case is the Inner Pr o duct ( IP n ) function in which Alice and Bo b need to determine the inner pro duct (mo d 2) of their n -bit input strings. F or IP n , a hard distribution is the uniform distribution which is pro duct, and hence I ( X : Y ) = 0, whereas log |Y | = n . How ever on the other hand (2) is also weak er than (3) in the s ense it o nly uppe r b ounds the public-coin r andomized co mplex it y , whereas (2) upp er b o unds the deterministic complexity of f . 2. Aarons on [Aar0 7] shows that fo r a total or partial bo olea n function f , R 1 ( f ) = O ( Q 1 ( f ) · log |Y | ) . (4) Again (2) is stronger than (4) in the sense that I ( X : Y ) could be m uch smaller than log |Y | depe nding on µ . A lso it is known that, Q 1 ( f ) = Ω ( VC ( f )) alwa ys, following from Nayak [Na y99], and Q 1 ( f ) could be muc h large r than VC ( f ). An example is the Greater Than ( GT n ) function for whic h Q 1 ( GT n ) = Ω ( n ), whereas V C ( GT n ) = O (1). On the other hand (2) o nly holds for total bo olea n functions wher eas (4) also ho lds for partial b o olean functions. 3. As men tioned b efore, for all tota l bo olea n functions f , R 1 , pub ( f ) = Ω ( VC ( f )), and R 1 , pub ( f ) could b e muc h larger than VC ( f ) (as in function GT n ). Now Eq. (2) s ays tha t in the la tter case , the mutual information I ( X : Y ) under any hard distribution µ must b e la rge. That is, a hard distribution µ m ust be hig hly correla ted. 4. It is known that for total bo olea n functions f , for which a har d distribution is pro duct, there is no separ ation b etw een the one-way public-coin randomized and quantum comm unicatio n com- plexities. Now our theorem gives a smo oth extension of th is fa ct to the functions whose hard distributions are not pro duct one s . A generaliza tion of the VC -dimension for non-b o olea n functions, is referr ed to as the pseudo- dimension (Def. 2, Sec. 2). F or a non-b o o lean function f : X × Y → { 1 , . . . , k } ( k ≥ 2 an integer), we show a similar upp er b ound on D 1 ,µ ǫ ( f ) in terms of k , I ( X : Y ) and the pseudo-dimension of f ′ def = f k . Theorem 3. L et k ≥ 2 b e an inte ger. L et f : X × Y → { 1 , . . . , k } and ǫ ∈ (0 , 1 / 6 ) b e a c onstant. L et f ′ : X × Y → [0 , 1] b e such that f ′ ( x, y ) = f ( x, y ) /k . L et µ b e a distribution (p ossibly non-pr o duct) on X × Y , a nd X Y b e joint r andom variables distribute d ac c or ding to µ . Then ther e is a universal c onstant κ such that, D 1 ,µ 3 ǫ ( f ) ≤ κ · k 4 ǫ 5 ·  log 1 ǫ + d lo g 2 dk ǫ  · ( I ( X : Y ) + log k ) wher e d def = P ǫ 2 576 k 2 ( f ′ ) is the ǫ 2 576 k 2 -pseudo-dimension of f ′ . Let us now dis c uss our o ther main result which w e show in the qua nt um one-way model. 1.2 Quan tum lo w er b ound F or a function f : X × Y → Z , a measure of its co mplex it y that is often very useful in unders tanding its classical ra ndo mized communication complexity , is what is referred to as the r e ctangle b ound (denoted b y rec ( f )), also often known as the c orruption b ound . The rectangle b o und rec ( f ) is actually defined first v ia a distributiona l version rec µ ( f ). It is a well studied mea sure and rec µ ( f ) is w ell known to form a lo wer b ound o n D µ ( f ) bo th in the one-wa y and t wo-wa y mode ls . In fact, in a celebrated r esult, Razborov [Raz92] provided optimal low er b o und o n the randomized co mm unicatio n complexity of the Set Disjoin tness function, b y ar guing a low er bound on its rectangle bound. It is natural to ask if this measure also forms a lo wer b ound on the quantum co mm unication complexity . W e answer in the positive for this question in the o ne-wa y mo del. W e show that, for a total or partial function, the quantum distributional o ne - wa y communication complexit y under a given pro duct distribution µ is lower b ounded by the corresp onding one-wa y rectangle b o und. Our precise result is as follows. Theorem 4. L et f : X × Y → Z b e a total function and let ǫ ∈ (0 , 1 / 2) b e a c onstant. L et µ b e a pr o duct distribution on X × Y and let rec 1 ,µ ǫ ( f ) > 2 · log(1 /ǫ ) . Then, Q 1 ,µ ǫ 3 / 8 ( f ) ≥ 1 2 · (1 − 2 ǫ ) · ( S ( ǫ/ 2) − S ( ǫ/ 4)) · ( ⌊ rec 1 ,µ ǫ ( f ) ⌋ − 1) = Ω ( rec 1 ,µ ǫ ( f )) , (5) wher e for p ∈ (0 , 1) , S ( p ) is t he binary en t r opy funct ion S ( p ) def = − p log p − (1 − p ) log(1 − p ) . If f : X × Y → Z ∪ {∗} is a p artial function t hen, Q 1 ,µ ǫ 6 / (2 · 15 4 ) ( f ) ≥ 1 2 · (1 − 2 ǫ ) · ǫ 2 300 · ( ⌊ rec 1 ,µ ǫ ( f ) ⌋ − 1) = Ω ( rec 1 ,µ ǫ ( f )) . Let us make a few imp ortant rema rks her e related to this result. 1. Recently , Jain, K lauck and Nayak [JK N08] show ed that for a ny relation f ⊆ X × Y × Z , the rectangle b ound of f tightly characterizes the r andomized one-wa y classical communication complexity of f . Theorem 5 ([JKN08]). L et f ⊆ X × Y × Z b e a r elation and let ǫ ∈ (0 , 1 / 2 ) . Then, R 1 , pub ǫ ( f ) = Θ ( rec 1 ǫ ( f )) . While showing Thm. 5, Jain, K lauck and Nay ak [JK N08] have shown that for all r elations f : X × Y → Z and for all distributions µ (pro duct and non-pro duct) on X × Y ; D 1 ,µ ǫ ( f ) = Ω ( rec 1 ,µ 4 ǫ ( f )). How ever in the quantu m setting w e are mak ing a similar statement only for (total or partial) functions f and only for pro duct distributions µ on X × Y . In fa c t it do es NOT hold if w e let µ to be non-pro duct. It c an be shown that there is a total f unction f and a non-pro duct distribution µ such that Q 1 ,µ ǫ ( f ) is exp onentially smaller than rec 1 ,µ ǫ ( f ). This fact is implicit in the work of Gavinsky et al. [GKK + 07]. W e make an explicit statement of this in Sec. A. in Appendix and s kip its pr o of for br e vity . 2. Let ǫ ∈ (0 , 1 / 4). Jain, Klauck a nd Na yak [JK N0 8] ha ve shown that for all relations g ⊆ X × Y × Z , R 1 , [] 2 ǫ ( g ) = O ( rec 1 , [] ǫ ( g )) . Here the supe r script [] represents ma ximization over a ll pro duct distributions. F rom Thm. 4 for a (total or pa rtial) function f we get, Q 1 , [] ǫ 6 / (2 · 15 4 ) ( f ) = Ω ( rec 1 , [] ǫ ( f )) . Since R 1 , [] ǫ ( f ) ≥ Q 1 , [] ǫ ( f ), combinin g everything we g et, Theorem 6. L et ǫ ∈ (0 , 1 / 4) . L et f : X × Y → Z ∪ {∗} b e a (p ossibly p artial and non-b o ole an) function. Then R 1 , [] ǫ 6 / (2 · 15 4 ) ( f ) ≥ Q 1 , [] ǫ 6 / (2 · 15 4 ) ( f ) = Ω ( R 1 , [] 2 ǫ ( f )) . It was kno wn earlier that for total b o olea n functions, Q 1 , [] ( f ) is tig htly bo unded by R 1 , [] ( f ). W e extend s uch a rela tionship here to apply for non-bo olean (par tial) functions as w ell. W e re mark that the earlier pro ofs for total b o olean functions used the VC -dimension re s ult, Thm. 1, of Kremer, Nisan and Ron [KNR9 5]. W e get the same r esult he r e without r equiring it. W e finally present an application of our result Thm. 4 in the context of studying security of extrac- tors a gainst quantu m adversaries . An extractor is a function that is used to extra ct almos t uniform randomness fr om a source of imp er fect r andomness. As v ery well studied ob jects, ex tr actors hav e found se veral uses in many cryptog raphic applicatio ns and also in complexity theor y . Recently , secu- rity of v arious extractors has b een increasingly studied in the presence of qua ntum adv ersar ies; since such secure e x tractors are then useful in several applications suc h a s priv a c y amplificatio n in quan- tum k ey distribution and k ey-expa nsion in quan tum bounded storage mo dels [KMR05,KR05,KT08]. In particular, K¨ onig and T erhal [K T08] hav e shown tha t any b o o le a n extracto r that c a n extract a uniform bit from so urces o f min-entr opy k is a lso secure against quantum a dversar ies with their memory bo unded b y a function o f k . W e get a similar statemen t for b o o lean extractors, a s a co rollary of our result Thm. 4 . W e obtain this corolla ry by obse rving a key connectio n b etw een the minimum min-ent ro py that an extracto r function f needs to extra c t a uniform bit and its rectangle b ound. The precise statemen t of our result, its relationship with the result o f [K T0 8], and o ther detailed discussio ns ar e de fer red to Sec. 5. 1.3 Organization In the following Sec. 2 we discuss v a rious informa tion theo retic prelimina ries a nd the mo del of one- wa y communication. In Sec. 3 we present the upp er b o unds in the classical setting. In the following Sec. 4 we pres ent the lower bounds in the quantum setting. The application concerning extractor s is dis cussed in Sec. 5 . W e finally co nclude with so me o pe n ques tions in Sec . 6. 2 Preliminaries 2.1 Information theory In this sectio n w e present so me information theoretic notations, definitions and facts that we use in the rest of the pap er. F or an in tro ductio n to classica l and quantum infor mation theory , we refer the reader to the texts b y Cov er and Thomas [CT9 1] and Nielsen and Chuang [NC00] resp ectively . Most of the facts stated in this section without proo fs may b e found in these bo oks. All loga rithms in this paper are taken with base 2, unles s other wise sp ecified. F or an integer t ≥ 1, [ t ] r epresents the set { 1 , . . . , t } . F or square matrices P , Q , by Q ≥ P we mean that Q − P is po sitive semi-definite. F or a ma trix A , k A k 1 def = T r ( √ A † A ) deno tes its ℓ 1 norm. F o r p ∈ (0 , 1), let S ( p ) def = − p log p − ( 1 − p ) log (1 − p ), denote the binar y en tropy function. W e ha ve the following fact. F act 1 F or δ ∈ [0 , 1 / 2 ] , S ( 1 2 + δ ) ≤ 1 − 2 δ 2 and S ( δ ) ≤ 2 √ δ . A quantum state, us ua lly represented by letters ρ, σ etc., is a p ositive semi-definite trace one op erator in a given Hilbert space. Sp ecializing from the q uantum case, we view a discrete proba bilit y distribution P a s a positive semi-definite tr ace o ne diag onal matrix indexed by its (finite) sample space. F or a distr ibution P with support on set X , and x ∈ X , P ( x ) denotes the ( x, x ) diagonal ent ry of P , and P ( E ) def = P x ∈E P ( x ) denotes the probability of the event E ⊆ X . A distribution P on X × Y is said to be pr o duct across X a nd Y , if it can b e written a s P = P X ⊗ P Y , wher e P X , P Y are distributions on X , Y resp ectively and ⊗ is the tensor op eration. Often for product distributions we do not mention the sets ac ross whic h it is product if it is clear from the context. Let X be a classica l random v a riable (or simply random v aria ble) taking v alues in X . F or a random v ariable X , we a lso let X represent its probability dis tribution. The ent r opy of X deno ted S ( X ) is defined to b e S ( X ) def = − T r X log X . Since X is clas sical an equiv alent definition would b e S ( X ) def = − P x ∈X Pr[ X = x ] lo g Pr[ X = x ] . Let X , Y be a correlated random v ar iables tak ing v alues in X , Y resp ectively . X Y are said to b e indep endent if their joint distribution is pr o duct. The mutual information betw een them, denoted I ( X : Y ) is defined to b e I ( X : Y ) def = S ( X ) + S ( Y ) − S ( X Y ) and c onditional entr opy deno ted S ( X | Y ) is defined to be S ( X | Y ) def = S ( X Y ) − S ( Y ). It is easily s een that S ( X | Y ) = E y ← Y [ S ( X | ( Y = y )]. W e ha ve the following facts. F act 2 F or al l r andom variables X , Y ; I ( X : Y ) ≥ 0 ; in o ther wor ds S ( X ) + S ( Y ) ≥ S ( X Y ) . If X , Y a r e indep endent then we have I ( X : Y ) = 0 ; in other wor ds S ( X Y ) = S ( X ) + S ( Y ) . The definitions and fa c ts stated in the ab ov e par agra ph for c lassical random v ariables a lso hold mutatis mutandis for quantum states as w ell. F or example for a qua nt um state ρ , its entropy is defined as S ( ρ ) def = − T r ρ log ρ . F or brevit y , w e avoid making all the corresp onding statements explicitly . As is the ca se with cla ssical random v ar iables, for a quan tum s y stem say Q , we also often let Q re present its quantum state. W e ha ve the following fact. F act 3 Any quantum state ρ in m -qubits has S ( ρ ) ≤ m . Also let X Q b e a joi nt classic al-quant um system with X b eing a classic al r andom variable, then I ( X : Q ) ≤ min { S ( X ) , S ( Q ) } . F or a system X Y M , let us define I ( X : M | Y ) def = S ( X | Y ) + S ( M | Y ) − S ( X M | Y ). If Y is a classical system then it is easily seen that I ( X : M | Y ) = E y ← Y [ I ( X : M | ( Y = y ))]. F or random v ariables X 1 , . . . , X n and a cor related (po ssibly quantum) system M , we hav e the following chain rule of mutual information , which will b e crucially used in our pr o ofs. I ( X 1 . . . X n : M ) = n X i =1 I ( X i : M | X 1 . . . X i − 1 ) (6) By con ven tion, conditioning on X 1 . . . X i − 1 for i = 1 means co nditio ning on the true even t. The follo wing is an imp or tant information theoretic fact kno wn as F ano’s ineq ua lity , which r elates the probability of disagreement for correlated random v ar iables to their mutual informa tion. Lemma 1 (F ano’s inequali ty). L et X b e a r andom varia ble taking values in X . Le t Y b e a c orr elate d r andom variable and let P e def = Pr( X 6 = Y ) . Then, S ( P e ) + P e log( |X | − 1 ) ≥ S ( X | Y ) . The VC-dimension of a b o ole a n function f is an importa nt com binatoria l concept and has close connections with the o ne-wa y co mmu nicatio n complexity of f . Definition 1 (V apnik-Cherv onenkis ( V C ) dimensio n). A set S ⊆ Y is said to b e shattered by a set G of b o ole an fu n ctions fr om Y to { 0 , 1 } , if ∀ R ⊆ S, ∃ g R ∈ G such t hat ∀ s ∈ S, ( s ∈ R ) ⇔ ( g R ( s ) = 1) . The lar gest value d for which ther e is a set S of size d that is shatter e d by G is t he V apnik-Chervonenkis dimension of G and is denote d by VC ( G ) . L et f : X × Y → { 0 , 1 } b e a b o ole an function. F or all x ∈ X let f x : Y → { 0 , 1 } b e define d as f x ( y ) def = f ( x, y ) , ∀ y ∈ Y . L et F def = { f x : x ∈ X } . Then the V apnik-Chervonenkis dimension of f , denote d by V C ( f ) , is define d to b e V C ( F ) . Let f and F be as defined in the above definition. W e call a function f trivial iff |F | = 1, in other w or ds iff the v a lue of the function, fo r all x , is determined only b y y . W e call f non-trivial iff it is not triv ial. Note th at a b o olean f is non-tr ivial if and only if VC ( f ) ≥ 1. Throughout t his pap er we assume all o ur functions to be non-trivia l. F ollowing is a useful fact, with sev eral applications, relating the V C-dimension of f to the size of F . It is usually attributed to Sauer [Sau7 2], ho wev er it has b een indep endently discovered b y several different p eople as w ell. Lemma 2 (Sauer’s Lemma [Sau72]). L et f : X × Y → { 0 , 1 } b e a b o ole an fu n ction. L et d def = V C ( f ) . L et m def = |Y | , then |F | ≤ d X i =0  m i  ≤ m d . The follo wing result from Blumer, Ehrenfeuch t, Haussler, and W armuth [B E HW89] is one of the most fundamen tal results from computational lear ning theory and in fact an imp or tant application of Sauer’s Le mma . Lemma 3. L et H b e class of b o ole an functions over a finite domain Y with VC-dimensio n d , let π b e an arbitr ary pr ob ability distribution over Y , and let 0 < ǫ, δ < 1 . L et L b e any algorithm t hat takes as input a set S ∈ Y m of m examples lab ele d ac c or ding to an unknown function h ∈ H , and outputs a hyp othesis funct ion h ′ ∈ H that is c onsistent with h on the sample S . If L r e c eives a r andom sample of size m ≥ m 0 ( d, ǫ, δ ) distribute d ac c or ding to π m , wher e m 0 ( d, ǫ, δ ) = c 0  1 ǫ log 1 δ + d ǫ log 1 ǫ  for some c onstant c 0 > 0 , then with pr ob abili ty at le ast 1 − δ over the r andom samples, P r π [ h ′ ( y ) 6 = h ( y )] ≤ ǫ . A s imilar learning result also ho lds for non-b o o lean functions. F or this let us first de fine the following gener a lization of the VC -dimension, known as the pseudo-dimension . Definition 2 (ps eudo-dime n s ion). A set S ⊆ Y is said t o b e γ -sha ttered by a set G of fun ctions fr om Y to Z ⊆ R , if ther e exists a ve ctor w = ( w 1 , . . . , w k ) ∈ Z k of di mension k = | S | fo r which the fol lowing holds. F or al l R ⊆ S, ∃ g R ∈ G such t hat ∀ s ∈ S, ( s ∈ R ) ⇒ ( g R ( s ) > w i + γ ) and ( s / ∈ R ) ⇒ ( g R ( s ) < w i − γ ) . The lar gest value d for which ther e is a set S of size d that is γ -shatter e d by G is the γ -pseudo -dimension of G and is denote d by P γ ( G ) . L et f : X × Y → Z b e a function. F or al l x ∈ X let f x : Y → Z b e define d as f x ( y ) def = f ( x, y ) , ∀ y ∈ Y . L et F def = { f x : x ∈ X } . Then the γ -pseudo-dimension of f , denote d by P γ ( f ) , is define d t o b e P γ ( F ) . F ollowing r esult of Ba r tlett, Lo ng and Williamson [B L W96] is similar to the lear ning lemma of Blumer et a l. [BEHW89] a nd concerns non-bo o lean functions. Theorem 7. L et G b e a class of functions over a finite domain Y into t he r ange [0 , 1] . L et π b e an arbitr ary pr ob ability distribution over Y a nd let ǫ ∈ (0 , 1 / 2) and δ ∈ (0 , 1) . L et d def = P ǫ 2 / 576 ( G ) . Then ther e exists a deterministic le arning algorithm L which has the fol lowing pr op erty. Given as input a set S ∈ Y m of m examples chosen ac c or ding to π m and lab ele d ac c or ding to an unknown function g ∈ G , L outputs a hyp othesis g ′ ∈ G such that if m ≥ m 0 ( d, ǫ, δ ) wher e m 0 ( d, ǫ, δ ) = c 0  1 ǫ 4 log 1 δ + d ǫ 4 log 2 d ǫ  for some c onstant c 0 > 0 , then with pr ob ability at le ast 1 − δ over t he r andom samples, X y ∈Y π ( y ) · | h ′ ( y ) − h ( y ) | ≤ ǫ. F ollowing is a v er y fundamental q ua ntum information theoretic fa ct shown by Holevo [Hol73]. Theorem 8 (T he Holevo b ound [Hol73]). L et X b e classic al r andom variable taki ng values in X . Le t M b e a c orr elate d quantum system and let Y b e a r andom variable obtaine d by p erforming a quantum m e asur ement on M . Then, I ( X : Y ) ≤ I ( X : M ) . (7) F ollowing is an interesting and useful information theoretic fact first sho wn b y Helstro m [Hel76]. Theorem 9 ([Hel76]). L et X Q b e joint classic al-quantum system wher e X is a classic al b o ole an r andom variabl e. F or a ∈ { 0 , 1 } , let the quant um state of Q when X = a b e ρ a . The optimal suc c ess pr ob ability of pr e dicting X with a me asur ement on Q is given by 1 2 + 1 2 · k P r[ X = 0] ρ 0 − Pr[ X = 1 ] ρ 1 k 1 . 2.2 One-wa y comm unication In this article we only consider the tw o-party one-w ay mo del of comm unicatio n. Let f ⊆ X × Y × Z be a relatio n. The r e lations we consider a re alwa ys total in the sense that for ev ery ( x, y ) ∈ X × Y , there is at least one z ∈ Z , such that ( x, y , z ) ∈ f . In a one-way proto co l P for computing f , Alice and Bob get input s x ∈ X and y ∈ Y resp ectively . Alice sends a single message to Bob , and their inten tion is to determine an a nswer z ∈ Z s uch that ( x, y , z ) ∈ f . In the o ne - wa y proto cols we co nsider, the single mes sage is alwa ys from Alice to Bob . A total function f : X × Y → Z , can b e view ed as a sp ecial type of re la tions in whic h for ev ery ( x, y ) there is a unique z , such that ( x , y, z ) ∈ f . A partial function is a s pe c ial type of relations such that for some inputs ( x, y ), there is a unique z , suc h that ( x, y , z ) ∈ f and for all o ther inputs ( x, y ), ( x, y , z ) ∈ f , ∀ z ∈ Z . W e view a partia l function f as a function f : X × Y → Z ∪ {∗} , such that the inputs ( x, y ) for whic h f ( x, y ) = ∗ a re exactly the ones for whic h ( x, y , z ) ∈ f , ∀ z ∈ Z . Let us first consider classica l communication proto cols. W e let D 1 ( f ) repres ent the deterministic one-wa y communication complexit y , that is the communication of the best deterministic proto col computing f co rrectly on all inputs. F or ǫ ∈ (0 , 1 / 2), let µ b e a probability distr ibution on X × Y . W e let D 1 ,µ ǫ ( f ) represen t the distributiona l one-wa y communication complexity of f under µ with exp ected erro r ǫ , i.e., the co mm unication of the b est pr iv ate-coin one- wa y pro to col fo r f , with distributional error (av era ge error over the coins and the inputs) at most ǫ under µ . It is easily noted that D 1 ,µ ǫ ( f ) is a lwa ys achiev ed b y a deterministic one-wa y proto col, and will henceforth restrict ourselves to deter ministic proto co ls in the context of distributiona l communication complexity . W e let R 1 , pub ǫ ( f ) represent the public-coin randomized o ne-wa y communication co mplex it y of f with worst case error ǫ , i.e., the communication o f the b es t public-coin r andomized one-wa y proto col for f with erro r for each input ( x, y ) b eing at most ǫ . The analo gous quan tity for priv ate c o in randomized proto cols is deno ted by R 1 ǫ ( f ). The public- and priv ate-coin ra ndomized communication complexities are not muc h different, as shown in Newman’s result [New91] that R 1 ( f ) = O ( R 1 ,pub ( f ) + log lo g |X | + log log |Y | ) . (8) The fo llowing result due to Y ao [Y ao 77] is a very useful fact connecting worst-case and distributiona l communication complexities. It is a consequence of the min-max theor em in game theor y [KN9 7, Thm. 3.20, pa g e 36]. Lemma 4 (Y ao’s principl e [Y ao77]). R 1 , pub ǫ ( f ) = max µ D 1 ,µ ǫ ( f ) . W e define R 1 , [] ǫ ( f ) def = max µ pro duct D 1 ,µ ǫ ( f ). Note that R 1 , [] ǫ ( f ) could be sig nificantly smaller than R 1 , pub ǫ ( f ) as is exhibited b y the Greater Than ( GT n ) function for whic h R 1 , pub ( GT n ) = Ω ( n ), whereas R 1 , [] ǫ ( f ) = O (1). In a o ne-wa y quantum co mm unicatio n proto col, Alice and Bob are allow ed to do quantum o p- erations and Alice c a n send a qua nt um message (qubits) to Bob . Given ǫ ∈ (0 , 1 / 2 ), the one- wa y quantum communication complexity Q 1 ǫ ( f ) is defined to b e the co mmunication of the b est one-wa y quantum proto co l with erro r at most ǫ o n a ll inputs. Given a distribution µ on X × Y , w e can similar ly define the quantum distributional one-wa y co mmu nicatio n complexity of f , denoted Q 1 ,µ ǫ ( f ), to b e the comm unicatio n o f the b est one-way quan tum proto col P for f such that the a verage error of P ov er the inputs drawn from the distribution µ is at mo st ǫ . W e define Q 1 , [] ǫ ( f ) def = max µ pro du ct Q 1 ,µ ǫ ( f ). 3 A new upp er bound on classical one-w ay distributional comm unication complexit y In this section we present the upper b ounds on the distributional communication complexit y , D 1 ,µ ǫ ( f ) for any dis tr ibution µ (pos sibly non-pro duct) on X × Y . W e b egin by restating the precise result for bo olean functions. Theorem 10. L et f : X × Y → { 0 , 1 } b e a b o ole an fu n ction and let ǫ ∈ (0 , 1 / 2) b e a c onstant. L et µ b e a distribution (p ossibly non-pr o duct) on X × Y . L et X Y b e joint r andom variables d istribute d ac c or ding to µ . Ther e is a u niversal c onstant κ such that, D 1 ,µ ǫ ( f ) ≤ κ · 1 ǫ log 1 ǫ ·  1 ǫ · I ( X : Y ) + 1  · VC ( f ) . In other wor ds, D 1 ,µ ǫ ( f ) = O (( I ( X : Y ) + 1) · V C ( f )) F o r showing this result we will crucially us e the following fact shown by Harsha , Jain, McAllester and Ra dhakrishnan [HJMR07] co ncerning co mm unicatio n r equired for gene r ating co rrelatio ns . W e beg in with the fo llowing definition. Definition 3 (Co rrelation proto col). L et ( X , Y ) b e a p air of c orr elate d r andom variables taking values in X × Y . L et Alice b e given x ∈ X , sample d ac c or ding to t he distribution X. Alice should tr ansmit a message to Bob , such that Ali ce and Bo b c an to gether gener ate a value y ∈ Y distribute d ac c or ding t o the c onditional distribution Y | X = x ; that is the p air ( x, y ) should have joint distribution ( X, Y ) . Alice and Bob ar e al lowe d t o use public r andomness. N ote that the gener ate d value y sho uld b e known to b oth Alice and Bo b . Harsha et al. [HJMR0 7] showed that the minimal exp ected num b er of bits that A lic e needs to send (in the presence of s hared ra ndomness), deno ted T R ( X : Y ), is characterized by the m utual information I ( X : Y ) as follows. Theorem 11 ([H JM R 07]). Ther e ex ists a universal p ositive c onstant l su ch that, I ( X : Y ) ≤ T R ( X : Y ) ≤ 4 I ( X : Y ) + l . W e will also need the following fact. Lemma 5. L et m ≥ 1 b e an inte ger. L et X Y b e c orr elate d r andom variables. L et µ x b e the distribu- tion of Y | X = x . L et X ′ Y ′ r epr esent joint r andom variables such that X ′ is distribute d i dentic al ly to X and the distribution of Y ′ | ( X ′ = x ) is µ ⊗ m x ( m indep endent c opie s of µ x ). Then, I ( X ′ : Y ′ ) ≤ m · I ( X : Y ) . Pr o of. Consider, I ( X ′ : Y ′ ) = S ( Y ′ ) − E x ← X ′ [ S ( Y ′ | X ′ = x )] = S ( Y ′ ) − m · E x ← X [ S ( Y | X = x )] ≤ m · S ( Y ) − m · E x ← X [ S ( Y | X = x )] = m · I ( X : Y ) The seco nd equality ab ove follows from F ac t 2 and s ince X ′ and X are identically distributed. Similarly the first inequa lit y ab ove follo ws from F a ct 2 by no ting that Y ′ is m -copies of Y . W e a re now r eady for the pro of of Thm. 10. Pro of of Thm. 10: Let m def = m 0 ( V C ( f ) , ǫ / 4 , ǫ / 4) = c 0 ·  1 ǫ/ 4 log 1 ǫ/ 4  · ( VC ( f ) + 1 ) as in Lem. 3. Let l b e the constant as in Thm. 1 1. Let c def = 4 m · I ( X : Y ) + l . W e exhibit a public c o in proto col P with inputs drawn from µ , in whic h Alice sends t wo messages M 1 and M 2 to Bob . The exp ected length of M 1 is at most c and the length of M 2 is alw ays at most m . The a verage error (o ver inputs and coins ) of P is at most ǫ/ 2. Let P ′ be the proto co l that sim ulates P but ab or ts a nd outputs 0 , whenev er the length of M 1 in P exceeds 2 c/ǫ . F ro m Markov’s inequality this happens with pr o bability at most ǫ/ 2. Hence the exp ected error of P ′ is at most ǫ/ 2 + ǫ / 2 = ǫ. F rom P ′ , w e finally get a deterministic proto col with communication bounded by 2 c/ǫ + m and dis tr ibutional error at most ǫ . This implies our result fr om definition o f D 1 ,µ ǫ ( f ) and b y setting κ appropria tely . F o r x ∈ X , let µ x be the distribution of Y | X = x . In P , o n rec e iv ing the input x ∈ X , Alice first sends a messag e M 1 to Bob , a ccording to the corresp onding cor relation proto col a s in Definition 3, and they tog ether sample from the distributio n of µ ⊗ m x . Let y 1 , . . . , y m be the sa m- ples generated. Note that fro m the pro p er ties of corr elation pro to col b oth Ali ce and Bob know the v a lues of y 1 , . . . , y m . Alice then sends to Bob the second message M 2 which is the v alues of f ( x, y 1 ) , . . . , f ( x, y m ). Bob then consider s the first x ′ (according to the increa sing or der) such that ∀ i ∈ [ m ] , f ( x ′ , y i ) = f ( x, y i ) a nd outputs f ( x ′ , y ), where y is his actual input . Using Lem. 3 , it is easy to verify that for every x ∈ X , the av erage error (ov er ra ndomness in the pro to col and inputs of Bob ) in this proto co l P will b e at most ǫ/ 2 . Hence also the ov erall a verage e r ror of P is at most ǫ/ 2. Also from Thm. 1 1 and Lem. 5, w e c a n verify that the exp ected length o f M 1 in P will be at most 4 m · I ( X : Y ) + l . ⊓ ⊔ F o llowing similar arguments and using Thm. 7 a nd Thm. 11, we obtain a simila r result for non-b o olean functions as follows. Theorem 12. L et k ≥ 2 b e an i nt e ger. L et f : X × Y → [ k ] b e a non-b o ole an function and let ǫ ∈ (0 , 1 / 6 ) b e a c onstant. L et f ′ : X × Y → [0 , 1] b e such that f ′ ( x, y ) = f ( x, y ) /k . L et µ b e a d istribution (p ossibly non-pr o duct) o n X × Y . L et X Y b e joint r andom variables distribute d ac c or ding to µ . Th er e is a universal c onstant κ su ch that, D 1 ,µ 3 ǫ ( f ) ≤ κ · k 4 ǫ 5 ·  log 1 ǫ + d lo g 2 dk ǫ  · ( I ( X : Y ) + log k ) wher e d def = P ǫ 2 576 k 2 ( f ′ ) is the ǫ 2 576 k 2 -pseudo-dimension of f ′ . Pr o of. Let m def = m 0 ( d, ǫ/k , ǫ ) = c 0  k 4 ǫ 4 log 1 ǫ + dk 4 ǫ 4 log 2 dk ǫ  as in Thm. 7. Let l be the constant as in Thm. 11. Let c def = 4 m · I ( X : Y ) + l . W e ex hibit a public coin proto col P for f , with inputs dr awn from µ , in which Alice sends t wo messag es M 1 and M 2 to Bob . The exp ected length of M 1 is at most c and the length of M 2 is alwa ys at most O ( m lo g k ). The av erag e err or (o ver inputs and coins) of P is at most 2 ǫ . L e t P ′ be the pro to col that sim ulates P but aborts a nd o utputs 0, whenev er the length of M 1 in P exceeds c/ǫ . F rom Marko v’s inequality this happ ens w ith pr obability at mos t ǫ . Hence the expec ted error of P ′ is at most 2 ǫ + ǫ = 3 ǫ. F ro m P ′ , w e finally get a deterministic proto col with communication b ounded by c/ǫ + O ( m log k ) a nd distributional erro r at most 3 ǫ . This implies o ur result from definition of D 1 ,µ 3 ǫ ( f ) and by setting κ appropriately . In P , Alice and Bob intend to first determine f ′ ( x, y ) a nd then output k f ′ ( x, y ). F or x ∈ X , let µ x be the distribution of Y | X = x . On re c e iving the input x ∈ X , Alice fir s t sends a messa ge M 1 to Bob , according to the corr esp onding correlation protocol as in Definition 3, and they together sample from the distributio n of µ ⊗ m x . Let y 1 , . . . , y m be the samples g enerated. Alic e then sends to Bob the second message M 2 which is the v alues o f f ′ ( x, y 1 ) , . . . , f ′ ( x, y m ) . Bob then co nsiders x ′ as obtained from the lear ning algor ithm L (as in Thm. 7) and then o utputs k f ′ ( x ′ , y ), where y is his actual input. Therefore from Thm. 7, with probability 1 − ǫ over the s amples y 1 , . . . , y m , X y ∈Y π ( y ) · | f ′ ( x ′ , y ) − f ′ ( x, y ) | ≤ ǫ/k . (9) Note that, ( f ′ ( x ′ , y ) 6 = f ′ ( x, y )) ⇒ | f ′ ( x ′ , y ) − f ′ ( x, y ) | ≥ 1 /k . Hence for samples y 1 , . . . , y m , for which (9 ) holds, using Mar ko v’s ineq uality , we ha ve Pr y ← µ x [ f ′ ( x ′ , y ) 6 = f ′ ( x, y )] ≤ ǫ . Therefore, for any fixed x , the error of P is at most 2 ǫ and hence also the ov erall erro r of P is at most 2 ǫ . F r om Thm. 11 and Lem. 5, w e can verify that the exp ected leng th of M 1 in P will b e at most 4 m · I ( X : Y ) + l . The length of M 2 is a t mos t O ( m lo g k ), s ince using a prefix fre e enco ding each f ′ ( x, y i ) can b e sp ecified in O (log k ) bits. 4 A new lo w er b ound on quan tum one-w ay distributional commu nication complexit y In this section w e present our low er b o und on the qua nt um one-wa y distributional communication complexity of a function f , in terms of the one-way re c tangle b ound of f . W e beg in with a few definitions leading to the definition of the one-wa y re c tangle bound. Definition 4 (R ectangle). A o ne-wa y rectangle R is a set S × Y , wh er e S ⊆ X . F or a distribution µ over X × Y , let µ R r epr esent the distribution arising fr om µ c onditione d on the event R and let µ ( R ) r epr esent the pr ob ability (under µ ) of t he event R . Definition 5 (O ne-w a y ǫ -mono chromatic). L et f ⊆ X × Y × Z b e a r elation. We c al l a dis- tribution λ on X × Y , one-wa y ǫ -mono chromatic for f if ther e is a function g : Y → Z such t hat Pr X Y ∼ λ [( X, Y , g ( Y )) ∈ f ] ≥ 1 − ǫ . Definition 6 (R ectangle b ound). L et f ⊆ X × Y × Z b e a r elation. F or distribution µ on X × Y , the one- wa y rectangle b ound is define d as: rec 1 ,µ ǫ ( f ) def = min { lo g 2 1 µ ( R ) : R is one-way r e ctangle and µ R is one-way ǫ -mono chr omatic } . The o ne-wa y rectangle b ound for f is define d as: rec 1 ǫ ( f ) def = max µ rec 1 ,µ ǫ ( f ) . We also define, rec 1 , [] ǫ ( f ) def = max µ :pro duct rec 1 ,µ ǫ ( f ) . W e r estate our precis e result here follow ed by its pro of. Theorem 13. L et f : X × Y → Z b e a total function and let ǫ ∈ (0 , 1 / 2) b e a c onstant. L et µ b e a pr o duct distribution on X × Y and let rec 1 ,µ ǫ ( f ) > 2(log (1 /ǫ )) . Then, Q 1 ,µ ǫ 3 / 8 ( f ) ≥ 1 2 · (1 − 2 ǫ ) · ( S ( ǫ/ 2) − S ( ǫ/ 4)) · ( ⌊ rec 1 ,µ ǫ ( f ) ⌋ − 1) . If f : X × Y → Z ∪ {∗} is a p artial function then, Q 1 ,µ ǫ 6 / (2 · 15 4 ) ( f ) ≥ 1 2 · (1 − 2 ǫ ) · ǫ 2 300 · ( ⌊ rec 1 ,µ ǫ ( f ) ⌋ − 1) . W e b eg in with the following information theoretic fact. Lemma 6. L et 0 ≤ d < c ≤ 1 / 2 . L et Z b e a binary r andom variable with min { Pr( Z = 0) , Pr( Z = 1) } ≥ c . L et M b e a c orr elate d qu antum system. L et Z ′ b e a classic al b o ole an r andom variable obtaine d by p erforming a me asur ement on M such t hat, Pr( Z 6 = Z ′ ) ≤ d , then I ( Z : M ) ≥ I ( Z : Z ′ ) ≥ S ( c ) − S ( d ) . Pr o of. The first inequalit y follo ws from the Holev o bound, Thm. 8. F or the second inequalit y w e note that S ( Z ) ≥ S ( c ) (since the binary e ntropy function is monotonically increasing in (0 , 1 / 2]) and from F ano ’s ine q uality , Lem. 1, we have S ( Z | Z ′ ) ≤ S ( d ). Therefore, I ( Z : Z ′ ) = S ( Z ) − S ( Z | Z ′ ) ≥ S ( c ) − S ( d ) . W e a re now r eady for the pro of of Thm. 13. Pro of of Thm . 13: F or total bo olean func tion s : F o r simplicity of the e x planation, we fir st present the pro of assuming f to b e a total b o olean function. Le t r def = ⌊ rec 1 ,µ ǫ ( f ) ⌋ or ⌊ rec 1 ,µ ǫ ( f ) ⌋ − 1 so as to make r ev en. Let P be the optimal one-wa y quantum proto co l for f with distributional err or under µ at most ǫ 3 / 4. (Although we ha ve ma de a stro nger assumption regarding the er ror in the statement of the Theorem, we do not need it her e and will only need it later while handling non- bo olean functions.) Let M represent the m def = Q 1 ,µ ǫ 3 / 4 ( f ) qubit quantum message of Alice in P . Let X Y be the random v ariables corres p o nding to Alice and Bob ’s inputs, jointly distributed according to µ . Our inten tion is to define binary r andom v aria ble s T 1 , . . . , T r / 2 such that they are determined b y X (and hence a specific v a lue for T 1 , . . . , T r / 2 would cor resp ond to a subset of X ) and ∀ i ∈ { 0 , . . . , r 2 − 1 } , I ( M : T i +1 | T 1 . . . T i ) ≥ (1 − 2 ǫ ) · ( S ( ǫ/ 2) − S ( ǫ/ 4)) . Therefore from F a c t 3 and the c hain rule of m utual information, Eq. (6), we ha ve, m ≥ S ( M ) ≥ I ( M : T 1 . . . T r / 2 ) = r / 2 − 1 X i =0 I ( M : T i +1 | T 1 . . . T i ) ≥ (1 − 2 ǫ ) · ( S ( ǫ/ 2 ) − S ( ǫ/ 4)) · r 2 . This completes our pr o of. W e define T 1 , . . . , T r / 2 in an inductive fashion. F or i ∈ { 0 , . . . , r 2 − 1 } , ass ume that w e hav e defined T 1 , . . . , T i and w e intend to define T i +1 . Let GOOD 1 be the set of strings t ∈ { 0 , 1 } i such that Pr( T 1 , . . . , T i = t ) > 2 − r . Then, Pr( T 1 , . . . , T i ∈ GOOD 1 ) ≥ 1 − 2 − r + i ≥ 1 − 2 − r / 2 − 1 . Let ǫ t be the err or of the proto co l P conditioned on T 1 , . . . , T i = t . No te that E [ ǫ t ] is the same as the ov erall exp ected error of P ; hence E [ ǫ t ] ≤ ǫ 3 / 4. No w using Markov’s inequality we get a set GOOD 2 ∈ { 0 , 1 } i such that Pr( T 1 . . . T i ∈ GOOD 2 ) ≥ 1 − ǫ and ∀ t ∈ GO OD 2 , ǫ t ≤ ǫ 2 / 4. Let GOOD def = GOOD 1 ∩ GOOD 2 . Therefore (s inc e r / 2 > log (1 /ǫ ), from the h yp othesis of the theo rem), Pr( T 1 . . . T i ∈ GOOD ) ≥ 1 − 2 − r / 2 − 1 − ǫ ≥ 1 − 2 ǫ. (10) F o r t ∈ { 0 , 1 } i and y ∈ Y , let δ t,y def = min { P r[ f ( X , y ) = 0 | ( T 1 . . . T i = t )] , Pr[ f ( X , y ) = 1 | ( T 1 . . . T i = t )] } . Also let, ǫ t,y be the exp ected err or o f P conditioned on Y = y a nd T 1 . . . T i = t . F o r t / ∈ GOOD , w e define T i +1 | ( T 1 . . . T i = t ) = 0. Let t ∈ GOOD from now on. Our inten tion is to iden tify a y t ∈ Y , s uch that ǫ t,y t ≤ ǫ/ 4 and δ t,y t ≥ ǫ/ 2. W e will then let T i +1 | ( T 1 . . . T i = t ) to be f ( X , y t ) | ( T 1 . . . T i = t ). Lem. 6 will now imply , I ( M : T i +1 | ( T 1 . . . T i = t )) ≥ S ( ǫ/ 2) − S ( ǫ / 4). Therefore, I ( M : T i +1 | T 1 . . . T i ) ≥ X t ∈ GOOD Pr( T 1 . . . T i = t ) · I ( M : T i +1 | ( T 1 . . . T i = t )) ≥ (1 − 2 ǫ ) · ( S ( ǫ / 2) − S ( ǫ/ 4)) (using Eq. 10) and w e would b e done. Now in or der to identify a desired y t , we pro ceed as follows. Since r ≤ rec 1 ,µ ǫ ( f ); from the definition of r e ctangle b ound and given that µ is a pro duct distribution we hav e the following. F or all S ⊆ X with µ ( S × Y ) > 2 − r or in other words with Pr[ X ∈ S ] > 2 − r , E y ← Y  min { Pr[ f ( X, y ) = 0 | X ∈ S ] , Pr[ f ( X, y ) = 1 | X ∈ S ] }  > ǫ. (11) Note that since t ∈ GOOD , Pr[ T 1 . . . T i = t ] > 2 − r . Hence (11) implies that E y ← Y [ δ t,y ] > ǫ . Now using Mar ko v’s inequality and the fact that, ∀ ( t, y ) , δ t,y ≤ 1 / 2, we get a set GOOD t ⊆ Y such that Pr[ Y ∈ GOOD t ] ≥ ǫ and ∀ y ∈ GOOD t , δ t,y ≥ ǫ/ 2. Since t ∈ GOOD , w e have ǫ t ≤ ǫ 2 / 4. Note that ǫ t = E y ← Y [ ǫ t,y ]. Using a Mar kov a rgument again we finally get a y t ∈ GOO D t , such that ǫ t,y t ≤ ǫ/ 4. Note that since y t ∈ GOO D t , we hav e δ t,y t ≥ ǫ/ 2 and w e are done. F or total no n-b o olean functio ns: Let f : X × Y → Z be a total non-b o olean function and let r be as b efor e. W e follo w the sa me inductiv e argument a s b efor e to define T 1 . . . T r / 2 . F or i ∈ { 0 , . . . , r 2 − 1 } , assume that we hav e defined T 1 . . . T i . As b efore w e ident ify a s e t GOOD ⊆ { 0 , 1 } i with Pr[ T 1 . . . T i ∈ GOOD ] ≥ 1 − 2 ǫ , such that ∀ t ∈ GOOD , Pr [ T 1 . . . T i = t ] > 2 − r and ǫ t ≤ ǫ 2 / 8. Since r ≤ rec 1 ,µ ǫ ( f ), from the definition of rectangle b ound and the fact that µ is pro duct, we hav e , ∀ S ⊆ X with µ ( S × Y ) > 2 − r , E y ← Y  max z ∈Z { Pr[ f ( X , y ) = z | X ∈ S ] }  < 1 − ǫ. (12) F o r t ∈ { 0 , 1 } i and y ∈ Y , let ǫ t,y be a s before and let, δ t,y def = max z ∈Z { Pr[ f ( X , y ) = z | ( T 1 . . . T i = t )] } . F o r t / ∈ GOOD , le t us define T i +1 | ( T 1 . . . T i = t ) to be 0. Let t ∈ GOOD from now on. Note that (12) implies E y ← Y [ δ t,y ] < 1 − ǫ . Using Markov’s inequality we get a set GOOD t ⊆ Y with Pr[ Y ∈ GOOD t ] ≥ ǫ/ 2 and ∀ y ∈ GOOD t , δ t,y ≤ 1 − ǫ/ 2 . Since E y ← Y [ ǫ t,y ] = ǫ t ≤ ǫ 2 / 8, a gain using a Marko v argument w e get a y t ∈ GOOD t , such that ǫ t,y t ≤ ǫ/ 4. Since δ t,y t ≤ 1 − ǫ/ 2 (and ǫ ∈ (0 , 1 / 2 )), observe that there w ould exist a set S t,y t ⊆ Z such that, min { Pr[ f ( X , y t ) ∈ S t,y t | ( T 1 . . . T i = t )] , Pr[ f ( X, y t ) ∈ Z − S t,y t | ( T 1 . . . T i = t )] } ≥ ǫ/ 2 . Let us now define T i +1 | ( T 1 . . . T i = t ) to b e 1 if a nd only if f ( X, y t ) ∈ S t,y t | ( T 1 . . . T i = t ) and 0 otherwise. Note that since ǫ t,y t ≤ ǫ/ 4, conditioned o n T 1 . . . T i = t , there exists a measurement on M , that can predict the v alue of T i +1 with success probability a t lea st 1 − ǫ/ 4. The rest of the pro of follows as b e fore. F or partial non-b o ol ean functions: Let f : X × Y → Z ∪ {∗ } b e a partial function and let r be as be fo re. Let i ∈ { 0 , . . . , r 2 − 1 } . W e follow a simila r inductive argument as in the case of total non-b o olean functions, except for th e definition of T i +1 | ( T 1 . . . T i = t ). As b efore we ident ify a se t GOOD ⊆ { 0 , 1 } i with Pr[ T 1 . . . T i ∈ GOOD ] ≥ 1 − 2 ǫ , such that ∀ t ∈ GOOD , P r[ T 1 . . . T i = t ] > 2 − r and ǫ t ≤ ǫ 5 / (2 · 15 4 ). Since r ≤ rec 1 ,µ ǫ ( f ), from the definition of rectangle b ound and the fact that µ is pr o duct, we hav e the following. F or all S ⊆ X with µ ( S × Y ) > 2 − r , E y ← Y  max z ∈Z { Pr[ f ( X , y ) = ( z or ∗ ) | X ∈ S ] }  < 1 − ǫ. (13) F o r t ∈ { 0 , 1 } i and y ∈ Y , let ǫ t,y be a s before and let δ t,y def = max z ∈Z { Pr[ f ( X , y ) = ( z or ∗ ) | ( T 1 . . . T i = t )] } . F o r t / ∈ GOO D , let us define T i +1 | ( T 1 . . . T i = t ) to b e 0. Let us assume t ∈ GOOD from now on. Let GOOD t ⊆ Y b e such that ∀ y ∈ GOOD t , δ t,y ≤ 1 − ǫ/ 2. Using Mark ov arguments as b efore we get a y t ∈ GOOD t , such tha t δ t,y t ≤ 1 − ǫ/ 2 and ǫ t,y t ≤ ( ǫ/ 1 5) 4 def = ǫ ′ . Since δ t,y t ≤ 1 − ǫ/ 2 it implies Pr[ f ( X , y t ) = ∗ ] ≤ 1 − ǫ/ 2. Obser ve no w that can w e get a s e t S t,y t ⊆ Z such that, min { Pr[ f ( X , y t ) ∈ S t,y t | ( T 1 . . . T i = t )] , P r[ f ( X , y t ) ∈ Z − S t,y t | ( T 1 . . . T i = t )] } ≥ ǫ/ 6 . (14) Let O be the o utput of Bob when Y = y t . All along the arg uments below we condition on T 1 . . . T i = t . Note that since Bob outputs some z ∈ Z even if f ( x, y ) = ∗ , let us assume without loss of genera lity that q def = P r[ O ∈ S t,y t ] ≥ 1 / 2 (otherwise similar a rguments would ho ld b y switching the roles of S t,y t and Z − S t,y t ). Let us define T i +1 to b e 1 if ( f ( X , y t ) ∈ S t,y t ∪ {∗ } ) and 0 otherwise. Note that Eq. (14) implies P r[ T i +1 = 1] ≤ 1 − ǫ/ 6. Now, q = Pr[ O ∈ S t,y t | ( T i +1 = 1)] · Pr[ T i +1 = 1] + Pr[ O ∈ S t,y t and T i +1 = 0] ≤ Pr[ O ∈ S t,y t | ( T i +1 = 1)] · Pr[ T i +1 = 1] + ǫ ′ ≤ Pr[ O ∈ S t,y t | ( T i +1 = 1)] · (1 − ǫ/ 6) + ǫ ′ This implies, Pr[ O ∈ S t,y t | ( T i +1 = 1)] ≥ q − ǫ ′ 1 − ǫ/ 6 ≥ ( q − ǫ ′ )(1 + ǫ / 6) = q + q ǫ/ 6 − ǫ ′ (1 + ǫ/ 6) ≥ q + ǫ/ 12 − ǫ (1 + 1 / 12) / (2 3 · 1 5 4 ) (since q ≥ 1 / 2 and ǫ ≤ 1 / 2 ) ≥ q + 0 . 08 ǫ Let us define O ′ = 1 iff O ∈ S t,y t and O ′ = 0 otherwise. Then, I ( M : T i +1 ) ≥ I ( O ′ : T i +1 ) = S ( O ′ ) − Pr[ T i +1 = 1] · S ( O ′ | ( T i +1 = 1)) − Pr[ T i +1 = 0] · S ( O ′ | ( T i +1 = 0)) ≥ S ( q ) − S ( q + 0 . 08 ǫ ) − S ( ǫ ′ ) ≥ 1 − S (0 . 5 + 0 . 08 ǫ ) − S ( ǫ ′ ) ≥ 1 − (1 − 2(0 . 08 ǫ ) 2 ) − 2( ǫ/ 15) 2 ≥ ǫ 2 / 300 The third inequality ab ove follo ws since the function S ( p ) is concav e a nd monotonically decreasing in [ 1 2 , 1]. The fourth ineq uality follows from F act 1. The rest of the pro o f follows as befor e. ⊓ ⊔ 5 Application: Security of b o olean extractors against quan tum adv ersaries In this sec tio n w e present a consequence our low er bound result Thm. 1 3 to prove security of extrac- tors against quan tum a dversar ies. In this s e c tion w e are only concerned with bo olean extractors . W e beg in with following definitions. Definition 7 (M in-entr opy). L et P b e a distribution on [ N ] . The min-entrop y of P d enote d S ∞ ( P ) is define d to b e − log max i ∈ [ N ] P ( i ) . Definition 8 (Strong extractor). L et ǫ ∈ (0 , 1 / 2) . L et Y b e uniformly distribute d on Y . A stro ng ( k , ǫ )-ex tr actor is a function h : X × Y → { 0 , 1 } such that for any r andom variable X distribute d on X which is indep endent of Y and with S ∞ ( X ) ≥ k we have, k h ( X , Y ) Y − U ⊗ Y k 1 < 2 ǫ, wher e U is the uniform distribution on { 0 , 1 } . In other wor ds, even given Y (and not X ); h ( X, Y ) is stil l close (in ℓ 1 distanc e) to b eing a uniform bit. Let X , Y , h be a s in the definition a b ov e. Let us c o nsider a r andom v aria ble M , taking v alues in some set M , correlated with X and indep endent of Y . Let us now limit the cor r elation that M has with X , in the s ense that ∀ m ∈ M , S ∞ ( X | M = m ) ≥ k . Since h is a stro ng ( k , ǫ )-extractor , it is easy to verify that in such a case, ∀ m ∈ M , k h ( X , Y ) Y | ( M = m ) − U ⊗ Y | ( M = m ) k 1 < 2 ǫ ⇒ k h ( X , Y ) Y M − U ⊗ Y M k 1 < 2 ǫ In other w ords, still close (in ℓ 1 distance) to be ing a uniform bit. Now let us ask what happ ens if the system M is a quantu m system. In that case, is it still true that given M and Y , h ( X, Y ) is clos e to b eing a uniform bit? This question has b een increas ing ly studied in recen t times specially for its applications for example in priv acy amplification in Quantu m key distribution proto co ls and in the Quant um bo unded storage models [KMR05,KR0 5,KT0 8]. How ever when M is a quantum system, the min-entropy of X , conditioned on M , is not easily captured since co nditioning on a quantum sy s tem needs to b e ca refully defined. An a lternate wa y to ca ptur e the cor relation b etw een X and M is via the guessing pr ob ability . Let us consider the following definition. Definition 9 (Gue s sing-entrop y). L et X b e a classic al r andom variable t aking values in X . L et M b e a c orr elate d quantum s yst em with the joint classic al-quantum state b eing ρ X M = P x Pr[ X = x ] | x ih x | ⊗ ρ x . Then the guessing-entropy of X given M , denote d S g ( X ← M ) is define d t o b e: S g ( X ← M ) def = − lo g max E X x Pr( X = x ) T r ( E x ρ x ) wher e the maximum is taken over al l POVM s E def = { E x : x ∈ X } . (Ple ase r efer to [NC00] for a definition of POVM s). The guessing -entrop y turns out to b e a useful notion in the quan tum co ntexts. Let h, X , Y , M b e as befo re, where M is a qua nt um system. K¨ onig a nd T er hal [KT08] hav e (r oughly) shown that if the guessing en tropy S g ( X ← M ), is at lea s t k , then given M and Y (and no t X ), h ( X, Y ) is still clos e to a uniform bit. W e state their precise result here. Theorem 14. L et ǫ ∈ (0 , 1 / 2 ) . L et h : X × Y → { 0 , 1 } b e a st ro ng ( k , ǫ ) -extr actor. L et U b e the uniform distribut ion on { 0 , 1 } . L et Y X M b e a classic al-quantum system with Y X b eing classic al and M quantum. L et Y b e uniformly distribute d and indep endent o f X M and, S g ( X ← M ) > k + log 1 /ǫ. Then, k h ( X , Y ) Y M − U ⊗ Y M k 1 < 6 √ ǫ. W e show a similar result as follo ws. Theorem 15. L et ǫ ∈ (0 , 1 / 2) . L et h : { 0 , 1 } n × { 0 , 1 } m → { 0 , 1 } b e a str ong ( k, ǫ ) -extr actor. L et U b e the uniform distribution on { 0 , 1 } . L et Y X M b e a classi c al-quantum system with Y X b eing classic al and M quantum. L et X b e uniformly distribute d on { 0 , 1 } n . L et Y b e uniformly distribute d on { 0 , 1 } m and indep endent of X M and, I ( X : M ) < b ( ǫ ) · ( n − k ) . (15) Then, k h ( X , Y ) Y M − U ⊗ Y M k 1 < 1 − a ( ǫ ) (16) wher e a ( ǫ ) def = 1 4 · ( 1 2 − ǫ ) 3 and b ( ǫ ) def = ǫ · ( S ( 1 4 − ǫ 2 ) − S ( 1 8 − ǫ 4 )) . Before proving Thm. 15, we will mak e a few p oints co mparing it with Thm. 14. 1. Let’s obser ve that if M is a classical system, then S g ( X ← M ) = − log E m ← M [2 − S ∞ ( X | M = m ) ] ≤ E m ← M [ S ∞ ( X | M = m ) · log e 2] ≤ E m ← M [ S ∞ ( X | M = m )] ≤ S ( X | M ) The fir st ineq ua lity follows from the co nv exity of the exp onential function. The last inequality follows easily from definitions. This implies , I ( X : M ) = S ( X ) − S ( X | M ) ≤ S ( X ) − S g ( X ← M ) . (17) So if M is clas sical, then the implication of Thm. 15 a ppea rs stronger than the implica tion in Thm. 14 (although b eing weak in terms o f the dependence on ǫ .) W e cannot show the inequality (17) when M is a quantum system but conjecture it to b e true. If the conjecture is true, Thm. 15 would hav e stronger implication than Thm. 14 in the qua nt um case as well. 2. The pr o of o f Thm. 1 4 in [KT0 8] crucially uses some prop erties o f the so called pr etty go o d me asur ements ( PGM s ). Our result follows here without using PG M s and via completely differen t arguments. 3. Often in applications concerning the Quantum bo unded stora ge mo del, an upp er b ound on the nu mber of qubits o f M is a v ailable. This implies the sa me uppe r b ound on I ( X : M ). If this bo und is sufficiently small suc h that it suffices the assumption of Thm. 15, then h could b e used to extract a pr iv ate bit successfully , in the presence of a quantum adversary . Let us return to the pro o f o f Thm. 15. W e b egin with the following key observ ation. It essentially states that a bo olea n function whic h can extract a bit from sources of lo w min-en tropy has high one-wa y r ectangle bound under the uniform distribution. Lemma 7. L et ǫ ∈ (0 , 1 / 2 ) . L et h : { 0 , 1 } n × { 0 , 1 } m → { 0 , 1 } b e a stro ng ( k , ǫ ) -ext r actor. L et µ def = U n ⊗ U m , wher e U n , U m ar e uniform distributions on { 0 , 1 } n and { 0 , 1 } m r esp e ctively. Then rec 1 ,µ 1 / 2 − ǫ ( h ) > n − k . Pr o of. Let R def = S × { 0 , 1 } m be any one-w ay rectangle where S ⊆ { 0 , 1 } n with µ ( R ) ≥ 2 − n + k which essentially means that | S | ≥ 2 k . Let X be uniformly distributed on S . This implies that S ∞ ( X ) ≥ k . Let Y b e uniformly dis tributed on { 0 , 1 } m . Since h is a strong extractor, from Definition 8 w e ha ve (where U is the uniform distribution on { 0 , 1 } ): k h ( X , Y ) Y − U ⊗ Y k 1 < 2 ǫ ⇔ E y ← Y [ k h ( X , y ) − U k 1 ] < 2 ǫ W e note that from Definition 5, above implies that µ R is not 1 / 2 + ǫ mono chromatic. Hence from the definition of the r ectangle b o und, Definition 6 we ha ve rec 1 ,µ 1 / 2 − ǫ ( h ) > n − k . W e will also need the following information theoretic fact. Lemma 8. L et RQ b e a joint classic al-quantum system wher e R is a classic al b o ole an r andom variable. F or a ∈ { 0 , 1 } , let the quantum state of Q when R = a b e ρ a . Then ther e is a me asur ement that c an b e done on Q to guess value of R with pr ob ability 1 2 + 1 2 · k R Q − U ⊗ Q k 1 . Pr o of. L et us note that k RQ − U ⊗ Q k 1 = k P r[ R = 0] ρ 0 − Pr[ R = 1] ρ 1 k 1 . Now Helstro m’s The or em (Thm. 9) imme diately helps u s c onclude the desir e d. W e a re now r eady for the pro of of Thm. 15. Pro of of Thm . 15: W e prove our r esult in the contrap ositive manner. Let, k h ( X , Y ) M Y − U ⊗ M Y k 1 > 1 − a ( ǫ ) . Note that this is equiv alent to: E y ← Y [ k h ( X , y ) M − U ⊗ M k 1 ] > 1 − a ( ǫ ) . (18) Let’s consider a one-way comm unication proto col P for h where the inputs X a nd Y of Alice and Bob r e s p e c tively ar e drawn independently from the uniform distributions on { 0 , 1 } n and { 0 , 1 } m resp ectively . Let µ b e the distribution o f X Y . No w let M be sent as the messa ge of Alice in P . Note that now (18) along with Lem. 8 implies that the distributional erro r of P will be at most a ( ǫ ) / 2 = 1 8 · ( 1 2 − ǫ ) 3 . Let ǫ ′ def = 1 / 2 − ǫ . Therefor e P has dis tributional error at most ǫ ′ 3 / 8. Arguing as in the pr o of of Thm. 13 we get that, I ( X : M ) ≥ 1 2 · (1 − 2 ǫ ′ )( S ( ǫ ′ / 2) − S ( ǫ ′ / 4)) · rec 1 ,µ ǫ ′ ( h ) = ǫ · ( S ( 1 4 − ǫ 2 ) − S ( 1 8 − ǫ 4 )) · rec 1 ,µ 1 / 2 − ǫ ( h ) = b ( ǫ ) · rec 1 ,µ 1 / 2 − ǫ ( h ) > b ( ǫ ) · ( n − k ) The last inequalit y follows from Lem. 7 s ince h is a strong ( k , ǫ )-e xtractor. ⊓ ⊔ 6 Conclusion In the wake of our quantum low er bound result, it is natur al to ask whether in the tw o-way model also, ther e is a similar relationship b etw een quantum distributional comm unication complexity of a function f , under pro duct distributions, and the cor resp onding rectangle b ound. 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Output: F o r a matching M and a string x , let M x repr esent the n bit string corr esp onding to the n edg es of M o btained as follows. F or an edge e def = ( i, j ) in M the bit included in M x is x i ⊕ x j , where x i , x j represent the i, j -th bit of x . Output bit b ∈ { 0 , 1 } if a nd o nly if the Hamming distance b etw een string s ( M x ) ⊕ b n and w is at most n/3. If there is no suc h bit b then output 0. Now let the no n-pro duct distribution µ on inputs of Alice and Bob b e as follows. Le t Ali ce be given x drawn uniformly from { 0 , 1 } n . Let Bob b e given matching M drawn uniformly from the set of all matchings on [2 n ]. With probability 1 / 2, Bob is given w uniformly from the set of all s trings with Ha mming distance at most n/ 3 from M x and with proba bilit y 1 / 2, he is given w uniformly from the s et of all strings with Hamming distance at most n/ 3 from ( M x ) ⊕ 1 n . Note that in µ ther e is correlation b etw een the inputs of Alice a nd Bob and hence µ is non-pro duct. No w w e ha ve the following. Theorem 16 ([GKK + 07], impli cit). L et n ≥ 1 b e a su fficiently lar ge inte ger and let ǫ ∈ (0 , 1 / 2 ) . L et NPM n and µ b e as describ e d ab ove. Then, rec 1 ,µ ǫ ( NPM n ) = Ω ( √ n ) wher e as Q 1 ,µ ǫ ( NPM n ) = O (log n ) .