Preparation contextuality as an essential feature underlying quantum communication advantage
PPreparation contextuality as an essential feature underlying quantum communicationadvantage
Debashis Saha
1, 2 and Anubhav Chaturvedi Institute of Theoretical Physics and Astrophysics, National Quantum Information Centre,Faculty of Mathematics, Physics and Informatics, University of Gda ´nsk, 80-952 Gda ´nsk, Poland Center for Theoretical Physics, Polish Academy of Sciences, Al. Lotników 32/46, 02-668 Warsaw, Poland
The study of ontology (hidden variables) provides for a vital ground on which significant non-classical features of quantum theory are revealed. One such non-classical ontic-feature is preparationcontextuality ( PC ) and advantage in oblivious communication tasks is one of its operational signa-tures. This article primarily pursues the ontic-feature underlying quantum advantage in communi-cation complexity ( CC ). We construct oblivious communication tasks tailored to given CC problems.We upper-bound the classical success probability of these oblivious communication tasks, obtain-ing preparation non-contextual inequalities. We use the very states and measurements responsiblefor advantage in CC problems along with the orthogonal mixtures of these states to orchestrate anadvantageous protocol for the associated oblivious communication tasks and the violation of the as-sociated inequalities, thereby unveiling PC . To showcase the vitality of our results, we find a criterionfor unbounded violation of these inequalities and demonstrate the same for two widely studied CC problems. I. INTRODUCTION
Quantum resources coupled with ingenious quantumprotocols have outshone their classical counterparts ina wide range of computation, communication, and in-formation processing tasks. But there is little insightinto what makes quantum theory stand out. The theory-specific features such as superposition do not make in-sightful answers for they are cyclic in the sense thatthey refer back to the operational quantum formalismwhich was a priori responsible for the advantageouspredictions. Therefore, any comprehensive approachto this question must arguably invoke a ground com-mon to both classical and quantum theories, on whichnon-classical features of the latter are unveiled. Thestudy of hidden variables (ontology) provides for sucha ground. Any ontological model that seeks to explainthe predictions of operational quantum formalism musthave certain non-classical features [1, 2]. Introduced in[3], the ontic-feature of preparation contextuality (
P C )discards any preparation non-contextual (
P N C ) mod-els as viable ontological descriptions of an operationaltheory. An ontological model is said to be
P N C ifit assigns identical ontic-distributions to operationallyindistinguishable preparations [3]. Quantum theorymanifests preparation contextuality (
P C ), i.e., it pos-tulates certain operationally indistinguishable prepara-tions which must have non-identical underlying ontic-distributions. Quantum protocols siphon this ontologi-cal distinguishability to an advantage in oblivious com-munication ( OC ) tasks i.e. any advantage in OC taskswitnesses P C [4–7].One of the predominant manifestations of quantumcommunication advantage is captured in communica-tion complexity ( CC ). The notion of CC , introduced inthe seminal paper [8], is an important aspect of complex-ity theory, which quantifies the amount of communica- tion required for distributed computation. Apart frommainstream applications in algorithmic mechanism de-sign, game theory and cryptography, lower bounds in CC can be used to prove lower bounds in decision treecomplexity, data structures, space-time trade-offs forTuring machines and more [9]. Quantum resources andstrategies have demonstrated supremacy in a multitudeof CC problems [10–15]. In this article, we substantiate afundamental link between quantum CC advantage and P C . Specifically, we establish that quantum advantagein CC manifests P C . We begin by constructing an OC task tailored to a given instance of the generic CC prob-lem. We orchestrate advantageous quantum strategiesfor the constructed OC tasks based on advantageous ( i ) one-way prepare and measure quantum CC protocols, ( ii ) two-way multi-round quantum CC protocols and, ( iii ) entanglement assisted classical communication CC protocols. These OC strategies utilize the same quan-tum set-up responsible for advantage in the CC task.Specifically, we provide a family of P N C inequalitiestailored to CC tasks and show that quantum CC advan-tage implies a violation of these inequalities, subject tocertain conditions. Additionally, we obtain a criterionfor unbounded violation of these P N C inequalities anddemonstrate it for two widely studied CC problems withexponential quantum advantage. We present an alter-native construction of the OC task and discuss the po-tential extension of our results to general probabilistictheories. Next, we use the machinery thus developed toprovide a complete proof of the fact (originally stated in[6]) that violation of (spatial or temporal) Bell inequal-ities [16–18] implies an advantage in an associated OC task. Finally, we conclude with a discussion of the im-plications of this work. a r X i v : . [ qu a n t - ph ] A ug II. PRIMITIVES
In this section, we lay down the framework we em-ploy in our investigation. Specifically, we introducethe generic formulations of CC problems and OC tasks,which form the key subjects of this article. A. Communication complexity problem
We begin with briefly introducing the generic for-mulation of CC problem. A typical CC problem en-tails two parties Alice and Bob, with inputs x ∈ [ n x ] , y ∈ [ n y ] (where [ n ] = {
0, 1, . . . , n − } ) respectively,distributed according to a prior probability distribution p ( x , y ) . Their task is to compute the value of a binaryoutput bivariate function, f ( x , y ) : [ n x ] × [ n y ] → {
0, 1 } by exchanging messages. Without loss of generality, weassume that Bob guesses the value of f ( x , y ) and hisguess is stored in an output bit z ∈ {
0, 1 } . They achievesuccess with probability, p = ∑ x , y p ( x , y ) p ( z = f ( x , y ) | x , y ) . (1)There are two inter-convertible metrics to gauge theirperformance: ( i ) maximal achievable success proba-bility ( p C d for classical resources and p Q d for quan-tum resources) given a bounded amount of communi-cation (bounded dimension d of the communicated sys-tem), and ( ii ) amount of communication (usually quan-tified in bits, denoted by C ( f , p S ) or qubits, denotedby Q ( f , p S ) ) required to achieve to achieve a specifiedprobability of success (denoted by p S ). Quantum CC ad-vantage implies p Q d > p C d or alternatively Q ( f , p S ) < C ( f , p S ) . B. Oblivious communication task
For this article, we need only invoke a subclass ofgeneral OC tasks (introduced in [5]) wherein Alice’s(sender) input comprises of a pair a = ( a , a ) with a ∈ [ n a ] , a ∈ [ n a ] . Bob (receiver) gets an input b ∈ [ n b ] andyields an output c ∈ [ n c ] . The inputs are distributed ac-cording to a prior probability distribution p ( a , b ) with anadditional condition p ( a | a , b ) = p ( a | a ) . Their task isto guess the value of a function g ( a , b ) : [ n a ] × [ n b ] → [ n c ] . In contrast to CC problems, there is no restrictionon the amount of communication. The communicationis constrained to be completely oblivious to the valueof a . They achieve success with probability defined as p = ∑ a , b p ( a , b ) p ( c = g ( a , b ) | a , b ) .In a classical OC protocol Alice prepares the message m employing an encoding scheme E which comprisesof conditional probability distributions of the form p E ( m | a ) . Bob outputs c based on his input b and the message m using a decoding scheme D entailing con-ditional probability distributions p D ( c | b , m ) . The obliv-ious constraint for classical encoding schemes E readsas, ∀ m , ∀ a , a (cid:48) ∈ [ n a ] , p E ( m ) = p E ( m | a ) = p E ( m | a (cid:48) ) ,(2)where p E ( m | a ) = ∑ a p ( a | a ) p E ( m | a , a ) . This condi-tion ensures that the same classical mixture is preparedfor all values of a . The expression for maximal classicalsuccess probability is, p N C = max {E}{D} (cid:26) ∑ m ∑ b p ( b ) (cid:18) ∑ a p ( a | b ) p E ( m | a ) p D ( g ( a , b ) | b , m ) (cid:19)(cid:27) , (3)where the message m can take arbitrary number of dis-tinct values. We use the subscript N C to reflect the factthat for OC tasks the maximal classical success probabil-ity is the same as the maximal P N C success probability[5, 6].On the other hand, quantum strategy for a OC task in-volves Alice transmitting states of arbitrary dimension, ρ a for input a , such that the same mixed state ρ is pre-pared for all values of a i.e., ∀ a , ∑ a p ( a | a ) ρ a , a = ρ .This ensures adherence to the oblivious condition.Upon receiving input b , Bob performs measurement { M bc } (where ∑ c M bc = I ) on the transmitted system.The average success probability is given by the expres-sion p Q = ∑ a , b p ( a , b ) Tr ( ρ a M bc = g ( a , b ) ) . III. ADVANTAGE IN CC IMPLIES ADVANTAGE IN OC In this section we present our main results. First, wemake a couple of essential observations concerning themaximal classical success probability of OC tasks. Next,we construct an OC task tailored to a given instance ofgeneric CC problem described in the previous section.We then formulate a P N C inequality by obtaining anupper-bound on the classical success probability of the OC task. We utilize the very resources responsible forquantum advantage in the given CC problem (pertain-ing to without prior entanglement i . one-way prepareand measure protocols, ii . two-way multi-round proto-cols and, iii . entanglement assisted classical communi-cation protocols) to orchestrate an advantageous quan-tum protocol for the associated OC task, thereby demon-strating the violation of the P N C inequality. Further,we present two instances of unbounded violations of
P N C inequalities based on CC problems with exponen-tial quantum advantage. Finally, we provide an alterna-tive construction of OC task tailored to given CC prob-lems and discuss the persistence of our results in generalprobabilistic theories. A. Bounding classical success in OC tasks In general, finding maximal classical success proba-bility for OC tasks is an arduous task as ( i ) the dimen-sion of the message is unbounded and, ( ii ) the encodingscheme may be probabilistic. In lieu of these issues weemploy the following lemmas (based on the observationin [5]) to facilitate an upper bound on classical successprobability of the OC task, Lemma 1.
For an instance of the subclass of OC tasks definedin Section II B, the classical success probability p N C is upperbounded in the following way,p
N C ≤ max { q a a } (cid:26) ∑ b p ( b ) max c (cid:26) ∑ a , a p ( a | b ) q a , a δ c , g ( a , b ) (cid:27)(cid:27) (4) where the outer maximization is over a set of variables { q a , a } satisfying the conditions,q a , a ≥ ∑ a q a , a =
1. (5)
Lemma 2.
The set of valid assignments of { q a , a } satisfy-ing the linear constraints (5) form a convex polytope. Theextremal points of this polytope resemble deterministic proba-bility distributions, i.e., any extremal point { q exta , a } is of theform: for each a , q exta , a = for all values of a except a spe-cific ˜ a for which q exta , ˜ a = . The proofs have been deferred to the Appendix. Itfollows from
Lemma { q a , a } . Let the extremalpoint yielding the maximal value be { q ext , maxa , a } . This ex-tremal point without loss of generality entails for each a , an ˜ a where q ext , maxa , ˜ a =
1. Let for each a , ˜ a = e a ,then we have q ext , maxa , a = δ a , e a . Similarly, for the innermaximization suppose that for this extremal point, foreach b the maximal value of ∑ a , a p ( a | b ) q a , a δ c , g ( a , b ) isobtained for c = c b . Consequently, we arrive at the fol-lowing distilled re-expression of (4), p N C ≤ ∑ b p ( b ) ∑ a , a p ( a | b ) δ e a , a δ c b , g ( a , b ) . (6) B. Tailoring OC tasks to given CC problems and PN C inequality
We now present the key ingredient of our modus-operandi, an OC task tailored to a given CC problem.Given an instance of the generic CC problem described above, we construct the following OC task (see Fig.1), a = ( a = x , a ) , b = y , c = z , p ( a , b ) = p ( x , a , y ) = p ( y ) p ( x | y ) p ( a | x ) ,where a ∈ {
0, 1 } , p ( a | x ) = (cid:40) d , if a = d − d , if a = g ( x , a , y ) = f ( x , y ) ⊕ a . (7)Recall, that in the OC task the oblivious condition con-strains the communicated system to not carry any re-trievable information about x . x yf ( x, y ) Alice Bob m ∈ { , , ..., d − } ρ ∈ P ( C d ) x, a y Alice Bob ∀ x, p ( m | x ) = p ( m ) ∀ x, P a p ( a | x ) ρ x,a = ρg ( x, a , y ) = a ⊕ f ( x, y ) CCOC
FIG. 1. Construction of OC task based on a given CC task. No-tice while the amount of communication in the CC task by thedimension d of the physical system, there is no such constrainton communication in the OC . Instead the communication isrestricted so as not to reveal any information about the oblivi-ous variable x . Next, by the means of the following propositionwhich upper-bounds the classical success probability forthe constructed OC task, we present a family of P N C inequalities tailored to CC problems. Proposition 1.
The
P N C success probability of the OC taskdescribed in (7) is upper bounded by the maximal classicalsuccess probability of the CC problem wherein Alice is re-stricted to communicate a two-leveled system, i.e.p N C ≤ p C . (8) Proof.
The proof involves obtaining an upper-bound forthe classical success probability of the OC -task (con-structed above) with the help of Lemma
Lemma CC problem whilst the dimension of the mes-sage is restricted to two. Note that when the communi-cation is restricted to be at-most two-dimensional, two-way multi-round CC protocols are equivalent to one-way CC protocols wherein only Alice is allowed to com-municate a two-level message m ∈ {
0, 1 } to Bob, deem-ing this inequality to be independent of the choice ofprotocol.The expression for maximal classical success probabilityof the CC task when Alice is restricted to transmit a bitof communication p C reads, p C = max { E }{ D } (cid:26) ∑ y p ( y ) (cid:18) ∑ m = ∑ x p ( x | y ) p E ( m | x ) p D ( z = f ( x , y ) | y , m ) (cid:19)(cid:27) ,(9)where Alice’s encoding scheme E entails conditionalprobability distributions of the form p E ( m | x ) and Bob’sdecoding scheme D entails conditional probability dis-tributions of the form p D ( z | y , m ) . On the other hand, itfollows from (6) that the classical success probability ofthe OC task is upper bounded as follows, p N C ≤ ∑ y p ( y ) ∑ x , a p ( x | y ) δ e x , a δ c y , a ⊕ f ( x , y ) = ∑ y p ( y ) ∑ x , a p ( x | y ) δ e x , a δ a , c y ⊕ f ( x , y ) = ∑ x , y p ( x , y ) δ e x , c y ⊕ f ( x , y ) . (10)To complete the proof we demonstrate that this upperbound (RHS of (10)) is achievable in the CC task employ-ing a two-leveled message m ∈ {
0, 1 } . To this end, wepresent the following classical CC protocol, p E ( m | x ) = δ m , e x , p D ( z | y , m ) = δ z , c y ⊕ m . (11)Inserting this strategy in (9), one obtains p C ≥ ∑ y p ( y ) ∑ x , m p ( x | y ) δ m , e x δ f ( x , y ) , c y ⊕ m = ∑ x , y p ( x , y ) δ e x , c y ⊕ f ( x , y ) , (12)which together with (10) yields the desired thesis (9). (cid:117)(cid:116) C. Violation of
PN C inequality from advantageousquantum CC protocols Notice that up-until this point our results are indepen-dent of the specifics of the CC protocol including the re-striction on the amount of communication, but dependonly on the problem itself. Now we take three distinctclasses of the advantageous quantum CC protocols andbased on these, we construct quantum strategies for the OC task to demonstrate the violation of the associated P N C inequalities.
1. One-way prepare and measure quantum CC protocols One-way quantum CC protocols without prior entan-glement are commonly referred to as prepare and measureprotocols . In such protocols, Alice’s state (a qu d it ρ x forinput x ) preparation and transmission is followed by abinary outcome measurement ( { M yz } upon receiving in-put y ) at Bob’s end. The quantum success probability isexpressed as, p Q d = ∑ x , y p ( x , y ) Tr ( ρ x M yz = f ( x , y ) ) . (13)Notice that here, quantum success probability p Q d is notrequired to be maximal. Now we present our result con-cerning P C manifest in advantageous prepare and mea-sure quantum CC protocols. Result 1.
Given a prepare and measure quantum CC proto-col, an advantage is obtained in the OC task described in (7) (p Q > p N C ) whenever the following condition holds, d ( p Q d + d − − χ ) > p C , (14) where χ = ∑ x , y p ( x , y ) Tr (cid:16) M yz = f ( x , y ) (cid:17) and { M yz } are Bob’smeasurements employed in quantum CC protocol.Proof. Our quantum strategy for the OC task describedin (7), involves Alice preparing the same states (as inthe quantum CC protocol described above) when a = ρ x , a = = ρ x and their orthogonal mixtures when a = ρ x , a = = I − ρ x d − . Alice’s preparations are there-fore oblivious to x , as ∀ x : ∑ a p ( a | x ) ρ x , a = I d . Bob’smeasurements remain unaltered from the quantum CC protocol. Plugging the expressions of p ( x , a , y ) from (7)and p Q d from (13), we obtain the following success prob-ability for this strategy, p Q = ∑ x , a = y p ( x , a , y ) Tr ( ρ x M yz = f ( x , y ) )+ ∑ x , a = y p ( x , a , y ) Tr (cid:18) I − ρ x d − M yz = ⊕ f ( x , y ) (cid:19) = d (cid:0) p Q d + d − − χ (cid:1) , (15)where χ = ∑ x , y p ( x , y ) Tr (cid:16) M yz = f ( x , y ) (cid:17) . Now our desiredresult simply follows from (8). (cid:117)(cid:116) Now, given that the CC protocol under considerationis advantageous, i.e. p Q d > p C d , if follows that a quan-tum advantage in the OC task is obtained ( p Q > p N C )whenever the following holds,1 d ( p C d + d − − χ ) ≥ p C . (16)To aid intuition and accessibility we simplify the abovecondition (14) employing two lemmas (the proofs aredeferred to the Appendix): Lemma 3.
For a given prepare and measure quantum CC protocol the following holds, χ ≤ dp G , (17) where χ = ∑ x , y p ( x , y ) Tr (cid:16) M yz = f ( x , y ) (cid:17) , d is dimension ofthe communicated system and p G is guessing probabilitywithout communication. Lemma 4.
Given a CC problem and a classical protocol usinga two-leveled classical message with a success probability p C ,the success probability of a protocol using a d-leveled classicalmessage is lower bounded in the following way,p C d ≥ − exp (cid:18) − log d p C ( p C − ) (cid:19) . (18) Corollary 1.
By substituting the upper bound of χ from (17) in the condition (14) , we find that p Q > p N C wheneverp Q > p C in any CC task with p G = . Corollary 2.
By imposing Lemma 3-4 into (16) , we find thatp Q > p N C whenever the following condition holds,d ( p C + p G − ) + (cid:18) − log d p C ( p C − ) (cid:19) ≤
1. (19)Notice, (19) relies only on classical success probabilityof the CC task with a two-leveled message p C andsuccess probability of the CC task without any commu-nication p G . This in-turn deems (19) to be independentof the specifics of the implementation of classical orquantum CC protocols including the dimension of thecommunicated system.
2. Two-way multi-round quantum CC protocols Even though one-way CC protocols form a predomi-nant subclass of quantum CC protocols, two-way multi-round CC protocols employ relatively more involvedfeatures of quantum theory to facilitate an advantage[12]. In two-way multi-round CC protocols, Alice andBob have access to local quantum memories and ex-change messages over multiple rounds of communica-tion. In each round they use local operations to store animprint of the message on their respective local mem-ories and prepare a message for the next round. Thisresults in complex pre-measurement states wherein Al-ice’s local memory may be entangled with Bob’s localmemory. Remarkably, our results hold intact for quan-tum advantage in CC tasks obtained via two-way multi-round CC protocols.We start by presenting a general two-way multi-round CC protocol denoted by P (first described in [19]). Al-ice and Bob have access to some quantum memory, thestates of respective quantum memory in the round r aresymbolized by A x , yr and B x , yr . These symbols serve for the convenience of description and for mere subscriptsof the quantum state ρ . Each round consists of trans-mission of a message from Alice to Bob and back. Wesymbolize the communicated quantum system from Al-ice to Bob and from Bob to Alice in the round r by α r and β r , respectively. Let the total number of rounds be R . The protocol proceeds as follows,1. Depending on the input x , Alice applies a local op-eration U x on the joint system of her initial mem-ory A and the blank message α to obtain an up-dated combined state ρ α , A x with local memory A x and the message α . Alice then sends the messagei.e. the reduced state ρ α to Bob. In general the up-dated local memory and the message may now beentangled.2. Depending on the input y , Bob applies a local op-eration U y on the joint system of his local memory B and the message from Alice α to obtain his up-dated combined system ρ β , B x , y with local memory B x , y and the message β which is then communi-cated back to Alice. As a result, Bob’s local mem-ory B x , y may be entangled with Alice’s local mem-ory A x .3. This marks the completion of the first round. Al-ice and Bob repeat these steps for R − ( r = R ) upon receiving the messagefrom Alice ( α R ) instead of sending a message backto Alice, Bob performs the measurement { M yz } onthe joint system of the message and Bob’s localmemory from the previous round ( B x , yR − ).Given an upper bound on total dimension of communi-cation d , they achieve success with probability p Q d = ∑ x , y p ( x , y ) Tr ( ρ α R , B x , yR − M yz = f ( x , y ) ) , where ρ α R , B x , yR − is thereduced density matrix corresponding to the joint sys-tem of the message from Alice ( α R ) and Bob’s local mem-ory from the penultimate round B x , yR − .To what follows, it is crucial to obtain an upper-boundon the dimension of Bob’s pre-measurement state. Weachieve this by employing the following steps,1. Following the methodology in [19], we first con-vert a given two-way multi-round quantum com-munication protocol P utilizing log d -qubit (i.e. d dimensional communication) communication toanother protocol ˜ P that employs 2 log d singlequbit exchanges. One can achieve this by splittinga q -qubit message from Alice to Bob (or the otherway round) into q rounds of one qubit exchanges.The new protocol ˜ P has a total of ˜ R = log d − α ˜ R and Bob instead ofsending back one, measures using another mea-surement { ˜ M yz } the joint system of her local mem-ory ˜ B x , y ˜ R − and the qubit message from Alice ˜ α ˜ R .The winning probability for ˜ P is equal to successprobability of P but has the expression p Q d = ∑ x , y p ( x , y ) Tr ( ρ ˜ α ˜ R , ˜ B x , y ˜ R − ˜ M yz ) . (20)2. In the protocol ˜ P , in each round r Bob applies aunitary ˜ U yr on the one qubit message from Alicefrom the previous round ˜ α r − and her local mem-ory ˜ B x , yr − . One can view the unitary operation asa controlled gate acting on the memory with onequbit message being the control. This observationimplies that for a fixed input x , for round r (i.e. af-ter r − r − orthogonal vectors (see Lemma 2 in [19]).This implies that for the last round Bob’s memoryin ˜ P requires at-most ˜ R − ρ ˜ α ˜ R , ˜ B x , y ˜ R − is at-most d -dimensional (or equivalentlylog d -qubits).Now we are prepared to present our result concern-ing P C manifest in advantageous two-way multi-roundquantum CC protocols, Result 2.
Given a two-way multi-round quantum CC proto-col P , an advantage is obtained in the OC task described in (7) with p ( a = | x ) = d n y (p Q > p N C ) whenever thefollowing condition holds, d n y ( p Q d + d n y − − d n y − χ ) > p C . (21) where χ = ∑ x , y p ( x , y ) Tr (cid:16) ˜ M yz = f ( x , y ) (cid:17) and { ˜ M yz } are Bob’smeasurements employed in the derived quantum CC protocol ˜ P .Proof. We begin by devising a quantum strategy for the OC task. We orchestrate a quantum strategy for the OC task based on the quantum two-way multi-round CC protocol. To achieve this we exploit the fact thatthere is no-restriction on the amount of communicationin the OC task. The core idea remains the same as inone-way CC case, Alice sends Bob’s pre-measurementstate when a = a =
1. We start with converting the given quan-tum two-way multi-round CC protocol P which uses d -dimensional communication in total, to one that uses2 log d qubits of communication ˜ P . There is still an is-sue with this approach, Alice does not know the value y in advance, and the pre-measurement state may de-pend on y . In order to deal with this issue, when a = y and sends a tensor product of thesestates as the message Θ x , a = = (cid:78) y ρ ˜ α ˜ R , ˜ B x , y ˜ R − . Recallthat the states ρ ˜ α ˜ R , ˜ B x , y ˜ R − are at-most d -dimensional. When a =
1, Alice sends the orthogonal mixture of Θ x , a = , Θ x , a = = I − (cid:78) y ρ ˜ α ˜ R , ˜ Bx , y ˜ R − d ny − . It is straightforward to see that Alice’s preparation are oblivious to x , as ∀ x , x (cid:48) ∈ [ n x ] , ∑ a p ( a , x ) Θ x , a = ∑ a p ( a , x (cid:48) ) Θ x (cid:48) , a = I d ny . Now,upon receiving the message from Alice, Bob performsthe measurement ˜ M yz on the relevant part (dependingon his input y ) of the message i.e. either ρ ˜ α ˜ R , ˜ B x , y ˜ R − or tr ¬ y ( Θ x , a = ) = d ny − I − ρ ˜ α ˜ R , ˜ Bx , y ˜ R − d ny − . This strategy yields thefollowing success probability, p Q = ∑ x , a = y p ( x , a , y ) Tr ( ρ ˜ α ˜ R , ˜ B x , y ˜ R − ˜ M yz = f ( x , y ) )+ ∑ x , a = y p ( x , a , y ) Tr d n y − I − ρ ˜ α ˜ R , ˜ B x , y ˜ R − d n y − M yz = ⊕ f ( x , y ) = d n y ( p C d + d n y − − d n y − χ ) , (22)where χ = ∑ x , y p ( x , y ) Tr (cid:16) ˜ M yz = f ( x , y ) (cid:17) and p Q d is givenby (20). Now our desired result simply follows from (8). (cid:117)(cid:116) Given quantum advantage in CC problem ( p Q d > p C d )and (22), an advantage is obtained in the OC task ( p Q > p N C ) described in (7) with p ( a = | x ) = d n y when-ever the following holds,1 d n y ( p C d + d n y − − d n y − χ ) > p C .
3. Entanglement assisted classical communication protocols
Another non-equivalent [20, 21] class of advanta-geous quantum CC protocols is that of entanglement as-sisted classical communication protocols, wherein Aliceand Bob share an entangled state ρ AB (a density opera-tor on H A ⊗ H B ), Alice performs a d outcome measure-ment ( { M xm } ) and sends her outcome m as the message.Upon receiving the message m , Bob performs a binaryoutcome measurement ( { M y , mz } ). The quantum guess-ing probability is expressed as p Q d = ∑ x , y p ( x , y ) d − ∑ m = Tr ( ρ AB M xm ⊗ M y , mz = f ( x , y ) ) . (23)Let the reduced density matrix of Bob’s part of the en-tangled state ρ B be of dimension e i.e. e = dim ( H B ) .A quantum strategy for the OC task (7) based on ad-vantageous entanglement assisted classical communica-tion CC protocols and the corresponding condition forretrieving an advantage is presented in the following re-sult, Result 3.
Given a entanglement assisted classical communi-cation CC protocol, an advantage is obtained in the OC task described in (7) with p ( a = | x ) = d (cid:48) (p Q > p N C )whenever the following condition holds, d (cid:48) ( p Q d + d (cid:48) − − χ ) > p C . (24) where χ = ∑ x , y p ( x , y ) Tr (cid:16) M yz = f ( x , y ) (cid:17) and { M yz } are Bob’smeasurements employed in the CC protocol, d (cid:48) = de and e isthe dimension of Bob’s local part of the shared entangled state.Proof. In this case, we capitalize over the fact that theamount of communication is unrestricted in the OC taskand convert the given entanglement assisted classicalcommunication protocol to a prepare and measure pro-tocol wherein Alice simply sends Bob the correspondingpre-measurement state (Bob’s marginal state along withthe classical message). This in-turn enables us to con-struct quantum strategies for the OC task employing theaforementioned methodology.In order to utilize the machinery developed so far wefirst construct a quantum prepare and measure protocoldeploying a d (cid:48) = de dimensional communicated sys-tem but with the same probability of success p Q d asthe given entanglement assisted classical communica-tion protocol. Upon receiving x Alice prepares the state ρ x = | m (cid:105)(cid:104) m | ⊗ ρ B where the state | m (cid:105)(cid:104) m | is simply thequantum encoding of the classical message m into d or-thogonal states. She accomplishes this feat by measur-ing { M xm ⊗ I } on the entangled state ρ AB to which weassume she has access to. The communicated system isof dimension d (cid:48) = de . Bob first retrieves the message byperforming the measurement { M m } on the appropriatesubsystem of the communicated system and depend-ing on it performs the measurement { M y , mz } on rest ofthe communicated system, captured conveniently in ajoint measurement { ˜ M yz = M m ⊗ M y , mz } . This yields thesame success probability p Q d . Now, we convert this pre-pare and measure protocol into an OC protocol utilizingthe methodology described in the proof of Result 1 andobtain the following lower bound on quantum successprobability for the OC task, p Q ≥ d (cid:48) (cid:0) p Q d + d (cid:48) − − χ (cid:1) where χ = ∑ x , y p ( x , y ) Tr (cid:16) M yz = f ( x , y ) (cid:17) and p Q d is givenin (23). This in-turn leads us to the condition for quan-tum advantage in the OC task (24). (cid:117)(cid:116) Notice that in a rather predominant subclass of en-tanglement assisted classical communication protocolsBob applies a completely-positive trace preserving map Λ m on his part of the entangled state ρ B and performsthe measurement { M yz } on Λ m ( ρ B ) . In such cases Alicehaving access to the message m sends ρ x = Λ m ( ρ B ) effectively reducing the dimension of the commu-nicated system in the prepare measure protocol to d (cid:48) = e , thereby improving the feasibility of the quantumadvantage in the OC task. D. Unbounded violation of
PN C inequalities
To demonstrate the vitality of the results obtainedso far we illustrate two examples of unbounded quan-tum violations of
P N C inequalities based of two widelystudied CC problems and associated prepare and mea-sure protocols with exponential quantum advantage.Let us re-write the P N C inequality (8) as α N C ≤ α C ,where α N C = p N C − , α C = p C − . Then a quan-tum advantage in a CC problem adhering to the con-dition (14) implies that there exists quantum protocolfor the OC task with α Q = d ( p Q d + d − − χ ) − .Quantum advantage in CC problems is prevalently re-ported in terms of the amount of communication re-quired to achieve a bounded probability of success p S ,i.e., Q ( f , p S ) < C ( f , p S ) . To apply our results to the in-numerable instances of quantum advantage reported inthis fashion, we employ the following lemma, Lemma 5.
Given a CC problem and a protocol which achievesa success probability p S using C ( f , p S ) bits, the success prob-ability of a protocol using a two-leveled classical message isupper bounded in the following way,p C ≤ + (cid:115) p S C ( f , p S ) . (25)The proof has been deferred to the Appendix. Corollary 3.
The ratio of quantum and
P N C values of α (denoted by β ) can be lower bounded with help of Lemma 3 inthe following way, β ≥ α Q α N C ≥ d ( p Q d + d − − χ ) − p C − ≥ (cid:112) C ( f , p S )( p Q d + d /2 − dp G − ) d (cid:112) p S . (26)To obtain an unbounded violation of the P N C in-equality α N C ≤ α C , it suffices to show that β could bearbitrarily large ( >>
1) [22]. We demonstrate the samefor two widely studied CC problems [23, 24] with expo-nential quantum advantage,1. Vector in a subspace:
Alice is given an n -dimensionalunit vector u and Bob is given a subspace of dimen-sion n /2, S with the promise that either u ∈ S or u ∈ S ⊥ . Their goal is to decide which is the case. Here p Q d = log n = Q ( f , 1 ) = log n , C ( f , p S = ) = Ω ( √ n ) (Theorem 4.2 in [23]) and a simple calculationyields χ = log n , p G = . Inserting these into (26) oneobtains an arbitrarily large lower bound for the ratio β ≥ Ω ( √ n log n ) .2. Hidden matching:
Alice is given a bit string x ∈ {
0, 1 } n of length n and Bob is given y ∈ M n ( M n denotes thefamily of all possible perfect matchings on n nodes).Their goal is to output a tuple ( i , j , t ) such that the edge ( i , j ) belongs to the matching y and t = x i ⊕ x j . Clearlythe hidden matching problem is not a typical CC prob-lem, specifically it is a relational problem. Nevertheless,we can find that the machinery developed so far includ-ing the Proposition
Corollary CC problem. Lemma 6.
For Hidden matching problem an OC task can beconstructed with a success probability p N C , such that p
N C ≤ p C . The proof is similar to the proof of
Proposition CC problems be-yond main-stream functional CC problems. For Hid-den matching p Q d = Q ( f , 1 ) = d = log n , p G = , χ = log n and C ( f , 1 ) = Ω ( √ n ) [24]. Inserting these ob-servations into (26) one obtains an even larger violationas the lower bound on β grows faster, i.e., β ≥ Ω ( √ n log n ) . E. Alternative construction of OC task An equivalent alternative construction of the OC tasktailored to a given CC problem is presented here. Givena general CC problem and an advantageous quantum CC protocol, i.e., p Q d > p C d , we construct the following OC task (shown is Fig. 2), a = ( y , z ) , b = x , c ∈ {
0, 1 } , p ( a , b ) = p ( y , z , x ) = p ( x ) p ( y | x ) p ( z | y ) ,where p ( z | y ) = Tr ( M yz ) d , g ( y , z , x ) = f ( x , y ) ⊕ z . (27)Here { M yz } are Bob’s measurements employed in thegiven quantum CC protocol under consideration, andthe oblivious condition constraints the communicatedsystem to not carry any information about y. Proposition 2.
The
P N C success probability of the OC taskdescribed in (27) is upper bounded by the maximal classicalsuccess probability of the CC problem wherein Alice is re-stricted to communicate a two-leveled system, i.e. p N C ≤ p C .Proof. We follow the same steps as in the proof of
Propo-sition
1. Again employing
Lemmas OC task described in (27), p N C ≤ ∑ x p ( x ) ∑ y , z p ( y | x ) δ e y , z δ c x , z ⊕ f ( x , y ) = ∑ x , y p ( x , y ) δ e y , c x ⊕ f ( x , y ) . (28)Let’s consider the following classical protocol employ-ing a two-leveled message m ∈ {
0, 1 } for the CC prob-lem, p E ( m | x ) = δ m , c x , p D ( z | y , m ) = δ z , m ⊕ e y . (29) x yf ( x, y ) Alice Bob m ∈ { , , ..., d − } ρ ∈ P ( C d ) y, z x Alice Bob ∀ y, p ( m | y ) = p ( m ) ∀ y, P z p ( z | y ) ρ y,z = ρg ( y, z, x ) = z ⊕ f ( x, y ) CCOC
FIG. 2. Alternative Construction of OC task based on a given CC task. The communication is restricted so as not to revealany information about a oblivious variable y . Inserting the above strategy in (9), one obtains the samesuccess probability in CC problem as given in the rightside of (28). (cid:117)(cid:116) The contrasting feature of this construction is thatthe exact duals of the states and measurements used inthe advantageous quantum CC protocol form the corre-sponding measurements and states respectively for thequantum OC protocol. That is, Alice’s preparation forthe OC task are ρ y , z = M yz Tr ( M yz ) and Bob’s measurementfor his input x is { ρ x , I − ρ x } . Clearly Bob remains obliv-ious to y due to the completeness of quantum measure-ments i.e. ∀ y , ∑ z p ( z | y ) ρ y , z = I d . Subsequently, plug-ging the expressions of p ( y , z , x ) from (27) and p Q d from(13), a simple calculation leads to the same expression asin (15), p Q = ∑ y , z = f ( x , y ) , x p ( y , z , x ) Tr (cid:32) ρ x M yz Tr ( M yz ) (cid:33) + ∑ y , z = ⊕ f ( x , y ) , x p ( y , z , x ) Tr (cid:32) ( I − ρ x ) M yz Tr ( M yz ) (cid:33) = d (cid:0) p Q d + d − − χ (cid:1) (30)where χ = ∑ x , y p ( x , y ) Tr (cid:16) M yz = f ( x , y ) (cid:17) . Thus, all theresults derived previously remain intact for this alter-native construction of OC task.This construction provides for our inference that ourmain results can be extended to general probabilistictheories with the feature of self-duality of states andmeasurement effects [1, 25]. This follows from the factthat the states and measurements that reveal P C in thealternative OC task are just the dual of the measure-ment effects and states employed in the CC . The prop-erty of self-duality emerges from a set of natural postu-lates in the framework of general probabilistic theories[25]. However, this implication is not true in any oper-ational theory. Here, we demonstrate a toy-theory andan ontic-model with CC advantage but no possibility of P C . Consider a well-known CC task, the ( → ) ran-dom access code [26] wherein Alice receives two ran-dom input bits x , x to be encoded into a two dimen-sional system and sends it to Bob. Bob receives a ran-dom input bit y along with the message from Alice andis required to guess x y . This theory, having only threepreparations and just two measurements, is a fragmentof quantum theory. This fragment of quantum theorydoesn’t adhere to self-duality. Clearly the theory admitsadvantage in this task as the average success probabil-ity p Q ≈ > p C = ψ x x which correspond to pure quan-tum preparations as, ψ = | (cid:105) , ψ = cos ( θ ) | (cid:105) + sin ( θ ) | (cid:105) , ψ = cos ( θ ) | (cid:105) − sin ( θ ) | (cid:105) where θ = π andtwo binary-outcome response schemes correspondingto Bob’s setting y =
0, 1 and measurements σ z , σ x respec-tively. However since this ontological model has onlythree ontic states, any mixed preparation in this theoryhas a unique decomposition, thus ruling out the possi-bility of P C [27]. This shows from the basis of the in-ference that self-duality of states and measurements is anecessary requirement for our results to persist in gen-eral probabilistic theories.
IV. BELL INEQUALITY VIOLATION IMPLIESADVANTAGE IN OC With the help of the tools developed so far we nowpresent the complete proof of the fact that Bell inequal-ity violations imply advantage in an associated OC task.For any Bell inequality an OC task can be constructedporting Bell-inequality violation to an advantageousstrategy for the OC task. For the space-like separatedscenario the collapsed state on Bob’s end is preparedand sent in the OC task and for the time-like separatedcase [18] the pre-measurement state at Bob’s end is pre-pared and sent in the OC task. This would make all Bell-inequality violation operationally reveal P C . However,there is a subtlety here, while deterministic encodingstrategies yield bounds on Bell inequalities, the
P N C bounds on the success parameter of the OC tasks mightspring from probabilistic encoding schemes [5]. An in-adequate attempt to prove the above thesis was madein [6], as the authors explicitly assume deterministic en-coding schemes for the constructed OC task. We use thetools developed in this article to provide the completeproof for the thesis.The set-up for a space-like separated Bell experimentdoes not involve any communication, instead two spa-tially separated parties Alice and Bob are provided with inputs x ∈ [ n x ] , y ∈ [ n y ] respectively. Their objective isto return outputs u ∈ [ n u ] , v ∈ [ n v ] respectively so as tomaximize an expression of the following form, B = ∑ u , v , x , y s x , y , u , v p ( x , y ) p ( u , v | x , y ) , (31)where s x , y , u , v ≥
0. The parties may share correlations(classical: shared randomness or quantum: entangledstates) which essentially yield advice in the form of con-ditional probability distributions p ( u , v | x , y ) . If Aliceand Bob share a local-realist (classical) correlation, themaximum they can achieve is, B L = ∑ λ , u , v , x , y s x , y , u , v p ( x , y ) p ( λ ) p λ ( u | x ) p λ ( v | y ) . (32)This fact is captured in Bell inequalities.Consider a quantum strategy which violates a Bell in-equality i.e. B Q > B L . The probability of gettingoutcome u when measurement x is performed on theshared quantum state is p Q ( u | x ) and the reduced quan-tum state on Bob’s subsystem is denoted by ρ Bu | x . Wefollow the construction of OC presented in [6], a = ( a , a ) = ( x , u ) , b = y , c = v , p ( a , b ) = p ( x , u , y ) = p ( y ) p ( x | y ) p Q ( u | x ) , (33)where communication is constrained to oblivious to x .The figure of merit in the OC is given by, p = ∑ u , v , x , y s x , y , u , v p ( x , u , y ) p ( v | x , u , y ) . (34) Proposition 3.
The non-contextual success probability of the OC task is upper bounded by the optimal local-realist value ofBell expression, i.e. p N C ≤ B L .Proof. It is straightforward to see that
Lemmas OC task and similar to(6) we retrieve an upper-bound on the associated p N C as follows, p N C ≤ ∑ y p ( y ) ∑ x , u , v p ( x | y ) s x , y , u , v δ e x , u δ v y , v = ∑ x , y , u , v p ( x , y ) s x , y , u , v δ e x , u δ v y , v . (35)Now, we detail the proof of the above observation. Theexpression for maximal classical success probability (34)is, p N C = max {E}{D} (cid:26) ∑ m ∑ y p ( y ) (cid:18) ∑ x , u p ( x | y ) p Q ( u | x ) s x , y , u , v p E ( m | x , u ) p D ( v | y , m ) (cid:19)(cid:27) ,(36)0and the oblivious constraints imply, ∀ m , ∀ x , x (cid:48) ∈ [ n x ] , p E ( m ) : = p E ( m | x )= ∑ u p Q ( u | x ) p E ( m | x , u )= p E ( m | x (cid:48) ) . (37)Now following the same argument as in the proof of Lemma p N C ≤ max { q x , u } (cid:26) ∑ y p ( y ) max v (cid:26) ∑ x , u p ( x | y ) q x , u s x , y , u , v (cid:27)(cid:27) ,(38)where ∀ x , u , q x , u ≥ ∑ u q x , u = Lemma
2, suppose the extremal pointyielding the optimal value of right-hand-side of (38)corresponds to u ext = e x for each x , i.e., q x , u = δ u , e x ,and for that extremal point max v { ∑ x , u p ( x | y ) q x , u s x , y , u , v } is achieved for v y for each y . Subsequently, (38) can beexpressed as (35).Now we propose a hidden variable model such that p λ ( u | x ) = δ u , e x , p λ ( v | y ) = δ v , v y . Plugging this localstrategy into (32), one obtains the same the expressionfor B L as the right-hand-side of (35), thus completingthe proof. (cid:117)(cid:116) A quantum strategy for the OC task can be easily con-structed from the states and measurements responsiblefor violation of Bell inequality: Alice sends ρ Bu | x for in-put ( x , u ) and Bob’s measurement settings are the sameas in the given Bell experiment. Adherence of obliviouscondition for this strategy simply follows from the no-signaling condition. Thus, we conclude p Q = B Q > B L ≥ p N C . V. CONCEPTUAL INSIGHT AND IMPLICATIONS
The early stages of the quantum information epochfocused primarily on finding communication, computa-tion and information processing tasks wherein quantumresources and protocols provide advantage over theirclassical counterparts. As a consequence, the quantumdeparture from classical limits in such tasks has beensignificantly substantiated in innumerable and varie-gated classes of tasks, this perception is now commonlyreferred to as the "quantum advantage". However, thereis little insight into what feature of quantum theory isunderneath such a remarkable feat. Consequently, fur-ther search for such tasks usually employs narrowingheuristic intuition. The answers to such questions carrywith them the potential of directing and broadening the search for tasks with quantum advantage. How-ever, this seemingly simple question turns out to be sub-stantially arduous and rich in complexity. We must be-gin by discarding the cyclic answers that inherently re-fer back to the operational quantum formalism whichwas apriori responsible for the advantageous predic-tions such as superposition of states. While these an-swers might lead to sharpening intuition, they don’tlead to any significant insights. To further insight, theanswers must arguably pertain to a ground commonto classical and quantum theory, where non-classicalfeatures underlying the quantum formalism are uncov-ered. The study of ontology or "underlying hiddenvariables" provides for such a ground. On the otherhand, quantum communication advantage has a vastvariety of manifestations, however, quantum CC ad-vantage and device-independent information process-ing form the most prominent of them. In this article, wesought to find the quantum ontic-feature that underliesquantum CC advantage.In a nutshell, this work exposes the essential connec-tion between operational quantum communication ad-vantage and the ontic-feature of P C , via operational OC tasks. In other words, we unveil a unifying connectionbetween quantum CC advantage and quantum advan-tage in OC tasks, where the later forms the operationalsignature of P C . We provide two intuitive ways of con-structing an OC task tailored to any given CC task (7)and (27). The OC tasks thus obtained have two salientfeatures: First, the maximal achievable classical successprobability in both OC tasks is bounded by the max-imal achievable classical success probability in the CC problem when the communicated system is restrictedto be two-dimensional. This in-turn provides for two-distinct P N C inequalities corresponding to every CC problem. Second: for any advantageous quantum ( i ) prepare and measure, ( iii ) two-way multi-round and, ( iii ) entanglement assisted classical communication CC protocols, we obtain quantum OC strategies which uti-lize the same states and measurements. An advantageis obtained in the constructed OC task revealing P C whenever the conditions (14),(21) and (24) are met re-spectively. It is a remarkable accomplishment of ourconstruction, that these conditions feature a compari-son between CC performance of quantum d -level and CC performance of classical 2-level systems. Notably,these conditions allow us to demonstrate first instancesof unbounded violation of P N C inequalities from ex-ponential quantum CC advantage. We remark that thereexists a trade-off between generality of our results andthe tightness of these conditions for higher dimensionalquantum CC protocols. Because in this work, we con-cern ourselves with general implications, these alreadysubstantially tight conditions might be tightened evenfurther by fine-tuning our constructions to specific CC problems and associated higher dimensional quantum CC protocols. For instance, these conditions base them-selves on the P N C inequality in
Proposition
P N C ) suc-cess probability of OC tasks. A tighter upper bound onmaximal classical success probability p N C or finding outthe exact value will further tighten the conditions underconsideration. In summary, not only do our results cap-ture
P C manifest in all predominant classes of advanta-geous quantum CC protocols but they also hold beyondmainstream functional CC problems i.e. even in case ofrelational CC problems (see proof of Lemma
P C is manifest in an advantageous CC protocol.An advantage in the first OC task (7) reveals P C man-ifest in the states from the CC protocol and their or-thogonal mixtures, using the same measurements fromthe CC protocol. Whereas an advantage in the second OC task (27) reveals P C manifest in the states corre-sponding to the measurement effects from the CC proto-col, with the aid of measurements corresponding to thestates employed in the CC task. The second construc-tion (27) enables a direct inference that our results andimplications can be extended beyond quantum theory ingeneral probabilistic theories with the property of self-duality of states and measurements effects.Concerning other ontic-features as plausible ground ofquantum CC advantage, the connection between quan-tum advantage in CC and non-locality has been exploredin [19, 28, 29]. Given any protocol offering a sufficientlylarge quantum CC advantage, [19, 29] provide a way forobtaining measurement statistics that violate some Bellinequality. These approaches basically employ an in-dependent teleportation subroutine to transmit Alice’spreparations (from quantum CC protocol). This in-turnimplies that the non-locality thus revealed stems fromadditional entangled states and measurements associ-ated with the teleportation protocol, which are unre-lated to the ones employed in the advantageous quan-tum CC protocol. Therefore, the assertion that quan-tum CC advantage implies non-locality is rather weak.Whereas, along with the very states and measurementsresponsible for the quantum CC advantage we use addi-tional preparations, but these preparations are orthog-onal mixtures of these states and therefore depend onthe advantageous protocol. Therefore in this sense, ourresults reveal a substantially more intimate connectionbetween quantum CC advantage and P C . Furthermore,we provide a complete proof of the fact that any Bell-inequality violation implies an advantage in an associ-ated OC task, thereby porting even the weak implicationalong with device-independent information processing operationally to P C . Moreover, [5] shows that all logi-cal proofs of Kochen-Specker contextuality yields an ad-vantage in the OC task. It is a well-known fact thatwhile a two dimensional quantum system is enoughto demonstrate P C , Kochen-Specker contextuality andnon-locality require at-least three and four dimensionalquantum systems respectively. In summary not only awide-spectrum of quantum communication advantagereveals
P C , even the operational witnesses of other wellknown ontic-features imply
P C . This leads us to ourtentative assertion that
P C is inmately related to quan-tum communication advantage.While our implications are ontological, our methodol-ogy is strictly operational and employs advantage in OC tasks as the intermediary between operational CC advantage and the ontic-feature of P C . Our resultstherefore indicate the fundamental significance of OC tasks to quantum advantage in communication. Fur-thermore, OC tasks form primitives for a range of cryp-tographic protocols [30, 31] and have found applicationsin privacy-preserving computation [32]. Apart from theaforementioned implications our methodology has ex-posed a large class of OC tasks with quantum advan-tage.The question “why quantum advantage?" is far fromsettled. While the results of this article point to P C , theyin no-way close the door to more fundamental ontolog-ical or causal features of quantum theory. A much morearduous question of whether
P C with self-duality (orsome other set of features) ensures a CC advantage re-mains to be addressed. Given the significance of OC tasks, it might prove worthwhile to consider their gener-alizations to multipartite scenarios and explore potentialapplication to the semi-device independent paradigm.Another natural direction for future research is to lookfor information theoretic principles [33] that restrict suc-cess in OC tasks to quantum maximum. VI. ACKNOWLEDGMENTS
We thank M. Pawłowski, M. Horodecki, M. Osz-maniec and C. M. Scandolo for helpful discussion. Thisresearch was conducted in
National Quantum Informa-tion Centre Gdansk . This work is supported by NCNgrants 2016/23/N/ST2/02817, 2014/14/E/ST2/00020and FNP grant First TEAM (Grant No. FirstTEAM/2016-1/5). [1]
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Nature , 1101 (2009). Appendix A: Proofs of Lemmas
In this appendix, we provide the proofs of all the
Lemmas used in the article.
Lemma A.1.
For an instance of the subclass of OC tasks defined in section II B, the classical success probability p N C is upperbounded in the following way, p
N C ≤ max { q a a } (cid:26) ∑ b p ( b ) max c (cid:26) ∑ a , a p ( a | b ) q a , a δ c , g ( a , b ) (cid:27)(cid:27) (A1) where the outer maximization is over a set of variables { q a , a } satisfying the conditions,q a , a ≥ ∑ a q a , a =
1. (A2)
Proof.
We follow the method introduced in [5]. Let us recall that the expression for maximal classical success proba-bility for the OC task described in Eq. (7) is, p N C = max {E}{D} (cid:26) ∑ m ∑ b p ( b ) (cid:18) ∑ a p ( a | b ) p E ( m | a ) p D ( c = g ( a , b ) | b , m ) (cid:19)(cid:27) , (A3)where the message m can take arbitrary number of distinct values. And we seek to obtain an upper bound of p N C under the oblivious constraints, ∀ m , ∀ a , a (cid:48) ∈ [ n a ] , p E ( m ) = p E ( m | a ) = ∑ a p ( a | a ) p E ( m | a , a ) = p E ( m | a (cid:48) ) . (A4)We proceed in two steps: first, we observe that given an encoding scheme, the optimal decoding scheme D ∗ for OC task is fixed and deterministic. Then we provide a technique for recovering an upper bound on p N C by finding theoptimal encoding scheme E ∗ for a single level of the message. Decoding in an OC task: In-order to attain the maximal success probability, Bob’s decoding strategy p D ( c | b , m ) is to output the most probable value g ( a , b ) given Alice’s message m pertaining to an encoding E and his input b .The right-hand-side of (A3) can be interpreted as the convex combination of elements ( ∑ a p ( a , b ) p E ( m | a )) with theweightage p D ( c | b , m ) for each pair of b , m . This in-turn implies that for a fixed encoding strategy Bob’s optimaldecoding strategy D ∗ is deterministic i.e., p D ∗ ( c | b , m ) =
1, if ∑ a | g ( a , b )= c p ( a , b ) p E ( m | a ) ≥ ∑ a | g ( a , b ) (cid:54) = c p ( a , b ) p E ( m | a ) ,0, else. (A5)This allows us to re-express (A3) as, p N C = max E (cid:26) ∑ m ∑ b p ( b ) max c (cid:18) ∑ a p ( a | b ) p E ( m | a ) δ c , g ( a , b ) (cid:19)(cid:27) . (A6) Encoding in an OC task: For any classical encoding strategy E define a set of non-negative parameters { q E , m ( a , a ) : = p ( a | a ) p E ( m | a , a ) p E ( m ) } . It follows from the oblivious constraint (A4) that, ∀ m , a , ∑ a q E , m ( a , a ) =
1. (A7)Using the additional condition p ( a | a , b ) = p ( a | a ) we may now re-write (A6) in terms of q E , m ( a , a ) as, p N C = max E (cid:26) ∑ m p E ( m ) ∑ b p ( b ) max c (cid:26) ∑ a , a p ( a | b ) q E , m ( a , a ) δ c , g ( a , b ) (cid:27)(cid:27) ≤ max { q a a } (cid:26) ∑ b p ( b ) max c (cid:26) ∑ a , a p ( a | b ) q a , a δ c , g ( a , b ) (cid:27)(cid:27) . (A8)The last inequality is implied by the fact that ∑ m p E ( m ) =
1. Specifically, the last inequality states that in-order toobtain an upper bound on p N C its enough to find the optimal encoding strategy E ∗ for a single level of the message,4which justifies the use of the symbol q a , a . The constraint (A7) along with the fact that ∀ a , a , q a , a ≥ q a , a form a convex polytope. Since the ‘max’ function is convex, hence with regardto find a upper bound on p N C it is sufficient to evaluate the expression (A8) at the extremal points of that polytopeand find the optimal. (cid:117)(cid:116)
Lemma A.2.
The set of valid assignments of { q a , a } satisfying the linear constraints (A2) form a convex polytope. The ex-tremal points of this polytope resemble deterministic probability distributions, i.e., any extremal point { q exta , a } is of the followingform: for each a , q exta , a = for all values of a except a specific ˜ a for which q exta , ˜ a = .Proof. Let us represent the variables by a n a × n a matrix whose ( a , a ) -th element is q a , a . Since ∑ a q a , a =
1, eachrow of such matrix sums to 1. The extremal points are described as follows. We consider a string ( e , e , ..., e n a − ) where e a ∈ {
0, ..., n a − } . Each extremal matrix is defined by this string such that q a , a = δ a , e a . There are n n a a number of such strings and each corresponds to an extremal point. One can check that, any arbitrary matrix whoseelements are ˜ q a , a can be obtained by the convex combination of these extremal points, in which the coefficient ofthe matrix corresponds to the string ( e , e , ..., e n a − ) is ∏ n a − i = ˜ q i , e i . (cid:117)(cid:116) Lemma A.3.
For a given quantum prepare and measure communication complexity protocol the following holds, χ ≤ dp G , (A9) where χ = ∑ x , y p ( x , y ) Tr (cid:16) M yz = f ( x , y ) (cid:17) , d is dimension of the communicated system and p G is guessing probability withoutcommunication.Proof. It is straightforward to see that, when there is no communication, given y the best strategy for Bob would beto output f ( x , y ) which is more likely according to the prior probability of the inputs, i.e., p G = ∑ y p ( y ) max ∑ x | f ( x , y )= p ( x | y ) , ∑ x | f ( x , y )= p ( x | y ) By denoting χ yz = Tr ( M yz ) , and imposing the fact χ y + χ y = d , one obtains, χ = ∑ x , y p ( x , y ) χ yz = f ( x , y ) = d ∑ y p ( y ) ∑ x | f ( x , y )= p ( x | y ) χ y d + ∑ x | f ( x , y )= p ( x | y ) χ y d ≤ d ∑ y p ( y ) max ∑ x | f ( x , y )= p ( x | y ) , ∑ x | f ( x , y )= p ( x | y ) = dp G . (cid:117)(cid:116) Lemma A.4.
Given a CC problem and a protocol using a two-leveled classical message with a success probability p C , thesuccess probability of a protocol using a d-leveled classical message is lower bounded in the following way,p C d ≥ − exp (cid:18) − p C log d ( p C − ) (cid:19) . (A10) Proof.
We have a communication complexity protocol P which uses a bit of communication to obtain a successprobability of p C . Now we shall use the pumping argument to discern the desired thesis (A10). Consider yet anotherprotocol P (cid:48) wherein Alice and Bob repeat protocol P log d times. They produce as their final outcome the majorityof outcomes obtained in log d runs of P . If (cid:100) log d (cid:101) is even they succeed if P succeeds (cid:100) log d (cid:101) + (cid:100) log d (cid:101) P succeeds (cid:100) log d (cid:101) times. Consider the event that the protocol P succeeds and the number ofsimultaneous occurrence of such event is captured in the variable τ . This allows us to lower bound p C d as, p C d ≥ p (cid:0) τ > (cid:24) log d (cid:25) (cid:1) = (cid:100) log d (cid:101) ∑ i = (cid:100) log d (cid:101) + (cid:18) (cid:100) log d (cid:101) i (cid:19) p i C ( − p C ) (cid:100) log d (cid:101)− i .The right hand side of the above equation is further lower bounded based on Chernoff’s inequality as, p (cid:0) τ > (cid:24) log d (cid:25) (cid:1) ≥ − exp (cid:18) − p C log d ( p C − ) (cid:19) . (cid:117)(cid:116) Lemma A.5.
Given a CC problem and a protocol which achieves a success probability p S using C ( f , p S ) bits, the successprobability of a protocol using a two-leveled classical message is upper bounded in the following way,p C ≤ + (cid:115) p S C ( f , p S ) . (A11) Proof.
We have a communication complexity protocol which achieves success probability p S using C ( f , p S ) bits ofcommunication. We know from the pumping argument used in the proof for Lemma 4 , p S ≥ − exp (cid:18) − p C C ( f , p S )( p C − ) (cid:19) .Now expanding the above exponential term in the above inequality and taking the first two terms one retrieves, p S ≥ (cid:18) p C C ( f , p S )( p C − ) (cid:19) .This is conveniently re-expressed as, 2 p S C ( f , p S ) ≥ ( p C − ) p C ≥ ( p C − ) ,where the second inequality follows from the observation that 0 ≤ p C ≤ (cid:117)(cid:116) Lemma A.6.
For Hidden matching problem an OC task can be constructed with a success probability p N C , such that p
N C ≤ p C .Proof. In the hidden matching task, Alice is given a bit string x ∈ {
0, 1 } n of length n and Bob is given y ∈ M n where M n denotes the family of all possible perfect matchings on n nodes. Their goal is to output a tuple z = ( i , j , t ) suchthat the edge ( i , j ) belongs to the matching y and t = x i ⊕ x j . Being a relational problem, given an input ( x , y ) ,Bob’s task is to return z from a set of possible relation, i.e., R ( x , y ) = { ( i , j , t ) } such that ( i , j ) ∈ y and t = x i ⊕ x j .Subsequently, the success probability is given by ∑ x , y p ( x , y ) ∑ z ∈ R ( x , y ) p ( z | x , y ) , and in classical communication withtwo-dimensional system p C = max { E }{ D } ∑ m = ∑ y p ( y ) (cid:18) ∑ x , z ∈ R ( x , y ) p ( x | y ) p E ( m | x ) p D ( z | y , m ) (cid:19) . (A12)We follow the same construction of the OC task described in Fig. 1. The corresponding OC is also a relationalproblem in which g ( a , b ) = (cid:40) R ( x , y ) for a = R ( x , y ) for a = R ( x , y ) = { ( i , j , 1 ⊕ t ) } such that ( i , j ) ∈ y and t = x i ⊕ x j . In other words, the hidden matching task isunaltered in the case of a =
0, while for a =
1, Bob’s objective is to output one edge ( i , j ) from the matching y i ⊕ j ⊕
1. Following
Lemmas
A.1 and A.2 we first state the expressionas given in (6), p N C = max {E}{D} (cid:26) ∑ m ∑ b p ( b ) (cid:18) ∑ a , c ∈ g ( a , b ) p ( a | b ) p E ( m | a ) p D ( c | b , m ) (cid:19)(cid:27) ≤ ∑ b p ( b ) ∑ a , a p ( a | b ) δ e a , a ∆ c b , g ( a , b ) , (A13)where ∆ c b , g ( a , b ) = c b ∈ g ( a , b ) , otherwise 0. Recall that in the proposed OC task a = x , b = y , c = ( i , j , t ) .Subsequently, by denoting c y = ( i ∗ , j ∗ , t ∗ ) y we re-write the above expression of p N C , p N C ≤ ∑ y p ( y ) ∑ x | e x = ( i ∗ , j ∗ , t ∗ ) y ∈ R ( x , y ) p ( x | y ) + ∑ y p ( y ) ∑ x | e x = ( i ∗ , j ∗ , t ∗ ) y ∈ ˜ R ( x , y ) p ( x | y ) . (A14)Further, consider the following classical strategy employing two-leveled message m ∈ {
0, 1 } , p E ( m | x ) = δ m , e x , p D ( i , j , t | y , m ) = δ ( i , j , t ) , ( i ∗ , j ∗ , m ⊕ t ∗ ) y .Inserting this strategy in (A12), and using the following feature of hidden matching problem, ∀ y , ( i , j , t ) , ∑ x | ( i , j ,1 ⊕ t ) ∈ R ( x , y ) p ( x | y ) = ∑ x | ( i , j , t ) ∈ ˜ R ( x , y ) p ( x | y ) ,one obtains the same expression of success probability in CC problem as given in the right side of (A14), p C ≥ ∑ y p ( y ) ∑ x | m = ( i ∗ , j ∗ , t ∗ ) y ∈ R ( x , y ) p ( x | y ) + ∑ y p ( y ) ∑ x | m = ( i ∗ , j ∗ ,1 ⊕ t ∗ ) y ∈ R ( x , y ) p ( x | y )= ∑ y p ( y ) ∑ x | m = ( i ∗ , j ∗ , t ∗ ) y ∈ R ( x , y ) p ( x | y ) + ∑ y p ( y ) ∑ x | m = ( i ∗ , j ∗ , t ∗ ) y ∈ ˜ R ( x , y ) p ( x | y ) ≥ p N C .Note that to show the quantum advantage in the OC task, we consider the same quantum strategy as describedin Result p Q = d ( p Q d + d − − χ ) where χ = ∑ x , y , z ∈ R ( x , y ) p ( x , y ) Tr ( M yz ) . Subsequently, one canshow the validity of Corollary3.