Does violation of a Bell inequality always imply quantum advantage in a communication complexity problem?

Quantum correlations which violate a Bell inequality are presumed to power better-than-classical protocols for solving some related communication complexity problems (CCPs). How general is this statement? We show that violations of correlation-type Bell inequalities allow advantages in CCPs, when communication protocols are tailored to emulate the Bell no-signaling constraint (by not communicating measurement settings). Abandonment of this restriction allows us to disprove the main result of, inter alia, [Brukner et. al., Phys Rev. Lett. 89, 197901 (2002)]; we show that a non-maximal quantum violation of the CGLMP Bell inequalities may not imply advantages in related CCPs. We also show that there exists quantum correlations, with nontrivial local marginal probabilities, which violate the $I_{3322}$ Bell inequality, but do not enable a quantum advantange in any CCP, for a scenario with a fixed number of inputs and outputs.

Introduction.-Correlations between outcomes of spacelike separated observers that violate a Bell inequality do not enable them to communicate. This could suggest that shared entanglement cannot be used to improve on classical communication protocols. That is however not the case [1]. The advantages of entanglement and Bell nonlocality are prominently manifested in the fact that they can reduce communication complexity of computation tasks beyond anything achievable by classical means [2].
Communication complexity problems (CCPs) are tasks in which separated parties collaborate to compute a function dependent on inputs distributed among them, while only being allowed a limited amount of communication. In their simplest form, such tasks can be viewed as games in which two parties Alice and Bob hold random inputs X and Y respectively and collaborate so that one of them (say Bob) can compute a function f (X, Y ). Alice communicates a classical message m(X) to Bob who outputs a guess g(m, Y ) for the value of f . If the guess is correct, the partnership earns a "point". Importantly, the communication is limited so that the alphabet of m is smaller than that of X, typically rendering perfect evaluations of f impossible. The CCP is to find for Alice and Bob a communication strategy maximising the score, i.e. the (over the inputs) averaged number of points.
If Alice and Bob use their inputs to measure a shared entangled state and violate a Bell inequality, they can sometimes increase the score beyond that achievable by classical means [2]. Notably, since classical protocols may be viewed as special instances of quantum protocols, one can always trivially find a quantum protocol that achieves the same score as that of the best possible classical protocol. However, to outperform classical models, Bell inequality violations are necessary. We illustrate this with an example [3]. Alice (Bob) has random inputs X = (x 0 , x) ∈ [2] 2 (Y = y ∈ [2]), where [s] denotes the set {0, . . . , |s| − 1}. Alice can send a binary message to Bob who earns a point if he gives g equal to f = x 0 + xy mod 2. Sharing an entangled state, they can use their inputs (x, y) as settings in a test of the Clauser-Horne-Shimony-Holt are their respective outcomes. The optimal message of Alice, for the classical and quantum case, is m(x 0 , x) = a + x 0 mod 2, which effectively hides both a and x by using x 0 as a scrambler. In the classical case, a is a bit-valued function of x. The optimal form of Bob's guess is g = m + b mod 2, where b = b(y). This gives the score S = 1/8 x0,x,y P (g = f |x 0 , x, y) = B. Hence, classically the score is limited by the local hidden variable (LHV) bound of the CHSH inequality. In contrast, by sharing a maximally entangled state, Alice and Bob can violate the CHSH inequality and achieve S > 3/4.
There are many more results showing that every probability distribution that violates specific Bell inequalities has the ability of enhancing a CCP beyond classical protocols. Examples include the Mermin inequalities [2,5], the Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequalities [6][7][8][9], the elegant Bell inequality [10], Bell inequalities for random access coding [11][12][13][14], the biased CHSH inequalities [15,16] and a large class of bipartite many-outcome Bell inequalities [17]. More generally, Ref. [18] showed that the violation of every multipartite correlation Bell inequality with binary outcomes implies beating the best possible classical score in a corresponding CCP constructed by generalising the above example based on the CHSH inequality. See the topical review [19] for further discussions. This fauna of results begs the question: does every nonlocal probability distribution (i.e. a probability distribution that violates a Bell inequality) lead to an advantage in a CCP? To show such advantages, one requires only an example of a CCP in which access to the nonlocal probability distribution is advantageous. However, proving that no such advantage is possible is significantly more challenging; one must rule out the possibility of an advantage in every possible CCP, i.e. no matter the number of inputs and outputs, the choice of score and the chosen classical communication strategy.
Whereas we do not provide a decisive answer to whether Bell nonlocality always implies advantages in CCPs, we ev-idence a negative answer. We first formalise classical and entanglement-assisted CCPs. Then, we show how to map multipartite d-outcome correlation Bell inequalities to corresponding CCPs, in particular reproducing the examples in the above discussion as special instances. We prove that a violation, when combined with well-choosen communication strategies which do not reveal the input the sender would use to define her measurement setting in a related Bell inequality test, implies beating the analogous classical protocols for the corresponding CCP (as in the CHSH-inspired example above). This additional restriction on the communication strategy is tacitly used in several previous works (see e.g. Refs. [7][8][9]). Our more complete analysis no longer sustains the generality of e.g. the main result of Ref. [7], that every violation of the CGLMP inequality implies an advantage in a related CCP. This leads us to consider the classical simulation of entanglement-assisted CCPs. We consider a situation with fixed number of inputs and outputs and show that there exists a quantum nonlocal probability distribution that does not enable better-than-classical communication complexity, regardless of the communication strategy and the choice of score. Our results are in opposition to the common belief that Bell nonlocality always is useful for better-than-classical communication complexity.
Formal scheme of the communication complexity problems analysis.-We mainly consider two-party protocols. These are formulated as games. In a given run Alice and Bob, each receive random inputs, respectively X ∈ [ Alice sends a message m ∈ [M ] (with M < N A ) to Bob who outputs g ∈ [G], which is awarded with t g X,Y points. We say that the tuple (N A , N B , M, G) corresponds to a choice of scenario. The score of a specific CCP within the chosen scenario is written as where p(g|X, Y ) is the probability of Bob's output for local inputs X, Y . Notice that the scoring function can incorporate nonuniform prior probabilities p(X, Y ), as we can always redefine it such that t g X,Y = s g X,Y p(X, Y ), where s g X,Y is the initial scoring function.
In a classical picture, Alice encodes her message with a function E : and Bob constructs his guess with a function D : . The choice of (E, D) can be coordinated via a shared random variable λ, with some probability distribution p(λ). Therefore, a classical model is of the form where the deterministic distribution is p λ (g|X, Y ) = m p(m|X, λ)p(g|m, Y, λ) with p(m|X, λ) = δ m,E λ (X) and p(g|m, Y, λ) = δ g,D λ (m,Y ) . Due to linearity in Eq. (1), the largest score is found with a deterministic communication strategy. We therefore define the optimal classical score in a CCP as In contrast, if Alice and Bob share an entangled state ρ, they may use their inputs to select measurement settings with associated outcomes a and b respectively. The statistics (possibly violating a Bell inequality) reads p(a, b|X, . Moreover, although shared randomness could be absorbed into the shared entangled state, we treat it separately in order to emphasise that it is a classical resource. Therefore, a quantum model is of the form where Note that classical models are special instances of quantum models. Furthermore, we assume that the Bell inequality test is performed before Bob receives Alice's message (which is in line with space-like separation), i.e. m does not influence Bob's measurement setting.

All violations of correlation Bell inequalities power advantages in constrained CCPs.-Consider a Bell scenario with
and any Bell inequality of the form . Each of these parties may send a message m i ∈ [d] after which O N , who has input X N = x N , produces a guess g ∈ [d] and earns a payoff c r The score is defined as ). To put the Bell inequality and the CCP on equal footing, let the N parties share an entangled state and use their inputs x to perform a mea- Here, in analogy with the previous CHSH-inspired example, x (i) 0 acts as a scrambler keeping O N unaware of the inputs x i . Also, notice that the parties O 1 , . . . , O N −1 only use part of their inputs for choosing a measurement setting. This is in analogy with previous litterature (e.g. Refs. [7,11,17,18]). We then find S = B.
Therefore, for such (additive) communication strategies we classically have S ≤ C. Hence, violation of Eq. (5) implies S > C. The above construction generalises results in Refs. [2, 7-9, 11, 13, 14, 16, 18]. Communication which does not allow one to reveal measurement settings (as above) is important in scenarios in which the task function should be calculated in a way which does not allow an eavesdropper to learn the inputs of a sender, or even in a more subtle situation in which a sender does not want the receiver to know her inputs.
For d = 2, the scenario reduces to that of Ref. [18], in which it was shown that messages of the form lead to the optimal classical score. However, the same does not have to be true for d > 2. We shall explicitly show that such is not the case using the specific example of Ref. [7]. Ref. [7] showed (via the above map) that every violation of the CGLMP inequality [6] implies an advantage in a corresponding CCP in which the communication is restricted to the "additive" communication strategies defined above. As we show next, this constraint effectively excludes the optimal classical strategy.
The CGLMP inequality and communication complexity.-Let us consider the CCP of Ref. [7] obtained by choosing (5) as the CGLMP inequality. This inequality is a facet Bell inequality when Alice and Bob have two settings x, y ∈ [2] and three outcomes; where f 1 = −xy and f 2 = −xy + (−1) x+y . Using two entangled qutrits, one can reach the maximal quantum violation B Q cglmp ≈ 0.7287 [20]. In the corresponding CCP, Alice (Bob) has random inputs x 0 ∈ [3] and x ∈ [2] (y ∈ [2]). Alice may send a ternary message m ∈ [3] to Bob who outputs a guess g ∈ [3]. The score (6) is For an additive communication strategy, the violation of (7) is necessary and sufficient for outperforming the corresponding classical value S cglmp = 1/2 [7]. We now relax the assumption of additive communication strategies (m = x 0 + a mod 3) and compute the optimal classical score (3). There are 3 12 deterministic encoding and guessing strategies. Interestingly, by separately considering all of them, we find that This can be saturated by Alice sending m(x 0 , x) = δ x,0 δ x0,2 + 2δ x,1 δ x0,1 mod 3 and Bob guessing g(m, y) = 2δ y,0 m + δ y,1 (m + 1) mod 3. Note that m = 1, 2 informs Bob of the value of x. Thus, the broad class of communication strategies considered in Ref. [7] is insufficient to find the optimal classical score. A large violation of the CGLMP inequality (2/3 < B cglmp ≤ B Q cglmp ) indeed does imply an advantage over general classical protocols for the CCP. However, weaker violations (1/2 < B cglmp ≤ 2/3) are insufficient to achieve the same feat. Similarly, we have found that analogous criticism and correction of the optimal classical score applies to the CCPs of Refs. [8,9] which extend the construction of Ref. [7] to the many-outcome CGLMP inequalities (see Appendix A).
A natural question is whether the limitation S cglmp ≤ B Q cglmp for additive messages assisted by entanglement can (in analogy with the classical case) be overcome when using a more general message. We have numerically searched for entangled states, local measurements and communication strategies, that maximise S cglmp (see Appendix B for details). The numerics return lower bounds on the maximal entanglement-assisted score. In every run, our numerics converges to S cglmp = B Q cglmp . Thus, we have found no improvement over the strategy in which Alice and Bob maximally violate the CGLMP inequality and the message is additive.
Departing from the particular CCP (8), we remind ourselves that entanglement-assisted advantages originate from the probability distribution p(g|x 0 , x, y). Evidently p(g|x 0 , x, y) may lack a classical model even when the specific score (8) does not exceed the classical bound (9). Does there exist some other CCP for which every probability distribution p(g|x 0 , x, y) obtained by a trit-communication and a violation of the CGLMP inequalities results in a higher score than classically possible? Presently, we answer this for the case of the additive communication strategy. Let Alice and Bob use their shared entanglement to generate a probability distribution of the form p(a, b|x, y) = vp cglmp (a, b|x, y) where p cglmp (a, b|x, y) maximally violates the CGLMP inequality and v ∈ [0, 1] is the protocol visibility parameter. This violates the CGLMP inequality when v > 0.6861. The probability distribution p Q v (g|x 0 , x, y), obtained via shared entanglement and an additive communication strategy, beats the classical bound (9) when v > 0.9149. We seek the largest v for which p Q v can be simulated by a classical model. This means solving the linear program By considering p λ (g|x 0 , x, y) for all possible deterministic strategies, we have found that the corresponding polytope of classical probability distributions has 47601 vertices. We have evaluated the linear program and found v ≈ 0.7943. Hence, probability distributions p Q v (g|x 0 , x, y) for 0.7943 < v ≤ 0.9149 indeed imply an advantage over classical protocols in some CCP despite the particular CCP (8) failing to detect it. However, when 0.6861 < v ≤ 0.7942 the CGLMP inequality is violated, but the probability distribution p Q v (g|x 0 , x, y) can be classically modelled.
Bell nonlocality without CCP advantages in fixed scenarios.-The above classical simulation focuses on entanglement-assisted correlations obtained via an additive communication strategy. Here, we prove a more general statement: that for a given scenario (that is, a fixed number of inputs and outputs) there exists a nonlocal probability distribution (realisable in quantum theory) which cannot be used to improve any CCP beyond classical constraints, regardless of the choice of communication strategy. Specifically, we find a nonlocal probability distribution that when combined with any communication strategy gives rise to a p(g|X, Y ) which can be simulated in a classical model for the given scenario.
To this end, we focus on the simplest Bell scenario going beyond that of the CHSH inequality. It has two parties, each with three settings x, y ∈ [3] and binary outcomes a, b ∈ [2]. Three settings are needed, as the two-setting scenario is fully characterised by the CHSH inequality, which is a correlation inequality and therefore implies advantages in a CCP whenever violated [3,18]. In the three-setting scenario, the facet Bell inequalities are the (lifted) CHSH inequality and the I 3322 inequality [23,24]. The I 3322 inequality reads (12) where P (x, y) is the probability of outputting a = b = 0, Notably, this inequality involves marginal probabilities and is therefore not a correlation Bell inequality. Consequently, it is not in the broad class of Bell inequalities whose violation necessarily implies advantages in CCPs [18].
Motivated by the choice of scenario in previous discussions for finding entanglement-assisted advantanges in CCPs from specific Bell inequality violations, we consider a communication scenario in which Alice has inputs x 0 ∈ [2] and x ∈ [3] and Bob has an input y ∈ [3]. Alice sends m ∈ [2] to Bob who outputs g ∈ [2]. To further motivate that this scenario is a good choice for revealing the CCP advantages of probability distributions that violate the I 3322 inequality, we have shown in Appendix C that a maximal violation of (12) (for qubits) implies better-than-classical communication complexity, and also that every probability distribution violating (12) obtained from mixing the optimal one with a uniform probability distribution (in analogy with Eq. (10)) also implies such an advantage. Note that such a scenario is a natural extension of the ones studied in Ref. [18].
Nevertheless, we show that there exists a nonmaximally entangled state and local measurements that give rise to a probability distribution that violates I 3322 inequality, that however is not advantageous in any CCP in the stated scenario. To this end, Alice and Bob can generate a Bell-like distribution of the form p(a, b|x 0 , x, y). In order to relate it to a test of the I 3322 inequality, we let its dependence on x 0 be trivial, i.e. p(a, b|x 0 , x, y) = p(a, b|x, y). Now, we can choose our candidate probability distribution p cand (a, b|x, y). This distribution has a quantum realisation. It also weakly violates Eq. (12) (I ≈ 0.0129), but importantly does not violate the CHSH inequality and hence cannot lead a betterthan-classical score in a CCP based on the CHSH inequality. The candidate probability distribution was originally pro-posed in Ref. [24] and we detail it and its quantum realisation in Appendix D. We show that for every possible communication strategy within the scenario, there exists no CCP in which p cand enables an advantage over classical protocols. We first note that since we have fixed the probability distribution in the Bell scenario to p cand , the set of distributions that Alice and Bob can generate in the communication scenario forms a polytope (4). Therefore, it suffices to show that all deterministic communication strategies with access to p cand can be classically modelled. Since Alice maps the twelve possible values of (a, x 0 , x) to her binary message m, and Bob maps the twelve values (m, b, y) to his binary output g, there exists a total of 2 24 deterministic communication strategies. For each of these (indexed by µ), we have evaluated the corresponding probability distribution p µ (g|x 0 , x, y) = a,b,m p(a, b|x, y)p(m|a, x 0 , x, µ)p(g|m, b, y, µ). We have found that the relevant polytope of probability distributions in the communication scenario has 8192992 vertices. To show that the probability distribution p µ (g|x 0 , x, y) can be simulated by a classical model for all vertices, we consider the mixture of each vertex probability distribution with random outcomes; p Q,v µ (g|x 0 , x, y) = vp µ (g|x 0 , x, y) + (1 − v)/2. Then, for each of the roughly eight million values of µ, we decide the possibility of a classical model by evaluating [25] a linear program analogous to Eq. (11). We find that for every choice of µ, the value of v is never smaller than one (up to machine precision). That is, every p µ (g|x 0 , x, y) can be classically modelled. Thus, we conclude that in the scenario in which Alice has X ∈ [6] and Bob has Y ∈ [3] and m and f are bit valued, there exists no CCP that can be improved beyond classical constraints by the parties sharing the Bell inequality violating probability distribution p cand (a, b|x, y).
Conclusions.-A substantial number of examples of quantum advantages in CCPs being powered by Bell inequality violations can be understood as different instances of a single map from Bell inequalities to CCPs. We found that a violation of the former implies an advantage in the latter for a simple class of communication strategies. As we explicitly showed, a complete analysis of classical communication complexity requires the revision of several previous claims in which violations of particular Bell inequalities where thought to imply advantages in CCPs. Going beyond that, we found that there exists quantum Bell inequality violations for which the statistics of every possible communication strategy in any possible CCP can be simulated by classical models in an input/output scenario that naturally extends previous works. This suggests that not all forms of Bell nonlocality are useful for better-thanclassical communication complexity. A definite proof of this statement would require an extension of our results to CCPs with any number of inputs and outputs. To this end, we believe that p cand is a good candidate. Our results also make relevant the question of characterising the (now seemingly nontrivial) relation between Bell nonlocality and entanglement-assisted communication complexity. Which Bell inequality violating probability distributions are useful for outperforming classical limitations in CCPs and which are not? to maximise the score S cglmp = 1 16 x0,x,y P x,y (g = x 0 + f 1 1 ) − P x,y (g = x 0 + f 1 2 ) computed modulo four. Alice (Bob) uses x (y) to measure an entangled pair with outcomes a ∈ [4] (b ∈ [4]). Then, using an additive communication strategy, i.e. m = x 0 + a mod 4 and g = m + b mod 4, one finds that S cglmp becomes identical to the Bell expression in the four-outcome CGLMP inequality [8,9] which has an LHV bound (in this form and normalisation) of 1/2. Therefore, under additive communication strategies Refs. [8,9] found S cglmp ≤ 1/2. However, the optimal classical score is not saturated with such a communication strategy. Since Alice maps eight inputs to four outputs, and similarly for Bob, there is a total of 4 16 pairs of encoding and guessing functions. We have evaluated the score (A1) for all such pairs and found that the optimal classical score is Thus, in full analogy with the discussion in the main text focused on ternary-outcome CGLMP inequalities, corrections also apply to its generalisation to more than three outcomes.
Appendix B: Numerical search for the optimal quantum score in the CCP based on the CGLMP inequality We present numerics in support of an additive communication strategy and a probability distribution that maximally violates the CGLMP inequality saturating the optimal entanglement-assisted score in the CCP based on the ternaryoutcome CGLMP inequality.
The joint state of Bob's local system (after Alice's measurement) and the classical message when averaged over Alice's outcome can be written where we encode the classical message in the computational basis state |m m|. For all deterministic messages of the restricted class m = m(a, x 0 ), we have evaluated the score and found that only those of the additive type lead to a better-thanclassical score. However, for a general deterministic message m = m(a, x 0 , x) such a brute-force approach is too timeconsuming. To address the general case, we can obtain upper bounds on the score by substituting |m m| in Eq. (B1) with a quantum system σ a,x0,x ∈ C 3 . Notice that this only serves as a tool towards treating the relevant problem in which the message is classical. Moreover, this substitution is far more constraining than allowing for general quantum communication assisted by shared entanglement. The substitution of the classical message for a quantum one comes with the advantage that one can efficiently run alternating convex searches for lower bounding the quantity S QC cglmp = max ρ,A,B,σ S cglmp 1 . For three respectively four dimensional entangled systems, we implemented the procedure by alternating semidefinite programs each optimising over the state, Alice's measurements, Bob's measurements and the quantum message respectively. We have repeated this 200 times for different initial points of the iteration. Without exception, we find that the optimisation converges to the value B Q cglmp , which is what is obtained by maximally violating the CGLMP inequality and then using an additive communication strategy. the entanglement-assisted probability distribution in the CCP has a classical model. We find that it returns v = 4/5, thus showing that every probability distribution of the form (C2) that violates the I 3322 inequality implies advantages in a CCP.