We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f ⊆ 𝒳×𝒴×𝒵. For any 0 < ε < δ < 1/2 and any k≥1, we show that Q¹_{1-(1-ε)^{Ω(k/log|𝒵|)}}(f^k) = Ω(k⋅Q¹_{δ}(f)), where Q¹_{ε}(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error ε and f^k denotes k parallel instances of f. As far as we are aware, this is the first direct product theorem for the quantum communication complexity of a general relation - direct sum theorems were previously known for one-way quantum protocols for general relations, while direct product theorems were only known for special cases. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao [Rahul Jain et al., 2014], and under anchored distributions due to Bavarian, Vidick and Yuen [Bavarian et al., 2017], as well as message compression for quantum protocols due to Jain, Radhakrishnan and Sen [Rahul Jain et al., 2005]. In particular, we show that a direct product theorem holds for the distributional one-way quantum communication complexity of f under any distribution q on 𝒳×𝒴 that is anchored on one side, i.e., there exists a y^* such that q(y^*) is constant and q(x|y^*) = q(x) for all x. This allows us to show a direct product theorem for general distributions, since for any relation f and any distribution p on its inputs, we can define a modified relation f̃ which has an anchored distribution q close to p, such that a protocol that fails with probability at most ε for f̃ under q can be used to give a protocol that fails with probability at most ε + ζ for f under p. Our techniques also work for entangled non-local games which have input distributions anchored on any one side, i.e., either there exists a y^* as previously specified, or there exists an x^* such that q(x^*) is constant and q(y|x^*) = q(y) for all y. In particular, we show that for any game G = (q, 𝒳×𝒴, 𝒜×ℬ, 𝖵) where q is a distribution on 𝒳×𝒴 anchored on any one side with constant anchoring probability, then ω^*(G^k) = (1 - (1-ω^*(G))⁵) ^{Ω(k/(log(|𝒜|⋅|ℬ|)))} where ω^*(G) represents the entangled value of the game G. This is a generalization of the result of [Bavarian et al., 2017], who proved a parallel repetition theorem for games anchored on both sides, i.e., where both a special x^* and a special y^* exist, and potentially a simplification of their proof.
We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f ⊆ 𝒳×𝒴×𝒵. For any 0 < ε < δ < 1/2 and any k≥1, we show that Q¹_{1-(1-ε)^{Ω(k/log|𝒵|)}}(f^k) = Ω(k⋅Q¹_{δ}(f)), where Q¹_{ε}(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error ε and f^k denotes k parallel instances of f. As far as we are aware, this is the first direct product theorem for the quantum communication complexity of a general relation - direct sum theorems were previously known for one-way quantum protocols for general relations, while direct product theorems were only known for special cases. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao [Rahul Jain et al., 2014], and under anchored distributions due to Bavarian, Vidick and Yuen [Bavarian et al., 2017], as well as message compression for quantum protocols due to Jain, Radhakrishnan and Sen [Rahul Jain et al., 2005]. In particular, we show that a direct product theorem holds for the distributional one-way quantum communication complexity of f under any distribution q on 𝒳×𝒴 that is anchored on one side, i.e., there exists a y^* such that q(y^) is constant and q(x|y^) = q(x) for all x. This allows us to show a direct product theorem for general distributions, since for any relation f and any distribution p on its inputs, we can define a modified relation f̃ which has an anchored distribution q close to p, such that a protocol that fails with probability at most ε for f̃ under q can be used to give a protocol that fails with probability at most ε + ζ for f under p. Our techniques also work for entangled non-local games which have input distributions anchored on any one side, i.e., either there exists a y^* as previously specified, or there exists an x^* such that q(x^) is constant and q(y|x^) = q(y) for all y. In particular, we show that for any game G = (q, 𝒳×𝒴, 𝒜×ℬ, 𝖵) where q is a distribution on 𝒳×𝒴 anchored on any one side with constant anchoring probability, then ω^(G^k) = (1 - (1-ω^(G))⁵) ^{Ω(k/(log(|𝒜|⋅|ℬ|)))} where ω^(G) represents the entangled value of the game G. This is a generalization of the result of [Bavarian et al., 2017], who proved a parallel repetition theorem for games anchored on both sides, i.e., where both a special x^ and a special y^* exist, and potentially a simplification of their proof.
We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f⊆X×Y×Z. For any 0 k ≥ 1, we show that [EQUATION] where Q1e(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error e and fk denotes k parallel instances of f. As far as we are aware, this is the first direct product theorem for the quantum communication complexity of a general relation - direct sum theorems were previously known for one-way quantum protocols for general relations, while direct product theorems were only known for special cases. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlenyi and Yao [24], and under anchored distributions due to Bavarian, Vidick and Yuen [4], as well as message compression for quantum protocols due to Jain, Radhakrishnan and Sen [29]. In particular, we show that a direct product theorem holds for the distributional one-way quantum communication complexity of f under any distribution q on X × Y that is anchored on one side, i.e., there exists a y* such that q(y*) is constant and q(x\y*) = q(x) for all x. This allows us to show a direct product theorem for general distributions, since for any relation f and any distribution p on its inputs, we can define a modified relation [EQUATION] which has an anchored distribution q close to p, such that a protocol that fails with probability at most e for [EQUATION] under q can be used to give a protocol that fails with probability at most e + ζ for f under p. Our techniques also work for entangled non-local games which have input distributions anchored on any one side, i.e., either there exists a y* as previously specified, or there exists an x* such that q(x*) is constant and q(y\x*) = q(y) for all y. In particular, we show that for any game G = (q,X×Y, A×B, V) where q is a distribution on X × Y anchored on any one side with constant anchoring probability, then [EQUATION] where ω* (G) represents the entangled value of the game G. This is a generalization of the result of [4], who proved a parallel repetition theorem for games anchored on both sides, i.e., where both a special x* and a special y* exist, and potentially a simplification of their proof.
We prove a direct product theorem for the one-way entanglement-assisted quantum communication complexity of a general relation f⊆X×Y×Z. For any ε,ζ>0 and any k≥1, we show that Q1−(1−ε)Ω(ζ6k/log∣Z∣)1(fk)=Ω(k(ζ5⋅Qε+12ζ1(f)−loglog(1/ζ))), where Qε1(f) represents the one-way entanglement-assisted quantum communication complexity of f with worst-case error ε and fk denotes k parallel instances of f. As far as we are aware, this is the first direct product theorem for quantum communication. Our techniques are inspired by the parallel repetition theorems for the entangled value of two-player non-local games, under product distributions due to Jain, Pereszlényi and Yao, and under anchored distributions due to Bavarian, Vidick and Yuen, as well as message-compression for quantum protocols due to Jain, Radhakrishnan and Sen. Our techniques also work for entangled non-local games which have input distributions anchored on any one side. In particular, we show that for any game G=(q,X×Y,A×B,V) where q is a distribution on X×Y anchored on any one side with anchoring probability ζ, then ω∗(Gk)=(1−(1−ω∗(G))5)Ω(log(∣A∣⋅∣B∣)ζ2k) where ω∗(G) represents the entangled value of the game G. This is a generalization of the result of Bavarian, Vidick and Yuen, who proved a parallel repetition theorem for games anchored on both sides, and potentially a simplification of their proof.
In this paper, we present the following quantum compression protocol `P': Let ρ,σ be quantum states, such that S (ρ∥σ)def= Tr(ρ log ρ - ρ log σ), the relative entropy between ρ and σ, is finite. Alice gets to know the eigendecomposition of ρ. Bob gets to know the eigendecomposition of σ. Both Alice and Bob know S(ρ∥σ) and an error parameter ε. Alice and Bob use shared entanglement and after communication of O((S(ρ∥σ) + 1)/ε4) bits from Alice to Bob, Bob ends up with a quantum state ̃ρ̃, such that F(ρ, ρ̃) ≥ 1-5ε, where F(·) represents fidelity. This result can be considered as a non-commutative generalization of a result due to Braverman and Rao where they considered the special case when ρ and σ are classical probability distributions (or commute with each other) and use shared randomness instead of shared entanglement. We use? to obtain an alternate proof of a direct-sum result for entanglement assisted quantum one-way communication complexity for all relations, which was first shown by Jain et al.. We also present a variant of protocol? in which Bob has some side information about the state with Alice. We show that in such a case, the amount of communication can be further reduced, based on the side information that Bob has. Our second result provides a quantum analog of the widely used classical correlated-sampling protocol. For example, Holenstein used the classical correlated-sampling protocol in his proof of a parallel-repetition theorem for two-player one-round games.
In this paper, we show a direct product theorm in the model of two-party bounded-round public-coin randomized communication complexity. For a relation f subset of X times Y times Z (X,Y,Z are finite sets), let R^{(t), pub}_e (f) denote the two-party t-message public-coin communication complexity of f with worst case error e. We show that for any relation f and positive integer k: R^{(t), pub}_{1 - 2^{-Omega(k/t^2)}}(f^k) = Omega(k/t (R^{(t), pub}_{1/3}(f) - O(t^2))) . In particular, it implies a strong direct product theorem for the two-party constant-message public-coin randomized communication complexity of all relations f. Our result for example implies a strong direct product theorem for the pointer chasing problem. This problem has been well studied for understanding round v/s communication trade-offs in both classical and quantum communication protocols. We show our result using information theoretic arguments. Our arguments and techniques build on the ones used in [Jain 2011], where a strong direct product theorem for the two-party one-way public-coin communication complexity of all relations is shown (that is the special case of our result when t=1). One key tool used in our work and also in [Jain 2011] is a message compression technique due to [Braverman and Rao 2011], who used it to show a direct sum theorem for the two-party bounded-round public-coin randomized communication complexity of all relations. Another important tool that we use is a correlated sampling protocol, which for example, has been used in [Holenstein 2007] for proving a parallel repetition theorem for two-prover games.
We show two new direct product results in two different models of communication complexity. Our first result is in the one-way public-coin model. Let f ⊆ X × Y × Z be a relation and ϵ > 0 be a constant. Let R 1,pub ϵ ( f ) represent the communication complexity of f , with worst-case error ϵ in this model. We show that if for computing f k ( k independent copies of f ) in this model, o ( k ċ R 1, pub 1/3 ( f )) communication is used, then the success is exponentially small in k . We show a new tight characterization of communication complexity in this model which strengthens the tight characterization shown in Jain et al. [2008]. We use this new characterization to show our direct product result and this characterization may also be of independent interest. Our second direct product result is in the model of two-way public-coin communication complexity. We show a direct product result for all relations in this model in terms of a new complexity measure that we define. Our new measure is a generalization to nonproduct distributions, of the two-way product subdistribution bound of Jain et al. [2008]. Our direct product result therefore generalizes to nonproduct distributions, their direct product result in terms of the two-way product subdistribution bound. As an application of our new direct product result, we reproduce (via completely different arguments) strong direct product result for the set-disjointness problem which was previously shown by Klauck [2010]. We show this by proving that our new complexity measure gives a tight lower bound of Ω( n ) for the set-disjointness problem on n -bit inputs (this strengthens the linear lower bound on the rectangle/corruption bound for set-disjointness shown by Razborov [1992]). In addition, we show that many previously known direct product results in this model are uniformly implied and often strengthened by our result.
We give a direct product theorem for the entanglement-assisted interactive quantum communication complexity in terms of the quantum partition bound for product distributions. The quantum partition or efficiency bound is a lower bound on communication complexity, a non-distributional version of which was introduced by Laplante, Lerays and Roland (2012). For a two-input boolean function, the best result for interactive quantum communication complexity known previously was due to Sherstov (2018), who showed a direct product theorem in terms of the generalized discrepancy. While there is no direct relationship between the maximum distributional quantum partition bound for product distributions, and the generalized discrepancy method, unlike Sherstov's result, our result works for two-input functions or relations whose outputs are non-boolean as well. As an application of our result, we show that it is possible to do device-independent quantum key distribution (DIQKD) without the assumption that devices do not leak any information after inputs are provided to them. We analyze the DIQKD protocol given by Jain, Miller and Shi (2020), and show that when the protocol is carried out with devices that are compatible with several copies of the Magic Square game, it is possible to extract a linear (in the number of copies of the game) amount of key from it, even in the presence of a linear amount of leakage. Our security proof is parallel, i.e., the honest parties can enter all their inputs into their devices at once, and works for a leakage model that is arbitrarily interactive, i.e., the devices of the honest parties Alice and Bob can exchange information with each other and with the eavesdropper Eve in any number of rounds, as long as the total number of bits or qubits communicated is bounded.