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28
The Communication Complexity of Correlation
"... Let X and Y be finite nonempty sets and (X, Y) a pair of random variables taking values in X × Y. We consider communication protocols between two parties, Alice and Bob, for generating X and Y. Alice is provided an x ∈ X generated according to the distribution of X, and is required to send a messa ..."
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Cited by 32 (10 self)
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Let X and Y be finite nonempty sets and (X, Y) a pair of random variables taking values in X × Y. We consider communication protocols between two parties, Alice and Bob, for generating X and Y. Alice is provided an x ∈ X generated according to the distribution of X, and is required to send a message to Bob in order to enable him to generate y ∈ Y, whose distribution is the same as that of Y X=x. Both parties have access to a shared random string generated in advance. Let T (X: Y) be the minimum (over all protocols) of the expected number of bits Alice needs to transmit to achieve this. We show that
STRONG DIRECT PRODUCT THEOREMS FOR QUANTUM COMMUNICATION AND QUERY COMPLEXITY
"... A strong direct product theorem (SDPT) states that solving n instances of a problem requires ˝.n / times the resources for a single instance, even to achieve success probability 2 ˝.n / : We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a singl ..."
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Cited by 17 (4 self)
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A strong direct product theorem (SDPT) states that solving n instances of a problem requires ˝.n / times the resources for a single instance, even to achieve success probability 2 ˝.n / : We prove that quantum communication complexity obeys an SDPT whenever the communication lower bound for a single instance is proved by the generalized discrepancy method, the strongest technique in that model. We prove that quantum query complexity obeys an SDPT whenever the query lower bound for a single instance is proved by the polynomial method, one of the two main techniques in that model. In both models, we prove the corresponding XOR lemmas and threshold direct product theorems.
Strengths and weaknesses of quantum fingerprinting
 In Proc. of the 21st Conf. on Computational Complexity (CCC
, 2006
"... We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical sharedrandomness SMP protocols by means of quantum SMP protocols without shared randomness (Q �protoco ..."
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Cited by 17 (2 self)
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We study the power of quantum fingerprints in the simultaneous message passing (SMP) setting of communication complexity. Yao recently showed how to simulate, with exponential overhead, classical sharedrandomness SMP protocols by means of quantum SMP protocols without shared randomness (Q �protocols). Our first result is to extend Yao’s simulation to the strongest possible model: every manyround quantum protocol with unlimited shared entanglement can be simulated, with exponential overhead, by Q �protocols. We apply our technique to obtain an efficient Q �protocol for a function which cannot be efficiently solved through more restricted simulations. Second, we tightly characterize the power of the quantum fingerprinting technique by making a connection to arrangements of homogeneous halfspaces with maximal margin. These arrangements have been well studied in computational learning theory, and we use some strong results obtained in this area to exhibit weaknesses of quantum fingerprinting. In particular, this implies that for almost all functions, quantum fingerprinting protocols are exponentially worse than classical deterministic SMP protocols.
On the role of shared entanglement
 Quantum Inf. Comput
, 2006
"... Despite the apparent similarity between shared randomness and shared entanglement in the context of Communication Complexity, our understanding of the latter is not as good as of the former. In particular, there is no known “entanglement analogue ” for the famous theorem by Newman, saying that the n ..."
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Cited by 14 (0 self)
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Despite the apparent similarity between shared randomness and shared entanglement in the context of Communication Complexity, our understanding of the latter is not as good as of the former. In particular, there is no known “entanglement analogue ” for the famous theorem by Newman, saying that the number of shared random bits required for solving any communication problem can be at most logarithmic in the input length (i.e., using more than O(log n) shared random bits would not reduce the complexity of an optimal solution). In this paper we prove that the same is not true for entanglement. We establish a wide range of tight (up to a logarithmic factor) entanglement vs. communication tradeoffs for relational problems. The “lowend ” is: for any t> 2, reducing shared entanglement from log t n to o(log t−1 n) qubits can increase the communication required for solving a problem almost exponentially, from O(log t n) to ω ( √ n). The “highend ” is: for any ε> 0, reducing shared entanglement from n 1−ε log n to o(n 1−ε) can increase the required communication from O(n 1−ε log n) to Ω(n 1−ε/2). The upper bounds are demonstrated via protocols which are exact and work in the simultaneous message passing model, while the lower bounds hold for boundederror protocols, even in the more powerful model of 1way communication. Our protocols use shared EPR pairs while the lower bounds apply to any sort of prior entanglement. We base the lower bounds on two new results in communication complexity: a strong direct product theorem for certain class of relations and a technique for lowerbounding the amount of entanglement required for a solution. We believe that the both results might have other applications. We also apply our entanglement bounding technique to the models of 2prover proof systems where the provers share entanglement but the number of qubits (as a function of the input length) is bounded by some polynomial fixed for the model. We show that the power of such proof systems equals NEXP, so these systems are equivalent to the classical 2prover systems (our technique gives a protocol for accepting NEXP, the matching upper bound has been shown
Direct product theorems for classical communication complexity . . .
, 2007
"... A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a ..."
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Cited by 10 (3 self)
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A basic question in complexity theory is whether the computational resources required for solving k independent instances of the same problem scale as k times the resources required for one instance. We investigate this question in various models of classical communication complexity. We introduce a new measure, the subdistribution bound, which is a relaxation of the wellstudied rectangle or corruption bound in communication complexity. We nonetheless show that for the communication complexity of Boolean functions with constant error, the subdistribution bound is the same as the latter measure, up to a constant factor. We prove that the oneway version of this bound tightly captures the oneway publiccoin randomized communication complexity of any relation, and the twoway version bounds the twoway publiccoin randomized communication complexity from below. More importantly, we show that the bound satisfies the strong direct product property under product distributions for both one and twoway protocols, and the weak direct product property under arbitrary distributions for twoway protocols. These results subsume and strengthen, in a unified manner, several recent results on the direct product question. The
Optimal direct sum and privacy tradeoff results for quantum and classical communication complexity
 CoRR
"... Abstract. We show optimal Direct Sum result for the oneway entanglementassisted quantum communication complexity for any relation f ⊆ X × Y × Z. We show: Q 1,pub (f ⊕m) = Ω(m · Q 1,pub (f)), where Q 1,pub (f), represents the oneway entanglementassisted quantum communication complexity of f with ..."
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Cited by 8 (3 self)
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Abstract. We show optimal Direct Sum result for the oneway entanglementassisted quantum communication complexity for any relation f ⊆ X × Y × Z. We show: Q 1,pub (f ⊕m) = Ω(m · Q 1,pub (f)), where Q 1,pub (f), represents the oneway entanglementassisted quantum communication complexity of f with error at most 1/3 and f ⊕m represents mcopies of f. Similarly for the oneway publiccoin classical communication complexity we show: R 1,pub (f ⊕m) = Ω(m · R 1,pub (f)), where R 1,pub (f), represents the oneway publiccoin classical communication complexity of f with error at most 1/3. We show similar optimal Direct Sum results for the Simultaneous Message Passing (SMP) quantum and classical models. For twoparty twoway protocols we present optimal Privacy Tradeoff results leading to a Weak Direct Sum result for such protocols. We show our Direct Sum and Privacy Tradeoff results via message compression arguments. These arguments also imply a new round elimination lemma in quantum communication, which allows us to extend classical lower bounds on the cell probe complexity of some data structure problems, e.g. Approximate Nearest Neighbor Searching (ANN) on the Hamming cube {0, 1} n and Predecessor Search to the quantum setting. In a separate result we show that Newman’s [New91] technique of reducing the number of publiccoins in a classical protocol cannot be lifted to the quantum setting. We do this by defining a general notion of blackbox reduction of prior entanglement that subsumes Newman’s technique. We prove that such a blackbox reduction is impossible for quantum protocols by exhibiting a particular oneround quantum protocol for the Equality function where the blackbox technique fails to reduce the amount of prior entanglement by more than a constant factor. In the final result in the theme of message compression, we provide an upper bound on the problem of Exact Remote State Preparation (ERSP). 1
New results in the simultaneous message passing model. Unpublished, available at arXiv:0902.3056
, 2009
"... Abstract—Consider the following Simultaneous Message Passing (SMP) model for computing a relation f ⊆ X ×Y×Z. In this model Alice, on input x ∈ X and Bob, on input y ∈ Y, send one message each to a third party Referee who then outputs a z ∈ Z such that (x, y, z) ∈ f. We first show optimal Direct su ..."
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Cited by 6 (3 self)
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Abstract—Consider the following Simultaneous Message Passing (SMP) model for computing a relation f ⊆ X ×Y×Z. In this model Alice, on input x ∈ X and Bob, on input y ∈ Y, send one message each to a third party Referee who then outputs a z ∈ Z such that (x, y, z) ∈ f. We first show optimal Direct sum results for all relations f in this model, both in the quantum and classical settings, in the situation where we allow shared resources (shared entanglement in quantum protocols and public coins in classical protocols) between Alice and Referee and Bob and Referee and no shared resource between Alice and Bob. This implies that, in this model, the communication required to compute k simultaneous instances of f, with constant success overall, is at least ktimes the communication required to compute one instance with constant success. This in particular implies an earlier Direct sum result, shown by Chakrabarti, Shi, Wirth and Yao [CSWY01] for the Equality function (and a class of other socalled robust functions), in the classical SMP model with no shared resources between any parties. Furthermore we investigate the gap between the SMP model and the oneway model in communication complexity and exhibit a partial function that is exponentially more expensive in the former if quantum communication with entanglement is allowed, compared to the latter even in the deterministic case.