We show optimal Direct Sum result for the one-way entanglement-assisted quantum com- munication complexity for any relation fX × Y × Z. We show: Q 1,pub (f ⊕m ) = (m · Q 1,pub (f)), where Q 1,pub (f), represents the one-way entanglement-assisted quantum communication complexity of f with error at most 1/3 and f ⊕m represents m-copies of f. Similarly for the one-way public-coin classical communication complexity we show: R 1,pub (f ⊕m ) = (m · R 1,pub (f)), where R 1,pub (f), represents the one-way public-coin 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 two-party two-way protocols we present optimal Privacy Trade-off results leading to a Weak Direct Sum result for such protocols. We show our Direct Sum and Privacy Trade-off 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 public-coins in a classical protocol cannot be lifted to the quantum setting. We do this by defining a general notion of black-box reduction of prior entanglement that subsumes Newman's technique. We prove that such a black-box reduction is impossible for quantum protocols by exhibiting a particular one-round quantum protocol for the Equality function where the black-box technique fails to reduce the amount of prior entanglement by more than a constant factor.
We show optimal Direct Sum result for the one-way entanglement-assisted quantum com- munication complexity for any relation fX × Y × Z. We show: Q 1,pub (f ⊕m ) = (m · Q 1,pub (f)), where Q 1,pub (f), represents the one-way entanglement-assisted quantum communication complexity of f with error at most 1/3 and f ⊕m represents m-copies of f. Similarly for the one-way public-coin classical communication complexity we show: R 1,pub (f ⊕m ) = (m · R 1,pub (f)), where R 1,pub (f), represents the one-way public-coin 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 two-party two-way protocols we present optimal Privacy Trade-off results leading to a Weak Direct Sum result for such protocols. We show our Direct Sum and Privacy Trade-off 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 public-coins in a classical protocol cannot be lifted to the quantum setting. We do this by defining a general notion of black-box reduction of prior entanglement that subsumes Newman's technique. We prove that such a black-box reduction is impossible for quantum protocols by exhibiting a particular one-round quantum protocol for the Equality function where the black-box technique fails to reduce the amount of prior entanglement by more than a constant factor.
Shared entanglement is a resource available to parties communicating over a quantum channel, much akin to public coins in classical communication protocols. Whereas shared randomness does not help in the transmission of information, or significantly reduce the classical complexity of computing functions (as compared to private-coin protocols), shared entanglement leads to startling phenomena such as ``quantum teleportation'' and ``superdense coding.'' The problem of characterising the power of prior entanglement has puzzled many researchers. In this paper, we revisit the problem of transmitting classical bits over an entanglement-assisted quantum channel. We derive a new, optimal bound on the number of quantum bits required for this task, for any given probability of error. All known lower bounds in the setting of bounded error entanglement-assisted communication are based on sophisticated information theoretic arguments. In contrast, our result is derived from first principles, using a simple linear algebraic technique.
We investigate the randomized and quantum communication complexities of the well-studied Equality function with small error probability ε, getting the optimal constant factors in the leading terms in various different models. The following are our results in the randomized model: - We give a general technique to convert public-coin protocols to private-coin protocols by incurring a small multiplicative error at a small additive cost. This is an improvement over Newman’s theorem [Inf. Proc. Let.'91] in the dependence on the error parameter. - As a consequence we obtain a (log(n/ε²) + 4)-cost private-coin communication protocol that computes the n-bit Equality function, to error ε. This improves upon the log(n/ε³) + O(1) upper bound implied by Newman’s theorem, and matches the best known lower bound, which follows from Alon [Comb. Prob. Comput.'09], up to an additive log log(1/ε) + O(1). The following are our results in various quantum models: - We exhibit a one-way protocol with log(n/ε) + 4 qubits of communication for the n-bit Equality function, to error ε, that uses only pure states. This bound was implicitly already shown by Nayak [PhD thesis'99]. - We give a near-matching lower bound: any ε-error one-way protocol for n-bit Equality that uses only pure states communicates at least log(n/ε) - log log(1/ε) - O(1) qubits. - We exhibit a one-way protocol with log(√n/ε) + 3 qubits of communication that uses mixed states. This is tight up to additive log log(1/ε) + O(1), which follows from Alon’s result. - We exhibit a one-way entanglement-assisted protocol achieving error probability ε with ⌈log(1/ε)⌉ + 1 classical bits of communication and ⌈log(√n/ε)⌉ + 4 shared EPR-pairs between Alice and Bob. This matches the communication cost of the classical public coin protocol achieving the same error probability while improving upon the amount of prior entanglement that is needed for this protocol, which is ⌈log(n/ε)⌉ + O(1) shared EPR-pairs. Our upper bounds also yield upper bounds on the approximate rank, approximate nonnegative-rank, and approximate psd-rank of the Identity matrix. As a consequence we also obtain improved upper bounds on these measures for a function that was recently used to refute the randomized and quantum versions of the log-rank conjecture (Chattopadhyay, Mande and Sherif [J. ACM'20], Sinha and de Wolf [FOCS'19], Anshu, Boddu and Touchette [FOCS'19]).
We prove a direct sum theorem for bounded round entanglement-assisted quantum communication complexity. To do so, we use the fully quantum definition for information cost and complexity that we recently introduced, and use both the fact that information is a lower bound on the communication, and the fact that a direct sum property holds for quantum information complexity. We then give a protocol for compressing a single copy of a protocol down to its quantum information cost, up to terms depending on the number of rounds and the allowed increase in error. Two important tools to derive this protocol are a smooth conditional min-entropy bound for a one-shot quantum state redistribution protocol, and the quantum substate theorem of Jain, Radhakrishnan and Sen (FOCS'02) to transform this bound into a von Neumann conditional entropy bound. This result further establishes the newly introduced notions of quantum information cost and complexity as the correct quantum generalisations of the classical ones in the standard communication complexity setting. Finding such a quantum generalization of information complexity was one of the open problem recently raised by Braverman (STOC'12).
We study the relative power of public randomness and quantum communication in the multiparty Simultaneous Message Passing (SMP) model. By analyzing a total Boolean function and a relational problem, we establish polynomial and exponential separations between classical public-coin protocols and quantum protocols without shared randomness or prior entanglement.
We first introduce XOR Equality, a total Boolean function in which each of k players receives an unrestricted string xj∈{0,1}n, and the referee must decide whether x1⊕⋯⊕xk=0n. For k≥8 and worst-case error at most 1/3, we prove quantum lower bounds of Ω(min{n,k2}) on total communication and Ω(min{n/k,k}) on maximum message length. A classical public-coin protocol uses two bits per player with error at most 1/4. For n=k2 and worst-case error at most 1/3, we prove that quantum protocols require maximum message length Ω(n) qubits, whereas two public-coin bits per player suffice.
We then turn to relations, for which we obtain exponential separations. Specifically, we study Multiparty Index Coordination, a k-player generalization of the two-player relation studied by Gavinsky, Kempe, Regev, and de Wolf (STOC 2006). We prove an unambiguous quantum lower bound of Ω(n1−1/k) on maximum message length, matched by a classical private-coin protocol. For any fixed error ε∈(0,1/8), we also prove a bounded-error quantum lower bound of Ω(n(k−1)/(k+1)) while we show that public-coin protocols solve the relation unambiguously with maximum message length O(logn), giving exponential separations in both regimes. The implicit constants are independent of k, and both quantum lower bounds become linear when k≥clogn for any fixed c>0.
We study the advantages of quantum communication models over classical communication models that are equipped with a limited number of qubits of entanglement. In this direction, we give explicit partial functions on n bits for which reducing the entanglement increases the classical communication complexity exponentially. Our separations are as follows. For every k≥1: Q∥∗ versus R2∗: We show that quantum simultaneous protocols with Θ~(k5log3n) qubits of entanglement can exponentially outperform two-way randomized protocols with O(k) qubits of entanglement. This resolves an open problem from [Gav08] and improves the state-of-the-art separations between quantum simultaneous protocols with entanglement and two-way randomized protocols without entanglement [Gav19, GRT22]. R∥∗ versus Q∥∗: We show that classical simultaneous protocols with Θ~(klogn) qubits of entanglement can exponentially outperform quantum simultaneous protocols with O(k) qubits of entanglement, resolving an open question from [GKRW06, Gav19]. The best result prior to our work was a relational separation against protocols without entanglement [GKRW06]. R∥∗ versus R1∗: We show that classical simultaneous protocols with Θ~(klogn) qubits of entanglement can exponentially outperform randomized one-way protocols with O(k) qubits of entanglement. Prior to our work, only a relational separation was known [Gav08]. Our techniques can also be used to show advantages of quantum communication models over hybrid classical-quantum models, i.e., models that have a large amount of both classical communication and quantum simultaneous communication.
The private simultaneous messages model is a non-interactive version of the multiparty secure computation, which has been intensively studied to examine the communication cost of the secure computation. We consider its quantum counterpart, the private simultaneous quantum messages (PSQM) model, and examine the advantages of quantum communication and prior entanglement of this model. In the PSQM model, k parties P1,…,Pk initially share a common random string (or entangled states in a stronger setting), and they have private classical inputs x1,…,xk. Every Pi generates a quantum message from the private input xi and the shared random string (entangled states), and then sends it to the referee R. Receiving the messages, R computes F(x1,…,xk). Then, R learns nothing except for F(x1,…,xk) as the privacy condition. We obtain the following results for this PSQM model. (1) We demonstrate that the privacy condition inevitably increases the communication cost in the two-party PSQM model as well as in the classical case presented by Applebaum, Holenstein, Mishra, and Shayevitz. In particular, we prove a lower bound (3−o(1))n of the communication complexity in two-party PSQM protocols with a shared random string for random Boolean functions of 2n-bit input, which is larger than the trivial upper bound 2n of the communication complexity without the privacy condition. (2) We demonstrate a factor two gap between the communication complexity of PSQM protocols with shared entangled states and with shared random strings by designing a multiparty PSQM protocol with shared entangled states for a total function that extends the two-party equality function. (3) We demonstrate an exponential gap between the communication complexity of PSQM protocols with shared entangled states and with shared random strings for a two-party partial function.