Exponential Separation of Information and Communication for Boolean Functions

Ganor, Anat; Kol, Gillat; Raz, Ran

doi:10.1145/2907939

Cited by 35 publications

(23 citation statements)

References 32 publications

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“…Another property of our construction is that the number of bits of input given to Alice is ≈ 7 log d. Thus, the lower bound on the expected communication cost is of the order of the input size. This may be contrasted with the well known exponential separations between information and communication [GKR14,GKR15] and their recent quantum counterpart [ATYY17], where the lower bound on the communication cost is doubly exponentially smaller than the input size.…”

Section: Some Fundamental Tasks In Quantum Information Theorymentioning

confidence: 98%

Expected Communication Cost of Distributed Quantum Tasks

Anshu¹,

Garg²,

Harrow³

et al. 2018

IEEE Trans. Inform. Theory

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A central question in classical information theory is that of source compression, which is the task where Alice receives a sample from a known probability distribution and needs to transmit it to the receiver Bob with small error. This problem has a one-shot solution due to Huffman, in which the messages are of variable length and the expected length of the messages matches the asymptotic and i.i.d. compression rate of the Shannon entropy of the source.In this work, we consider a quantum extension of above task, where Alice receives a sample from a known probability distribution and needs to transmit a part of a pure quantum state (that is associated to the sample) to Bob. We allow entanglement assistance in the protocol, so that the communication is possible through classical messages, for example using quantum teleportation. The classical messages can have a variable length and the goal is to minimize their expected length. We provide a characterization of the expected communication cost of this task, by giving a lower bound that is near optimal up to some additive factors.A special case of above task, and the quantum analogue of the source compression problem, is when Alice needs to transmit the whole of her pure quantum state. Here we show that there is no one-shot interactive scheme which matches the asymptotic and i.i.d. compression rate of the von Neumann entropy of the average quantum state. This is a relatively rare case in quantum information theory where the cost of a quantum task is significantly different from its classical analogue. Further, we also exhibit similar results for the fully quantum task of quantum state redistribution, employing some different techniques. We show implications for the one-shot version of the problem of quantum channel simulation.with the environment (which we shall refer to as the Reference system). A well known example of such tasks is quantum state merging [HOW07], which provided the first operational interpretation to the negativity of quantum conditional information. A generalization of this is the task of quantum state redistribution, which was first introduced in [DY08, YD09] and applied to the setting of quantum communication complexity in [Tou15].Quantum state redistribution: Alice (A,C), Bob(B) and Reference(R) share a joint pure quantum state |Ψ RBCA . Alice needs to transfer the register C to Bob such that the final state between Alice (A), Bob (B,C) and Reference (R) is Ψ ′ RBCA . It is required that the distance between Ψ ′ RBCA and Ψ RBCA is smaller than ε ∈ (0, 1). The worst case communication cost of this task in the asymptotic and i.i.d. setting is characterized by the conditional quantum mutual information I(C : R|B) Ψ [DY08, YD09]. All of the above tasks are illustrated in Figure 1.Coherent quantum tasks such as quantum state merging [HOW05, HOW07] have also found applications to the problems related to quantum channel coding, through the notion of quantum channel simulation. This task is informally defined as follows.Entanglement-assisted quantum chan...

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Section: Some Fundamental Tasks In Quantum Information Theorymentioning

confidence: 98%

Expected Communication Cost of Distributed Quantum Tasks

Anshu¹,

Garg²,

Harrow³

et al. 2018

IEEE Trans. Inform. Theory

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show abstract

“…Hence, we wish to bound the amount of information about Y j that Alice has in any round, conditioned on some fixed values of x ≤j j. In all previous works [GKR16, GKR15,RS15b] on exponential separation between information and communication, the proof relied on a statement of the form "the information Alice has about the j-th part of Bob's input is upper bounded by 2 O( ) n ". This holds even when conditioning on some j playing a role similar to the hidden index j here, and also on some part of Alice's input corresponding to j. is the total number of bits of communication in the protocol, and n is the number of parts of Alice's input (usually related to the depth of some underlying communication tree), of size exponentially larger than the desired communication bound.…”

Section: Lemma 32 Consider a Quantum State |ψmentioning

confidence: 99%

Exponential separation of quantum communication and classical information

Anshu

Touchette

Yao

et al. 2017

Proceedings of the 49th Annual ACM SIGACT Symposium on Theory of Computing

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We exhibit a Boolean function for which the quantum communication complexity is exponentially larger than the classical information complexity. An exponential separation in the other direction was already known from the work of Kerenidis et. al. [SICOMP 44, pp. 1550-1572, hence our work implies that these two complexity measures are incomparable. As classical information complexity is an upper bound on quantum information complexity, which in turn is equal to amortized quantum communication complexity, our work implies that a tight direct sum result for distributional quantum communication complexity cannot hold. The function we use to present such a separation is the Symmetric k-ary Pointer Jumping function introduced by Rao and Sinha [ECCC TR15-057], whose classical communication complexity is exponentially larger than its classical information complexity. In this paper, we show that the quantum communication complexity of this function is polynomially equivalent to its classical communication complexity. The high-level idea behind our proof is arguably the simplest so far for such an exponential separation between information and communication, driven by a sequence of round-elimination arguments, allowing us to simplify further the approach of Rao and Sinha.As another application of the techniques that we develop, we give a simple proof for an optimal trade-off between Alice's and Bob's communication while computing the related GreaterThan function on n bits: say Bob communicates at most b bits, then Alice must send n 2 O(b) bits to Bob. This holds even when allowing pre-shared entanglement. We also present a classical protocol achieving this bound.arXiv:1611.08946v1 [quant-ph]

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“…Put differently, is it always possible to compress the communication cost of a protocol to its information cost? For the two-party case it is known that perfect compression is not possible for single shot protocols [GKR15a,GKR15b] (unless they are restricted to a constant number of rounds [JRS03]). Still, several interesting compression results are known.…”

Section: Introductionmentioning

confidence: 99%

Multi-Party Protocols, Information Complexity and Privacy

Kerenidis

Rosen

Urrutia

2019

ACM Trans. Comput. Theory

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We introduce a new information theoretic measure that we call Public Information Complexity (PIC), as a tool for the study of multi-party computation protocols, and of quantities such as their communication complexity, or the amount of randomness they require in the context of information-theoretic private computations. We are able to use this measure directly in the natural asynchronous messagepassing peer-to-peer model and show a number of interesting properties and applications of our new notion: the Public Information Complexity is a lower bound on the Communication Complexity and an upper bound on the Information Complexity; the difference between the Public Information Complexity and the Information Complexity provides a lower bound on the amount of randomness used in a protocol; any communication protocol can be compressed to its Public Information Cost; an explicit calculation of the zero-error Public Information Complexity of the k-party, n-bit Parity function, where a player outputs the bit-wise parity of the inputs. The latter result also establishes that the amount of randomness needed by a private protocol that computes this function is Ω(n).Our main goal is to introduce novel information-theoretical measures for the study of number-in-hand, message-passing multi-party protocols, coupled with a natural model that, among other things, allows private protocols (which is not the case for, e.g., the coordinator model).We define the new measure of Public Information Complexity (PIC), as a tool for the study of multiparty computation protocols, and of quantities such as their communication complexity, or the amount of randomness they require in the context of information-theoretic private computations. Intuitively, our new measure captures a combination of the amount of information about the inputs that the players leak to other players, and the amount of randomness that the protocol uses. By proving lower bounds on PIC for a given multi-party function f , we are able to give lower bounds on the multi-party communication complexity of f and on the amount of randomness needed to privately compute f . The crucial point is that the PIC of functions, in our multi-party model, is not always 0, unlike their IC.Our new measure works in a model which is a slight restriction of the most general asynchronous model, where, for a given player at a given time, the set of players from which that player waits for a message can be determined by that player's own local view. This allows us to have the property that for any protocol, the information which is leaked during the execution of the protocol is at most the communication cost of the protocol. Note that in the multi-party case, the information cost of a protocol may be higher than its communication cost, because the identity of the player from which one receives a message might carry some information. We are able to define our measure and use it directly in a natural asynchronous peer-to-peer model (and not, e.g., in the coordinator model used in most works studying...

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Exponential Separation of Information and Communication for Boolean Functions

Cited by 35 publications

References 32 publications

Expected Communication Cost of Distributed Quantum Tasks

Expected Communication Cost of Distributed Quantum Tasks

Exponential separation of quantum communication and classical information

Multi-Party Protocols, Information Complexity and Privacy

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