Multidimensional extensions of the Grothendieck inequality and applications

Blei, Ron C.

doi:10.1007/bf02385457

Cited by 22 publications

(22 citation statements)

References 7 publications

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“…In fact, it is possible to reduce the instance of Theorem 6 related to the NOF model to the NIH case. We feel, however, that Theorem 6 is superior to statements in [Ble79], [Ton78] as it works in the broadest generality and moreover the proof we give provides the clearest insight to the machinery behind the scenes.…”

Section: Dimensionmentioning

confidence: 86%

Lower Bounds on Quantum Multiparty Communication Complexity

Lee

Schechtman

Shraibman

2009

2009 24th Annual IEEE Conference on Computational Complexity

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Abstract-A major open question in communication complexity is if randomized and quantum communication are polynomially related for all total functions. So far, no gap larger than a power of two is known, despite significant efforts.We examine this question in the number-on-the-forehead model of multiparty communication complexity. We show that essentially all lower bounds known on randomized complexity in this model also hold for quantum communication. This includes bounds of size Ω(n/2 k ) for the k-party complexity of explicit functions, bounds for the generalized inner product function, and recent work on the multiparty complexity of disjointness. To the best of our knowledge, these are the first lower bounds of any kind on quantum communication in the general number-on-the-forehead model.We show this result in the following way. In the twoparty case, there is a lower bound on quantum communication complexity in terms of a norm γ2, which is known to subsume nearly all other techniques in the literature. For randomized complexity there is another natural bound in terms of a different norm µ which is also one of the strongest techniques available. A deep theorem in functional analysis, Grothendieck's inequality, implies that γ2 and µ are equivalent up to a constant factor. This connection is one of the major obstacles to showing a larger gap between randomized and quantum communication complexity in the two-party case.The lower bound technique in terms of the norm µ was recently extended to the multiparty number-on-the-forehead model. Here we show how the γ2 norm can be also extended to lower bound quantum multiparty complexity. Surprisingly, even in this general setting the two lower bounds, on quantum and classical communication, are still very closely related. This implies that separating quantum and classical communication in this setting will require the development of new techniques. The relation between these extensions of µ and γ2 is proved by a multi-dimensional version of Grothendieck's inequality.

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Section: Dimensionmentioning

confidence: 86%

Lower Bounds on Quantum Multiparty Communication Complexity

Lee

Schechtman

Shraibman

2009

2009 24th Annual IEEE Conference on Computational Complexity

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“…Based on a multilinear version of Grothendieck's inequality as given in [3,11] and [36], the third-named author has proved the following result in [32]: Let λ 1 , . . .…”

Section: Extendibility and Summability: The Case Nmentioning

confidence: 99%

Hahn–Banach extension of multilinear forms and summability

Jarchow

Palazuelos

Pérez-Garcı́a

et al. 2007

Journal of Mathematical Analysis and Applications

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“…The existence of such fi and h ( is implied by the existence of bounded multi-linear forms on a Hilbert space which are projectively unbounded [1, Definition 1.1]: such forms had been displayed first by Varopoulos in [17], and subsequently characterized in [1].…”

Section: An Analogous Characterization Of T 3 (% X%x%)mentioning

confidence: 99%