Anurag Anshu

Separations in Communication Complexity Using Cheat Sheets and Information Complexity

While exponential separations are known between quantum and randomized communication complexity for partial functions (Raz, STOC 1999), the best known separation between these measures for a total function is quadratic, witnessed by the disjointness function. We give the first super-quadratic separation between quantum and randomized communication complexity for a total function, giving an example exhibiting a power 2.5 gap. We further present a 1.5 power separation between exact quantum and randomized communication complexity, improving on the previous ≈ 1.15 separation by Ambainis (STOC 2013). Finally, we present a nearly optimal quadratic separation between randomized communication complexity and the logarithm of the partition number, improving upon the previous best power 1.5 separation due to Goos, Jayram, Pitassi, and Watson. Our results are the communication analogues of separations in query complexity proved using the recent cheat sheet framework of Aaronson, Ben-David, and Kothari (STOC 2016). Our main technical results are randomized communication and information complexity lower bounds for a family of functions, called lookup functions, that generalize and port the cheat sheet framework to communication complexity.

Quantum Communication Using Coherent Rejection Sampling

We show near optimal bounds on the worst case quantum communication of single-shot entanglement-assisted one-way quantum communication protocols for the {\em quantum state redistribution} task and for the sub-tasks {\em quantum state splitting} and {\em quantum state merging}. Our bounds are tighter than previously known best bounds for the latter two sub-tasks. A key technical tool that we use is a {\em convex-split} lemma which may be of independent interest.

A lower bound on the crossing number of uniform hypergraphs

In this paper, we consider the embedding of a complete d-uniform geometric hypergraph with n vertices in general position in Rd, where each hyperedge is represented as a (d1)-simplex, and a pair of hyperedges is defined to cross if they are vertex-disjoint and contain a common point in the relative interiors of the simplices corresponding to them. As a corollary of the Van KampenFlores Theorem, it can be seen that such a hypergraph contains (2dd)n2d crossing pairs of hyperedges. Using Gale Transform and Ham Sandwich Theorem, we improve this lower bound to (2dlogdd)n2d.

A One-Shot Achievability Result for Quantum State Redistribution

We study the problem of state redistribution both in the classical (shared randomness assisted) and quantum (entanglement assisted) one-shot settings and provide new upper bounds on the communication required. Our classical bounds are in terms of smooth-min and max relative entropies and the quantum bounds are in terms of max relative entropy and Renyi relative entropy of order

A Generalized Quantum Slepian–Wolf

\frac{1}{2}

On the Rectilinear Crossing Number of Complete Uniform Hypergraphs

. We also consider a special case of this problem in the classical setting, previously studied by Braverman and Rao (2011). We show that their upper bound is optimal. In addition we provide an alternate protocol achieving a priory different looking upper bound. However, we show that our upper bound is essentially the same as their upper bound and hence also optimal. For the quantum case, we show that our achievability result is upper bounded by the achievability result obtained in Berta, Christandl, Touchette (2016).We study the problem of entanglement-assisted quantum state redistribution in the one-shot setting and provide a new achievability result on the quantum communication required. Our bounds are in terms of the max-relative entropy and the hypothesis testing relative entropy. We use the techniques of convex split and position-based decoding to arrive at our result. We show that our result is upper bounded by the result obtained in Berta et al. (2016).

Exponential separation of quantum communication and classical information

In this paper, we consider a quantum generalization of the task considered by Slepian and Wolf regarding distributed source compression. In our task, Alice, Bob, Charlie, and Reference share a joint pure state. Alice and Bob wish to send a part of their respective systems to Charlie without collaborating with each other. We give achievability bounds for this task in the one-shot setting and provide the asymptotic and independent identically distributed analysis in the case when there is no side information with Charlie. Our result implies the result of Abeyesinghe et al., who studied a special case of this problem. As another special case wherein Bob holds trivial registers, we recover the result of Devetak and Yard regarding quantum state redistribution.

A composition theorem for randomized query complexity

In this paper, we consider a generalized version of the rectilinear crossing number problem of drawing complete graphs on a plane. The minimum number of crossing pairs of hyperedges in the

Contextuality in Multipartite Pseudo-Telepathy Graph Games

d

Lower Bound on Expected Communication Cost of Quantum Huffman Coding

-dimensional rectilinear drawing of a