Marc Kaplan

The communication complexity of non-signaling distributions

We study a model of communication complexity that encompasses many well-studied problems, including classical and quantum communication complexity, the complexity of simulating distributions arising from bipartite measurements of shared quantum states, and XOR games. In this model, Alice gets an input x, Bob gets an input y, and their goal is to each produce an output a, b distributed according to some pre-specified joint distribution p(a, b|x, y). Our results apply to any non-signaling distribution, that is, those where Alices marginal distribution does not depend on Bobs input, and vice versa. By taking a geometric view of the non-signaling distributions, we introduce a simple new technique based on affine combinations of lower-complexity distributions, and we give the first general technique to apply to all these settings, with elementary proofs and very intuitive interpretations. Specifically, we introduce two complexity measures, one which gives lower bounds on classical communication, and one for quantum communication. These measures can be expressed as convex optimization problems. We show that the dual formulations have a striking interpretation, since they coincide with maximum violations of Bell and Tsirelson inequalities. The dual expressions are closely related to the winning probability of XOR games. Despite their apparent simplicity, these lower bounds subsume many known communication complexity lower bound methods, most notably the recent lower bounds of Linial and Shraibman for the special case of Boolean functions. We show that as in the case of Boolean functions, the gap between the quantum and classical lower bounds is at most linear in the size of the support of the distribution, and does not depend on the size of the inputs. This translates into a bound on the gap between maximal Bell and Tsirelson inequality violations, which was previously known only for the case of distributions with Boolean outcomes and uniform marginals. It also allows us to show that for some distributions, information theoretic methods are necessary to prove strong lower bounds. Finally, we give an exponential upper bound on quantum and classical communication complexity in the simultaneous messages model, for any non-signaling distribution. One consequence of this is a simple proof that any quantum distribution can be approximated with a constant number of bits of communication.

Kolmogorov complexity and combinatorial methods in communication complexity

We introduce a method based on Kolmogorov complexity to prove lower bounds on communication complexity. The intuition behind our technique is close to information theoretic methods. We use Kolmogorov complexity for three different things: first, to give a general lower bound in terms of Kolmogorov mutual information; second, to prove an alternative to Yaos minmax principle based on Kolmogorov complexity; and finally, to identify hard inputs. We show that our method implies the rectangle and corruption bounds, known to be closely related to the subdistribution bound. We apply our method to the hidden matching problem, a relation introduced to prove an exponential gap between quantum and classical communication. We then show that our method generalizes the VC dimension and shatter coefficient lower bounds. Finally, we compare one-way communication and simultaneous communication in the case of distributional communication complexity and improve the previous known result.

Kolmogorov Complexity and Combinatorial Methods in Communication Complexity

We introduce a method based on Kolmogorov complexity to prove lower bounds on communication complexity. The intuition behind our technique is close to information theoretic methods [1,2]. Our goal is to gain a better understanding of how information theoretic techniques differ from the family of techniques that follow from Linial and Shraibmans work on factorization norms [3]. This family extends to quantum communication, which prevents them from being used to prove a gap with the randomized setting. We use Kolmogorov complexity for three different things: first, to give a general lower bound in terms of Kolmogorov mutual information; second, to prove an alternative to Yaos minmax principle based on Kolmogorov complexity; and finally, to identify worst case inputs. We show that our method implies the rectangle and corruption bounds [4], known to be closely related to the subdistribution bound [2]. We apply our method to the hidden matching problem, a relation introduced to prove an exponential gap between quantum and classical communication [5]. We then show that our method generalizes the VC dimension [6] and shatter coefficient lower bounds [7]. Finally, we compare one-way communication and simultaneous communication in the case of distributional communication complexity and improve the previous known result [7].

Non-Local Box Complexity and Secure Function Evaluation

A non-local box is an abstract device into which Alice and Bob input bits x and yrespectively and receive outputs a and b, where a, b are uniformly distributed and a+b =x∧y. Such boxes have been central to the study of quantum or generalized non-locality, aswell as the simulation of non-signaling distributions. In this paper, we start by studyinghow many non-local boxes Alice and Bob need in order to compute a Boolean functionf. We provide tight upper and lower bounds in terms of the communication complexityof the function both in the deterministic and randomized case. We show that non-localbox complexity has interesting applications to classical cryptography, in particular tosecure function evaluation, and study the question posed by Beimel and Malkin [1] ofhow many Oblivious Transfer calls Alice and Bob need in order to securely compute afunction f. We show that this question is related to the non-local box complexity of thefunction and conclude by greatly improving their bounds. Finally, another consequenceof our results is that traceless two-outcome measurements on maximally entangled statescan be simulated with 3 non-local boxes, while no finite bound was previously known.

Provably secure key establishment against quantum adversaries

At Crypto 2011, some of us had proposed a family of cryptographic protocols for key establishment capable of protecting quantum and classical legitimate parties unconditionally against a quantum eavesdropper in the query complexity model. Unfortunately, our security proofs were unsatisfactory from a cryptographically meaningful perspective because they were sound only in a worst-case scenario. Here, we extend our results and prove that for any e > 0, there is a classical protocol that allows the legitimate parties to establish a common key after O(N) expected queries to a random oracle, yet any quantum eavesdropper will have a vanishing probability of learning their key after O(N^{1.5-e}) queries to the same oracle. The vanishing probability applies to a typical run of the protocol. If we allow the legitimate parties to use a quantum computer as well, their advantage over the quantum eavesdropper becomes arbitrarily close to the quadratic advantage that classical legitimate parties enjoyed over classical eavesdroppers in the seminal 1974 work of Ralph Merkle. Along the way, we develop new tools to give lower bounds on the number of quantum queries required to distinguish two probability distributions. This method in itself could have multiple applications in cryptography. We use it here to study average-case quantum query complexity, for which we develop a new composition theorem of independent interest.