Jamie Sikora

Akshay Bansal, Ian George, Soumik Ghosh et al.

In many quantum tasks, there is an unknown quantum object that one wishes to learn. An online strategy for this task involves adaptively refining a hypothesis to reproduce such an object or its measurement statistics. A common evaluation metric for such a strategy is its regret, or roughly the accumulated errors in hypothesis statistics. We prove a sublinear regret bound for learning over general subsets of positive semidefinite matrices via the regularized-follow-the-leader algorithm and apply it to various settings where one wishes to learn quantum objects. For concrete applications, we present a sublinear regret bound for learning quantum states, effects, channels, interactive measurements, strategies, co-strategies, and the collection of inner products of pure states. Our bound applies to many other quantum objects with compact, convex representations. In proving our regret bound, we establish various matrix analysis results useful in quantum information theory. This includes a generalization of Pinsker's inequality for arbitrary positive semidefinite operators with possibly different traces, which may be of independent interest and applicable to more general classes of divergences.

Ashwin Nayak, Jamie Sikora, Levent Tunçel

We focus on a family of quantum coin-flipping protocols based on bit-commitment. We discuss how the semidefinite programming formulations of cheating strategies can be reduced to optimizing a linear combination of fidelity functions over a polytope. These turn out to be much simpler semidefinite programs which can be modelled using second-order cone programming problems. We then use these simplifications to construct their point games as developed by Kitaev. We also study the classical version of these protocols and use linear optimization to formulate optimal cheating strategies. We then construct the point games for the classical protocols as well using the analysis for the quantum case. We discuss the philosophical connections between the classical and quantum protocols and their point games as viewed from optimization theory. In particular, we observe an analogy between a spectrum of physical theories (from classical to quantum) and a spectrum of convex optimization problems (from linear programming to semidefinite programming, through second-order cone programming). In this analogy, classical systems correspond to linear programming problems and the level of quantum features in the system is correlated to the level of sophistication of the semidefinite programming models on the optimization side. Concerning security analysis, we use the classical point games to prove that every classical protocol of this type allows exactly one of the parties to entirely determine the coin-flip. Using the relationships between the quantum and classical protocols, we show that only "classical" protocols can saturate Kitaev's lower bound for strong coin-flipping. Moreover, if the product of Alice and Bob's optimal cheating probabilities is 1/2, then one party can cheat with probability 1. This rules out quantum protocols of this type from attaining the optimal level of security.

Ashwin Nayak, Jamie Sikora, Levent Tunçel

Coin-flipping is a cryptographic task in which two physically separated, mistrustful parties wish to generate a fair coin-flip by communicating with each other. Chailloux and Kerenidis (2009) designed quantum protocols that guarantee coin-flips with near optimal bias. The probability of any outcome in these protocols is provably at most $1/\sqrt{2} + δ$ for any given $δ> 0$. However, no explicit description of these protocols is known, and the number of rounds in the protocols tends to infinity as $δ$ goes to 0. In fact, the smallest bias achieved by known explicit protocols is $1/4$ (Ambainis, 2001). We take a computational optimization approach, based mostly on convex optimization, to the search for simple and explicit quantum strong coin-flipping protocols. We present a search algorithm to identify protocols with low bias within a natural class, protocols based on bit-commitment (Nayak and Shor, 2003) restricting to commitment states used by Mochon (2005). An analysis of the resulting protocols via semidefinite programs (SDPs) unveils a simple structure. For example, we show that the SDPs reduce to second-order cone programs. We devise novel cheating strategies in the protocol by restricting the semidefinite programs and use the strategies to prune the search. The techniques we develop enable a computational search for protocols given by a mesh over the parameter space. The protocols have up to six rounds of communication, with messages of varying dimension and include the best known explicit protocol (with bias 1/4). We conduct two kinds of search: one for protocols with bias below 0.2499, and one for protocols in the neighbourhood of protocols with bias 1/4. Neither of these searches yields better bias. Based on the mathematical ideas behind the search algorithm, we prove a lower bound on the bias of a class of four-round protocols.

André Chailloux, Gus Gutoski, Jamie Sikora

Oblivious transfer is a fundamental cryptographic primitive in which Bob transfers one of two bits to Alice in such a way that Bob cannot know which of the two bits Alice has learned. We present an optimal security bound for quantum oblivious transfer protocols under a natural and demanding definition of what it means for Alice to cheat. Our lower bound is a smooth tradeoff between the probability B with which Bob can guess Alice's bit choice and the probability A with which Alice can guess both of Bob's bits given that she learns one of the bits with certainty. We prove that 2B + A is greater than or equal to 2 in any quantum protocol for oblivious transfer, from which it follows that one of the two parties must be able to cheat with probability at least 2/3. We prove that this bound is optimal by exhibiting a family of protocols whose cheating probabilities can be made arbitrarily close to any point on the tradeoff curve.