I. M. George

Ian George, Marco Tomamichel

The maximal correlation coefficient measures the linear correlation in a bipartite distribution and contraction coefficients measure how much information is lost under a noisy channel. Remarkably, Raginsky established a close relation between these two concepts by showing that the $χ^2$ contraction coefficient equals the maximal correlation coefficient of the joint input/output distribution of the channel. In quantum theory, several generalizations of these concepts have been proposed, but none recover all the classical properties. Here we construct a framework in which the classical theory extends to the quantum setting. We introduce families of quantum maximal correlation coefficients and show that many impose limits on converting quantum states under local operations. We establish a family of quantum contraction coefficients are efficiently computable, yielding a generic efficient algorithm for mixing times of quantum channels with a full rank fixed point. Furthermore, we establish a quantum analogue of Raginsky's classical correspondence that relates these two families of quantities. To do this, we develop the operator-theoretic approach to Petz's family of non-commutative $L^{2}(p)$ spaces that extend the data processing inequality for variance to quantum theory.

Chen-Xun Weng, Ming-Yang Li, Nai-Rui Xu et al.

Zero-knowledge proofs (ZKPs) are widely applied in digital economies, such as cryptocurrencies and smart contracts, for establishing trust and privacy between untrusted parties. Classical ZKPs rely on computational assumptions and are vulnerable to quantum attacks. While a recent advance suggests quantum-sound symmetric relativistic ZKPs for the graph three-coloring problem without computational assumptions, the high round complexity, which leads to unachievable runtime and overall randomness cost, renders them impractical for real-life deployment. To overcome this, we develop an efficient asymmetric relativistic ZKP protocol using relativistic bit commitments, and prove its quantum soundness by relating it to the nonlocal Clauser-Horne-Shimony-Holt (CHSH) game. Our protocol achieves a linear relationship between the round complexity and the number of edges, and thus significantly improves practical feasibility. In addition, we implement a proof-of-principle experiment which completes all interactive rounds in about 0.22 seconds and requires an overall randomness cost of 430.81 MB. Our work illustrates the powerful potential of integrating special relativity with quantum theory in trustless cryptography, paving the way for robust applications against quantum attacks in distrustful Internet environments.

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.