# Research

My research area is quantum information science, an interdisciplinary field at the intersection of quantum physics, mathematics and computer science. My main focus is to use the tool of convex optimization and information entropies to solve problems that arise in quantum information processing. My contributions range from areas of quantum resource theory, quantum Shannon theory to the mathematical foundations of quantum information. My PhD thesis is available here.

# Featured works

### Quantum resource theory

Quantum resource theory provides a highly versatile and powerful framework to study various quantum information processing tasks. It aims to characterize and quantify the manipulation of quantum “resources” such as entanglement, coherence and magic states, under restricted operations. In practice, a particularly important kind of manipulation is to “purify” the quantum resources — a key subroutine in quantum communication and computation that helps to extract high-quality resources better suited for applications.

**No-Go Theorems for Quantum Resource Purification**

**Kun Fang**, Zi-Wen Liu

*Physical Review Letters (Editors’ Suggestion & Featured in Physics), QIP 2021 contributed talk*

It proves fundamental limitations on how effectively generic noisy
resources can be purified enforced by the laws of quantum mechanics, which universally apply to *any* reasonable kind of quantum resource. In particular, it induces the *first* explicit lower bounds on the resource cost of magic state distillation, a leading scheme for realizing scalable fault-tolerant quantum computation.

**Non-asymptotic Entanglement Distillation**

**Kun Fang**, Xin Wang, Marco Tomamichel, and Runyao Duan

*IEEE Transaction on Information Theory, AQIS 2017 long talk, top 10 of all ~140 submissions*

It studies the practical scenario of distilling entanglement from *finite* copies of given states and introduces an efficiently computable framework to estimate the distillation rate via semidefinite programming.

**Probabilistic Distillation of Quantum Coherence**

**Kun Fang**, Xin Wang, Ludovico Lami, Bartosz Regula, and Gerardo Adesso

*Physical Review Letters, AQIS 2018 contributed talk*

It developes a general framework of probabilistic distillation of quantum coherence from a single instance of unstructured quantum state. Together with another PRL work — *One-Shot Coherence Distillation, (Bartosz Regula, Kun Fang, Xin Wang, and Gerardo Adesso)*, we contribute to a better understanding of the operational power of different free operation classes.

### Quantum Shannon theory

Quantum Shannon theory generalizes the classical theory, incorporating quantum phenomena like entanglement that have the potential to enhance communication capabilities. The theory of quantum channels is much richer but the ultimate performance of quantum channel communication is less well-understood than its classical counterpart.

**Geometric Rényi Divergence and its Applications in Quantum Channel Capacities**

**Kun Fang**, Hamza Fawzi

*Communications in Mathematical Physics, QIP 2020 contributed talk*

It gives a systematic study of the geometric Rényi divergence from the point of view of quantum information theory. It also provides the *first* sequence of applications of Belavkin-Staszewski relative entropy in quantum theory, significantly improving previous results based on the max-relative entropy.

**Semidefinite Programming Converse Bounds for Quantum Communication**

Xin Wang, **Kun Fang**, and Runyao Duan

*IEEE Transaction on Information Theory, QIP 2018 contributed talk*

It provides improved efficiently computable upper bounds for one-shot as well as asymptotic quantum capacity.

**On Converse Bounds for Classical Communication over Quantum Channels**

Xin Wang, **Kun Fang**, and Marco Tomamichel

*IEEE Transaction on Information Theory, QIP 2018 contributed talk*

It provides a finite resource analysis of classical communication over quantum channels, including the *first* tight second-order expansion beyond entanglement-breaking channels.

### Mathematical foundations and Optimization theory

The development of quantum information science often accompanies by a more thorough understanding of the mathematical framework underlying it. Of particular relevance are the theory of information entropies and convex optimization. The former usually provides the way for the information quantification and the latter helps to do the evaluation in practice.

**Chain Rule for the Quantum Relative Entropy**

**Kun Fang**, Omar Fawzi, Renato Renner and David Sutter

*Physical Review Letters, QIP 2020 contributed talk*

It proves a chain rule inequality for the quantum relative entropy in terms of channel relative entropies. The new chain rule allows us to solve an open problem in the context of asymptotic quantum channel discrimination: surprisingly, adaptive protocols cannot improve the error rate for asymmetric channel discrimination compared to non-adaptive strategies.

**The Sum-of-Squares Hierarchy on the Sphere, and Applications in Quantum Information Theory**

**Kun Fang**, Hamza Fawzi

*Mathematical Programming Series A, ICCOPT 2019 organized talk*

It considers the problem of maximizing a homogeneous polynomial on the unit sphere and its hierarchy of Sum-of-Squares (SOS) relaxations. It gives a quadratic improvement of the known convergence rate by Reznick and Doherty & Wehner, solving a problem left open by de Klerk & Laurent (arXiv:1904.08828). It also gives an exact duality relation that connects the SOS hierarchy and the set of quantum extendable states.

(More detailed research statement available upon request)