北京雁栖湖应用数学研究院

In quantum mechanics, macroscopic observables may be represented by bounded self-adjoint operators $T_1, T_2, ..., T_n$ on a Hilbert space $H$. Commutators $T_jT_i-T_iT_j$ are related to the uncertainty principle in their measurements and small commutators indicate more precise measurements. In his recent book, David Mumford proposed to study “near eigenvectors” for some set of human observables which are called “Approximately Macroscopically Unique” states. This talk will present some answers to Mumford’s questions.

We introduce recent progresses in the theory of quantum information and related physics, including quantum coherence, quantum correlations, quantum uncertainty relations, as well as quantum measurement enhanced quantum battery capacity.

Fisher information is a measure of the amount of information that an observable random variable $X$ carries about an unknown parameter $\theta$. One important application of classical Fisher information is the sufficient statistic: a statistic $T=T(X)$ is sufficient for $X_{\theta}$ w.r.t the parameter $\theta$ if and only if the Fisher information is preserved by $T$. In this talk, I talk about the sufficiency about quantum Fisher information. It turns out that the sufficiency (i.e. the recoverability by a quantum channel) are not guaranteed by the preservation of SLD or RLD Fisher Information, which are the two most considered definitions in the literature. Nevertheless, the sufficiency is equivalent to the preservation of a large family of “regular” Fisher information, including BKM Fisher Information, just as the classical case.

The classical Borsuk-Ulam theorem may be seen as a statement about the complexity of spheres as principal Z/2Z-bundles via the antipodal action. I will talk about introducing the local triviality dimension, a generalization of G-index for noncommutative principal bundles.

This talk is part of a programme to understand quantum symmetries through subfactors and twisted equivariant K-theory and their applications in conformal field theory. Here I discuss the question of constructing actions of these quantum symmetries on the irrational rotation algebras and more generally noncommutative tori. This is based on joint work with Corey Jones.

Firstly, we will introduce the notion of free independence, which comes from Voiculescu’s probabilistic method to attack the free group von Neumann algebra isomorphism problem. Then, we introduce free analogues of certain classical groups, which are compact quantum groups in the sense of Woronowicz. There is a canonical way to define symmetric invariants on operator algebras with faithful states from compact quantum groups. With those symmetric conditions, we are able to determine the relations between generators of given von Neumann algebras conditionally by Kostler, Speicher, Curran, etc. These results are called de Finetti type theorems. In my recent work, we will provide a full classification of de Finetti type theorems for non-selfadjoint generators in both the commutative and free case. If time permits, we will explain the possible symmetries between classical and free case.

The classical shadow estimation is a sample-efficient protocol for learning the properties of a quantum system through randomized measurements. For qubit systems, this approach is efficient due to the Clifford group's unitary 3-design property. We show similar efficiency for qudit systems.

I will introduce the quantum generalization of Rényi’s information divergence and its use in quantum information theory, discussing its operational interpretation and error exponents in quantum information.

The von Neumann’s inequality states that for a contraction operator T on a Hilbert space and an analytic polynomial p, the norm of p(T) is controlled by the supremum norm of p on the unit disc. I will share a new proof and a method for generating counterexamples for the polydisc case.

Quantum walks are the quantum mechanical counterpart of classical random walks. I present a systematic approach to chiral quantum walks, introducing a full characterization of all possible Hamiltonians describing time evolution over a given topology.

Quantum codes achieving approximate quantum error correction (AQEC) are important but lack a systematic understanding. I will establish connections between quantum circuit complexity and AQEC properties, and propose O(k/n) as a boundary for AQEC codes.

We provide a unified approach to explain important notions in quantum information theory, such as separability/entanglement and Schmidt numbers of bipartite states, using bilinear pairings and Choi matrices. We extend these notions to infinite dimensional analogues.

We will talk about the asymptotic decay of the Fourier coefficients of the Mandelbrot canonical cascade measure and more general cascade measures. Our method is to put the analysis of these Fourier series into the framework of vector-valued martingales.

We give a new complete description theorem of the intermediate operator algebras, unifying the discrete Galois correspondence and crossed product splitting results. We obtain a Galois-type result for Bisch—Haagerup type inclusions arising from isometrically shift-absorbing actions.

I will present recent progress and open problems on control theory for stochastic partial differential equations, highlighting new phenomena and difficulties in controllability and optimal control problems.

We propose a method for simulating non-unitary dynamics as a linear combination of Hamiltonian simulation problems, achieving optimal state preparation cost. We demonstrate an application for open quantum dynamics simulation using near-optimal parameters.

Quantum adversarial machine learning is an emergent interdisciplinary research frontier that studies the vulnerability of quantum learning systems in adversarial scenarios and the development of potential countermeasures to enhance their robustness against adversarial perturbations. In this talk, I will first make a brief introduction to this field and review some recent progresses. I will show, through concrete examples, that typical quantum classifiers are extremely vulnerable to adversarial perturbations: adding a tiny amount of carefully crafted noises into the original legitimate samples may lead the classifiers to make incorrect predictions at a high confidence level. I will talk about possible defense strategies against adversarial attacks.<br><br>I will also talk about a recent experimental demonstration of quantum adversarial learning with programmable superconducting qubits.<br><br>Ref:<br>[1] S.-R. Lu, L. M. Duan, and D.-L. Deng, Phys. Rev. Research 2, 033212 (2020)<br>[2] W.-Y. Gong and D.-L. Deng, National Science Review 9, nwab130 (2022)<br>[3] W.-H. Ren et al., Nature Computational Science 2, 711 (2022)<br>[4] H.-L. Zhang et al., Nature Communications 13, 4993 (2022)

We propose a unified framework to extract an intrinsic error threshold from approximate QEC conditions, providing insights into error thresholds across QEC codes and models. This sharpens understanding of error thresholds for different QEC codes.

I will introduce recent results in noncommutative algebraic characterizations of nonlocal games and discuss algebraic reformulations of Connes’ embedding problem and counterexamples for polynomials in noncommuting variables.

Quantum k-SAT is known to be QMA_1-complete for k ≥ 3. I will show that quantum k-SAT can be solved in randomized polynomial time given that the instance is either satisfiable by any state or far from satisfiable by a product state.

Alterfold theory of dimension three, is a three dimensional generalization of Jones’ Planar algebra of finite depth. It has the advantage of describing/discovering/proving nontrivial equalities and inequalities in the theory of tensor categories and subfactors. In this talk, we will review the basic setting of the three dimensional alterfold theory, for a pair of Morita equivalent spherical fusion categories. In addition, if one of the category is braided, we show how modular invariant matrix can be read from the certain alterfold diagram. As a corollary, we show an obstruction for a modular invariant matrix to be physical.

We review the theory of 2+1 TQFT and its construction from spherical fusion categories. We introduce a unified framework to generalize the 2+1 theory to non-semisimple and higher dimensional cases. We will construct the Ising 3+1 TQFT and a non-semisimple one.