Theme

The focus of this conference revolves around the mathematical intricacies of quantum theory, including operator algebras, vertex operator algebras, quantum field theory, noncommutative analysis etc.

Description of the aim

The workshop unites global young mathematicians exploring quantum theory’s diverse facets. It fosters collaboration, shares insights, and advances our understanding of quantum theory’s intricate math. Through engaging interactions, participants inspire and contribute to this dynamic field.

Hosts

Tsinghua University

Beijing Institute of Mathematical Sciences and Applications (BIMSA)

Contact

wjs@bimsa.cn

Venue

BIMSA A6-101, Huairou, Beijing

Organizers

Bin Gui (归斌), Tsinghua University

Zhengwei Liu (刘正伟), Tsinghua University

Jinsong Wu (吴劲松), BIMSA

Accommodation

Aloft Beijing Huairou (北京怀柔雅乐轩酒店)， Building No.3, Compound No.9, Yong Le North 2 Street, Huairou District, Beijing, China（北京怀柔区永乐北二街 9 号院 3 号楼）

Hotel Website: http://aloft-beijing-huairou-aloft.marriottcn.cn/en

Online Booking (English): ah.bjshl.rsvn@marriott.com

Hotel Reserve (Chinese):+0086 01056736558

Tentative Schedule

Jan 06, 2025

Registration

Jan 07, 2025

08:45 ~ 09:00		Open Ceremony and Photo
09:00 ~ 09:45	Yasu Kawahigashi	Operator algebras, tensor networks and quantum 6j-symbols
09:55 ~ 10:40	Xin Sun	Annulus crossing formulae for critical planar percolation
Tea Time
11:00 ~ 11:45	Guo Chuan Thiang	Exact quantization and large-scale index theory
Lunch
14:15 ~ 15:00	Guixiang Hong	An overview over noncommutative maximal inequalities
15:10 ~ 15:55	Sayan Das	Poisson boundaries of groups and von Neumann algebras
Tea Time
16:15 ~ 17:00	Baojun Wu	Conformal bootstrap in boundary Liouville CFT
17:10 ~ 17:55	Yuguo Shao	Pauli path integral for simulating noisy variational quantum algorithm
Dinner

Jan 08, 2025

09:00 ~ 09:45	Chunlan Jiang	Gelfand and Helly inequality and von Neumann invariant subspace problem
09:55 ~ 10:40	Penghui Wang	Optimal control of quantum stochastic systems in fermion fields: Pontryagin maximal principle
Tea Time
11:00 ~ 11:45	Kai Wang	Recent progress in the central limit theorem of determinantal point processes
Lunch
14:15 ~ 15:00	Yongjiang Duan	Essential normality for the weighted Bergman modules
15:10 ~ 15:55	Yuanhang Zhang	Stability relations for Hilbert space operators and a problem of Kaplansky
Tea Time
16:15 ~ 17:00	Jinghao Huang	Isometric embeddings of noncommutative $L_p$-spaces into noncommutative symmetric spaces
17:10 ~ 17:55	Hua Wang	Topological quantum groups via locally convex Hopf algebras
Dinner

Jan 09, 2025

09:00 ~ 09:45	Shunlong Luo	Quantifying incompatibility between POVMs via negativity of Jordan product
09:55 ~ 10:40	Simeng Wang	A pictorial construction of quantum group actions arising from subfactor planar algebras
Tea Time
11:00 ~ 11:45	Oleg Aristov	In search of quantum Lie groups
Lunch
14:15 ~ 15:00	Heon Lee	First-order differential calculi and Laplacians on $q$-deformations of compact semisimple Lie groups
15:10 ~ 15:55	Ting Lu	The Haar state of O(SU_q(3)) on a monomial basis
Tea Time
Discussion
Dinner

Speaker:

Oleg Aristov (Harbin Institute of Technology)

Title:

In search of quantum Lie groups

Abstract:

Locally compact quantum group is a well-established concept in operator algebra theory. We propose, however, to define a quantum Lie group using objects beyond the scope of operator algebras, such as Arens-Michael algebras (i.e., projective limits of Banach algebras) in the complex case and projective limits of Banach algebras of polynomial growth in the real case. Our approach leads to slightly different results in the complex and the real case.

Speaker:

Sayan Das (East China Normal University)

Title:

Poisson boundaries of groups and von Neumann algebras

Abstract:

Furstenberg introduced the notion of Poisson boundary of groups in 1960s. Since then, Poisson boundaries have played a pivotal role in ergodic theory, and measured group theory to study various rigidity phenomenon associated with groups and their actions.

Speaker:

Yongjiang Duan (Jinan University)

Title:

Essential normality for the weighted Bergman modules

Abstract:

Let \(\omega\) be a radial weight on the the unit ball \(B_d\) that satisfies certain doubling condition, we study the essential normality of the weighted Bergman module itself, as well as the essential normality of some kind submodules.

Speaker:

Guixiang Hong (Harbin Institute of Technology)

Title:

An overview over noncommutative maximal inequalities

Abstract:

In this talk, I shall try to give a survey over the maximal inequalities in noncommutative analysis including noncommtuative harmonic analysis, noncommutative ergodic theory and quantum probability.

Speaker:

Jinghao Huang (Harbin Institute of Technology)

Title:

Isometric embeddings of noncommutative \(L_p\)-spaces into noncommutative symmetric spaces

Abstract:

We establish a noncommutative version of a familiar Johnson–Maurey–Schechtman–Tzafriri Theorem, by showing that for any 0 < p < 2 and a (not necessarily semifinite) von Neumann algebra M on a separable Hilbert space, if a symmetric quasi-Banach function space E(0,1) containing the function t↦t^(1/p), 0<t≤1, then there exists a noncommutative probability space (N ,σ) such that L_p (M) is isometric to a subspace of E(N ,σ). In particular, this answers two questions raised by Randrianantoanina in 2006.

Speaker:

Chunlan Jiang (Hebei Normal University)

Title:

Gelfand and Helly inequality and von Neumann invariant subspace problem

Abstract:

In the 1920s, Von Neumann proposed the famous invariant subspace problem: Does every bounded linear operator acting on the infinite dimensional Hilbert space have a non trivial invariant subspace? Gelfand and Helly gave the following result in the 1950s: For operators with a single point spectrum, If the n power of such operators is polynomial growth, the answer to the Von Neumann ’s invariant subspace problem is “yes”. In this report, we extend the depth of Gelfand and Helly inequality, thereby proving that a larger class of operators have non trivial invariant subspace. The spectrum of this type of operator that may be infinitely set.

Speaker:

Yasu Kawahigashi (University of Tokyo)

Title:

Operator algebras, tensor networks and quantum 6j-symbols

Abstract:

We discuss roles of tensor networks in studies of two-dimensional topological order from a view point of operator algebras. We present certain 4-tensors in the setting of generalized quantum 6j-symbols and flat bi-unitary connections.

Speaker:

Heon Lee (Harbin Institute of Technology)

Title:

First-order differential calculi and Laplacians on \(q\)-deformations of compact semisimple Lie groups

Abstract:

In this talk, we suggest a simple definition of Laplacian on a compact quantum group (CQG) associated with a first-order differential calculus (FODC) on it. Applied to the classical differential calculus on a compact Lie group, this definition yields classical Laplacians, as it should. Moreover, on the CQG $ K_q $ arising from the $ q $-deformation of a compact semisimple Lie group \(K\), we can find many interesting linear operators that satisfy this definition, which converge to a classical Laplacian on $ K $ as $ q $ tends to 1. In the light of this, we call them $ q $-Laplacians on $ K_q $ and investigate some of their operator theoretic properties. In particlar, we show that the heat semigroups generated by these are not completely positive, suggesting that perhaps on the CQG $ K_q $, stochastic processes that are most relevant to the geometry of it are not quantum Markov processes. This work is based on the preprint arXiv:2410.00720.

Speaker:

Ting Lu (Harbin Institute of Technology)

Title:

The Haar state of O(SU_q(3)) on a monomial basis

Abstract:

The quantum group O(SU_q(3)) is a Hopf algebra dual to the Drinfeld-Jimbo quantum group U(sl_q(3)). It can be considered as a q-deformation (q is a real number and q^2 ) of the algebra of functions on the Lie group SU(3). The Haar state on O(SU_q(3)) is an invariant linear functional which can be considered as a q-deformation of the normalized integration on SU(3) with respect to the Haar measure. In this talk, I will introduce a monomial basis on O(SU_q(3)) and give an explicit expression of the Haar state of the monomial basis as a rational polynomial in variable q. Then I will prove that when q goes to 1, the limiting Haar state on O(SU_1(3)) coincides with the normalized integration on SU(3) with respect to the Haar measure.

Speaker:

Shunlong Luo (Academy of Mathematics and Systems Science)

Title:

Quantifying incompatibility between POVMs via negativity of Jordan product

Abstract:

A fundamental feature of quantum mechanics is the incompatibility between quantum measurements, which leads to the Heisenberg uncertainty relations and is intrinsically related to detection of many other quantum features such as nonlocality and steering. Given the significance and ubiquity of incompatibility, it is desirable to quantify the degree of incompatibility between two quantum measurements which are mathematically described by positive operator valued measures (POVMs). The commutator (Lie product) between operators has widely been used in quantifying incompatibility (non-commutativity) and plays a crucial role in quantum mechanics. In contrast, the anti-commutator (Jordan product) between operators has been relatively less employed in the study of quantum mechanics. In this work, we explore the role of the Jordan product in characterizing measurement incompatibility. The key observation lies in that while the product of two non-negative classical observables is automatically non-negative, this is not the case for quantum observables. Indeed the Jordan product of two non-negative observables is not necessarily non-negative. This negativity of the Jordan product is a purely quantum phenomenon and is here exploited to quantify a kind of measurement incompatibility, which interpolates between non-commutativity and joint measurability. We reveal basic features of this approach, make some comparisons with existing quantifiers of incompatibility.

Speaker:

Yuguo Shao (Tsinghua University)

Title:

Pauli path integral for simulating noisy variational quantum algorithm

Abstract:

Large-scale variational quantum algorithms (VQA) are widely recognized as a potential pathway to achieve practical quantum advantages. However, the presence of quantum noise might suppress and undermine these advantages, which blurs the boundaries of classical simulability. To examine noise effects on VQAs, we introduce a new classical simulation approach based on the Pauli path integral. This approach enables approximate calculation of operator expectation values under single-qubit Pauli noise, with controllable truncation error and polynomial computational cost. Furthermore, we apply this method to simulate IBM’s 127-qubit quantum computations, yielding consistent results. This work is a collaboration with Fuchuan Wei, Song Chen, and Zhengwei Liu.

Speaker:

Xin Sun (Peking University)

Title:

Annulus crossing formulae for critical planar percolation

Abstract:

This is a joint work with Shengjing Xu and Zijie Zhuang. We derive exact formulae for three basic annulus crossing events for the critical planar Bernoulli percolation in the continuum limit. The first is for the probability that there is an open path connecting the two boundaries of an annulus of inner radius r and outer radius R. The second is for the probability that there are both open and closed paths connecting the two annulus boundaries. These two results were predicted by Cardy based on non-rigorous Coulomb gas arguments. Our third result gives the probability that there are two disjoint open paths connecting the two boundaries. Its leading asymptotic as r/R→0 is captured by the so-called backbone exponent, a transcendental number recently determined by Nolin, Qian and two of us. This exponent is the unique non-trivial real root to the an elementary equation. Besides the real roots, this equation has countably many complex roots. Our third result shows that these roots appear exactly as exponents of the subleading terms in the crossing formula. This suggests that the backbone exponent is part of a conformal field theory (CFT) whose bulk spectrum contains this set of roots. Expanding the same crossing probability as r/R→1, we obtain a series with logarithmic corrections at every order, suggesting that the backbone exponent is related to a logarithmic boundary CFT. Our proofs are based on the coupling between SLE curves and Liouville quantum gravity (LQG). The key is to encode the annulus crossing probabilities by the random moduli of certain LQG surfaces with annular topology, whose law can be extracted from the dependence of the LQG annuli partition function on their boundary lengths.

Speaker:

Guo Chuan Thiang (Peking University)

Title:

Exact quantization and large-scale index theory

Abstract:

Quantum mechanics has extremely precise manifestations in macroscopic phenomena. Indeed, Planck’s constant is accessed most accurately via a “classical” measurement of the electrical Hall conductance of a macroscopic 2D electron gas. Forty years ago, this conductance was famously discovered to be exactly quantized. Although such exact quantization phenomena are informally understood as “topological” phases, I will instead explain the analytic story, and relate the two perspectives via a macroscopic version of index theory. A simple mechanism for exact fractional quantization will also be sketched.

Speaker:

Simeng Wang (Harbin Institute of Technology)

Title:

A pictorial construction of quantum group actions arising from subfactor planar algebras

Abstract:

It is well-known that any subfactor planar algebra gives rise to a subfactor inclusion. Pinzari and Roberts have proven that one can derive ergodic actions of compact quantum groups from inclusions of II_1 factors. In 2008 Vaughan R. Jones asked if one may establish an explicit connection from subfactor planar algebras to the aforementioned ergodic actions. This talk will present an answer to this question. Our construction of the action is very simple and purely relies on the pictorial languages of planar algebras and universal quantum groups. The talk is based on previous joint work with Amaury Freslon and Frank Taipe, as well as an ongoing joint project with Frank Taipe.

Speaker:

Hua Wang (Harbin Institute of Technology)

Title:

Topological quantum groups via locally convex Hopf algebras

Abstract:

In his famous ICM address, Drinfel’d already had the observation that a quantum group should be described by some sort of Hopf algebras where the underlying tensor product can be topological instead of purely algebraic. Later rapid development in the direction of locally compact quantum groups, however, uses heavily the theory of operator algebras, and the resulting formulation of the general theory diverges further and and further from the axioms of Hopf algebras. As a pursuit of topological quantum groups from Drinfel’d’s original insight, I will describe a framework of locally convex Hopf algebras, as well as some duality results. I will focus on how to include all compact and discrete quantum groups into this framework, and time permitting, I will also describe some new interesting topological quantum groups that go beyond the locally compact case.

Speaker:

Kai Wang (Fudan University)

Title:

Recent progress in the central limit theorem of determinantal point processes

Abstract:

In this talk, we will survey some recent progress the central limit theorem for linear statistics over Pfaffian point processes. As an application, we obtain the Gaussian limit for the number of points over Sine4-process under a scaling. This is joint work with Dr. Mei Xu.

Speaker:

Penghui Wang (Shandong University)

Title:

Optimal control of quantum stochastic systems in fermion fields: Pontryagin maximal principle

Abstract:

In this talk, I will introduced some recent development on the optimal control of quantum stochastic systems. Such a quantum stochastic system is an noncommutative analogue of stochastic differential equations, and the well-definedness of L^p solution is based on the BG inequality obtained by Pisier and Xu. By using dynamic programming principle, the optimal control problem was studied when the drift term has no control. An in this talk, we will introduced the Pontryagin maximal principle of such a controlled quantum stochastic system with control in the drift term. The talk is based on the joint work with S.Wang

Speaker:

Baojun Wu (Peking University)

Title:

Conformal bootstrap in boundary Liouville CFT

Abstract:

In this talk, we present recent developments in boundary Liouville conformal field theory. We begin with the probabilistic construction of correlation functions, delving into the Hamiltonian and its spectrum. We then demonstrate how Ward identities can be derived from variations in the semigroups of half-annuli. As an application, we explain the relationship between the conformal blocks of open surfaces and the chiral conformal blocks of closed surfaces. This is based on the joint work with Colin Guillarmou and Rémi Rhodes.

Speaker:

Yuanhang Zhang (Jilin University)

Title:

Stability relations for Hilbert space operators and a problem of Kaplansky

Abstract:

In his monograph on Infinite Abelian Groups, Kaplansky raised three ``test questions" concerning their structure and multiplicity. Here, we are interested in the operator theoretic version of Kaplansky’s second question which asks: if \(A\) and \(B\) are operators on an infinite-dimensional, separable Hilbert space and \(A \oplus A\) is equivalent to \(B \oplus B\) in some (precise) sense, is \(A\) equivalent to \(B\)? We examine this question under a strengthening of the hypothesis, where a “primitive” square root \(J_2(A)\) of \(A\oplus A\) is assumed to be equivalent to the corresponding square root \(J_2(B)\) of \(B \oplus B\). When “equivalence” refers to similarity of operators and \(A\) is a compact operator, we deduce from this stronger hypothesis that \(A\) and \(B\) are similar. Also, we provide a large family of examples of unitary operators \(U\) which are unitarily equivalent to \(J_n(U)\), a “primitive” \(n^{th}\)-root of \(U \oplus U \oplus \cdots \oplus U\).

This is a joint work with Laurent Marcoux, Heydar Radjavi and Sascha Troscheit.

We invite anyone interested in joining our upcoming workshop to register by emailing wjs@bimsa.cn. This is a great opportunity to learn, connect, and engage with others in the field. Graduate students are especially encouraged to attend—don’t miss out!