Mathematical Aspects of Quantum Theory

January 12 - 17, 2024

Theme

The focus of this conference revolves around the mathematical intricacies of quantum theory, including operator algebras, vertex operator algebras, quantum field theory, conformal field theory, quantum information 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)
Tsinghua Sanya International Mathematical Forum (TSIMF)

Organizers

Bin Gui (归斌), Tsinghua University
Zhengwei Liu (刘正伟), Tsinghua University
Jinsong Wu (吴劲松), BIMSA

Venue

Room A-110, TSIMF, Sanya, Hainan

清华三亚国际数学论坛 A-110教室

海南省三亚市天涯区清华路100号

Jan 12, 2024

Registration

Jan 13, 2024

09:40 ~ 10:00		Open Ceremony and Photo
10:00 ~ 10:50	Sen Zhu	Irreducible approximation of Hilbert space operators
Tea Break
11:10 ~ 12:00	Qiang Zeng	Hessian spectrum at the global minimum of locally isotropic Gaussian random fields
Lunch
14:00 ~ 14:50	Rui Shi	Reducible operators and Gamma Property in von Neumann algebras
14:50 ~ 15:40	Yongle Jiang	On invariant von Neumann subalgebras rigidity and related problems
Tea Break
16:00 ~ 16:50	Robert McRae	Tensor categories from vertex operator algebras

Jan 14, 2024

09:30 ~ 10:20	Shilin Yu	Quantization of coadjoint orbits
10:20 ~ 11:10	Jinwei Yang	Ribbon categories for the singlet algebras and their extensions
Tea Break
11:25 ~ 12:05	Zishuo Zhao	Relative entropy for quantum channels
Lunch
14:00 ~ 14:50	Haonan Zhang	An inequality in Fourier analysis and its application to learning theory: classical and quantum
Tea Break
15:10 ~ 16:00	Yuanyuan Xu	Universality of spectral radius of a large non-Hermitian random matrix

Jan 15, 2024

09:30 ~ 10:20	Xuanzhong Dai	Vertex algebras arising from congruence subgroups
10:20 ~ 11:10	Qing Wang	Orbifold theory of the parafermion vertex operator algebra $K(osp(1\|2),k)$
Tea Break
11:25 ~ 12:05	Fan Lu	Classification of exchange relation planar algebras of rank 5
Lunch
14:00 ~ 17:00		Free Discussion

Jan 16, 2024

09:30 ~ 10:20	Guixiang Hong	Pointwise convergence of noncommutative Fourier series
10:20 ~ 11:10	Ke Li	Operational interpretation of the sandwiched Rényi divergence of order $1/2$ to $1$ as strong converse exponents
Tea Break
11:25 ~ 12:05	Yongsheng Yao	Large deviation analyses for quantum channels
Lunch
14:00 ~ 14:50	Baojun Wu	Irreducibility of Virasoro representations in Liouville CFT
Tea Break
15:10 ~ 16:00	Jiawen Zhang	Quasi-local algebras for groupoids

Jan 17, 2024

09:30 ~ 10:20	Qianfeng Hu	Non-commutative operator-valued measures for Banach spaces, frames and related quantum problems
Tea Break
10:40 ~ 11:20	Hao Zhang	Conformal blocks and tensor categories of $C_2$-cofinite vertex operator algebras
Lunch
14:00 ~ 17:00		Free Discussion

Speaker:: Xuanzhong Dai (Kyoto University)
Title:: Vertex algebras arising from congruence subgroups
Abstract:: Chiral de Rham complex constructed by Malikov et al. in 1998, is a sheaf of vertex operator algebras on any complex manifold or nonsingular algebraic variety $X$. Let $G$ be a connected compact semisimple Lie group and $K$ be a closed subgroup of $G$. It is natural to consider the case $X = Γ\G/K$, where $G/K$ is a Hermitian symmetric space and Γ is an arithmetic subgroup of G. Particular interest would be given to the case $SL(2,R)/SO(2)$,as modular forms appears on the global section of chiral de Rham complex. For any congruence subgroup $\Gamma$, we will see that the vertex operations of $\Gamma$-invariant global sections can be expressed by the Rankin-Cohen brackets. A slight modification would also be implemented to derive a family of simple vertex operator algebras from congruence subgroups by relaxing cuspidal conditions.

Speaker:: Guixiang Hong (Harbin Institute of Technology)
Title:: Pointwise convergence of noncommutative Fourier series
Abstract:: The study of convergence of Fourier series goes back to the very beginning of Fourier analysis around one hundred years ago, and it, in particular the Bochner-Riesz mean conjecture, has been still stimulating the development of harmonic analysis and beyond. But the noncommutative (or quantum) harmonic analysis was born just recently. In this talk, I shall present the first pointwise convergence of the Fourier series affiliated to von Neumann algebras generated by non-Abelian groups. Without doubt, this induces many problems. This is based on a joint work with Simeng Wang and Xumin Wang.

Speaker:: Qianfeng Hu (Hebei University of Technology)
Title:: Non-commutative operator-valued measures for Banach spaces, frames and related quantum problems
Abstract:: The quantum measure theory has its origin in mathematical formalism of quantum mechanics and is often viewed as non-commutative or quantum analog of classical measure theory. The recently developed dilation theory for operator-valued measures (OVMs) leads to a Banach dilation theory for the operator-valued quantum measures (OVQMs) on projection lattices of von Neumann algebras (vN-algebras). In this talk, we present the countable-additive minimal dilation for OVQMs from finite vN-algebras without a Type $I_2$ direct summand to $B(X)$, where the Banach space $X$ is $l_p$ $(1< p< 2)$ or has the Schur property. Meanwhile, positive operator-valued measures (POVMs) are basic mathematical tool in quantum information theory (QIT), and have a natural connection with Hilbert space frame theory. Through POVM, we introduce this powerful tool, including continuous frames (also called coherent states) and also the multi-window Gabor frame, etc., into quantum state tomography, phase retrieval and quantum detection problems.

Speaker:: Yongle Jiang (Dalian University of Technology)
Title:: On invariant von Neumann subalgebras rigidity and related problems
Abstract:: Let $G$ be a countable discrete group. If every $G$-invariant von Neumann subalgebra $M$ in $L(G)$ is of the form $M=L(H)$ for some normal subgroup $H$ of $G$, then we call this group $G$ has the invariant von Neumann subalgebras rigidity (ISR) property. In this talk, we discuss the problem of determining which group $G$ has this ISR property and more generally, the problem of characterizing all $G$-invariant von Neumann subalgberas in $L(G)$. Part of this talk is based on joint work with Dr. Tattwamasi Amrutam.

Speaker:: Ke Li (Harbin Institute of Technology)
Title:: Operational interpretation of the sandwiched Rényi divergence of order $1/2$ to $1$ as strong converse exponents
Abstract:: We find the precise operational interpretation for the sandwiched Rényi divergence of order $\alpha \in (1/2,1)$, as well as its induced quantum information quantities, in characterizing the strong converse exponents of quantum tasks. The sandwiched Rényi divergence is well defined on $(1/2,1)∪(1,\infty)$ and the two intervals $(1/2,1)$ and $(1,\infty)$ are related by the Hölder-type duality. However, while the operational interpretation to the sandwiched Rényi divergence of order $\alpha \in (1,\infty)$ has been known much earlier, it had been open when $\alpha\in (1/2,1)$. Our result has filled this gap. In addition, I will also review and comment on the status of the quantum generalization of the Rényi divergence.

Speaker:: Fan Lu (Tsinghua University)
Title:: Classification of exchange relation planar algebras of rank 5
Abstract:: Exchange relation planar algebras are natural generalizations of irreducible depth-2 subfactor planar algebras, and the exchange relation is a kind of skein relation a biprojection must satisfy. I will introduce some combinatorial characterizations of the exchange relation, which help us rule out a large portion of possible candidates. By this method, we classify exchange relation planar algebras with 5-dimensional 2-boxes. They extend the classification of singly generated planar algebras obtained by Bisch, Jones, and Zhengwei Liu. This is a joint work with Zhengwei Liu.

Speaker:: Robert McRae (Tsinghua University)
Title:: Tensor categories from vertex operator algebras
Abstract:: This will be an overview talk on braided tensor categories obtained from representation categories of vertex operator algebras. I will describe how the tensor category structure is defined and also discuss what is currently known about the existence and further properties (especially rigidity) of this tensor category structure

Speaker:: Rui Shi (Dalian University of Technology)
Title:: Reducible operators and Gamma Property in von Neumann algebras
Abstract:: In this talk, we recall several equivalent characterizations about Property Gamma in type II_1 factors. Then we introduce some recent developments of Property Gamma. In particular, we apply Property Gamma to consider density of reducible operators in type II$_1$ factors.

Speaker:: Qing Wang (Xiamen University)
Title:: Orbifold theory of the parafermion vertex operator algebra $K(osp(1|2),k)$
Abstract:: In this talk, we present our recent progress about the orbifold theory of parafermion vertex operator algebras $K(osp(1|2),k)$ associated to the affine vertex operator superalgebra $L_{\widehat{osp(1|2)}}(k,0)$ with any positive integer $k$. In particular, we classify the irreducible modules for the orbifold of the parafermion vertex operator algebra $K(osp(1|2),k)$.

Speaker:: Baojun Wu (Peking University)
Title:: Irreducibility of Virasoro representations in Liouville CFT
Abstract:: In this talk, I will review the recent progress of Liouville conformal field theory from the path integral point of view developed by Vargas, Rhodes, Guillarmou and Kupiainen: including the structure constant (The DOZZ formula), the Segal axiom, also its representation theory. An initial assumption in the physics treatment of Liouville theory, proposed by Zamolodchikov brothers’, is the highest weight representation is always irreducible. We construct the Hilbert space of Liouville theory and prove this conjecture. This is based on joint work with Guillaume Baverez.

Speaker:: Yuanyuan Xu (Academy of Mathematics and Systems Science, CAS)
Title:: Universality of spectral radius of a large non-Hermitian random matrix
Abstract:: We will report on recent progress regarding the universality of the extreme eigenvalues of a large random matrix with i.i.d. entries. Beyond the radius of the celebrated circular law, we will establish a precise three-term asymptotic expansion for the largest eigenvalue (in modulus) with an optimal error term. Based on this result, we will further show that the properly normalized largest eigenvalue converges to a Gumbel distribution as the dimension goes to infinity. Similar results also apply to the rightmost eigenvalue of the matrix. These results are based on several joint works with Giorgio Cipolloni, Laszlo Erdos, and Dominik Schroder.

Speaker:: Jinwei Yang (Shanghai Jiao Tong University)
Title:: Ribbon categories for the singlet algebras and their extensions
Abstract:: In this talk, we summarize our recent work on the tensor categories for the singlet algebras, including the tensor structure on the category of the atypical modules, as well as on the full category of $C_1$-cofinite modules. We will also apply these results to study representation theory of vertex algebras that are extensions of the singlet algebras, especially the $B_p$-algebra and super $B_p$-algebra. This talk is based on a series of joint work with T. Creutzig and R. McRae.

Speaker:

Yongsheng Yao (Harbin Institute of Technology)

Title:

Large deviation analyses for quantum channels

Abstract:

Recently, there has been increasing interest in the studying of the exponential behavior of errors in quantum Shannon theory. Here we are concerned with the exact error exponents of quantum channels when unlimited shared entanglement is available. Our main results areas follows:

(a)We determine the exact strong converse exponent for entanglement-assisted communication. This also enables us to obtain the exact strong converse exponent for quantum-feedback-assisted communication.

(b)We address the reverse Shannon simulation of quantum channels with classical communication and unlimited free entanglement. We have derived the exact formula for the direct error exponent (reliability function) when the simulation cost is below a critical value. For high-rate case, we provide meaningful upper and lower bounds.

The above results are given in terms of the sandwiched Rényi divergence $D_{\alpha} (\rho \| \sigma) =\frac{\log Tr(\sigma^{(1-\alpha)/2\alpha}\rho \sigma^{(1-\alpha)/2\alpha} )^\alpha } {\alpha-1}$ defined for quantum states $\rho$, $\sigma$ and $\alpha > 0$. More specifically, the channel’s sandwiched Rényi mutual information defined as follows, will play a crucial role:

$I_{\alpha} (\mathcal{N}):=\max_{\varphi_{RA}} ⁡I_{\alpha} (R:B)_{\mathcal{N}(\varphi_{RA})}$

where the maximization is over all pure state $\varphi\_{RA}$, and $I_{\alpha}:=\min_{\sigma_B }⁡D_{\alpha} (\mathcal{N}(\varphi_{RA} )\| \varphi_R\otimes \sigma_B)$ is the sandwiched Rényi mutual information of a bipartite state.

Speaker:: Shilin Yu (Xiameng University)
Title:: Quantization of coadjoint orbits
Abstract:: The coadjoint orbit method of Kirillov and Kostant suggests that irreducible unitary representations of a Lie group can be constructed as quantization of coadjoint orbits of the group. Later Vogan reformulated the orbit method in algebro-geometric language for noncompact reductive Lie groups. Namely, it is conjectured that some Harish-Chandra (HC) modules can be attached to vector bundles on nilpotent orbits under certain assumptions. I will sketch the construction of these HC modules via deformation quantization. This is based on joint work with Conan Leung and a preprint coauthored with Ivan Losev.

Speaker:: Qiang Zeng (Academy of Mathematics and Systems Science, CAS)
Title:: Hessian spectrum at the global minimum of locally isotropic Gaussian random fields
Abstract:: Locally isotropic Gaussian random fields were first introduced by Kolmogorov in 1941. Such models were used to describe various phenomena in statistical physics. In particular, they were introduced to model a single particle in a random potential by Engel, Mezard and Parisi in 1990s. Using Parisi’s award winning replica trick, Fyodorov and Le Doussal predicted the high dimensional limit of the Hessian spectrum at the global minimum of these models, and discovered phase transitions according to different levels of replica symmetry breaking. In this talk, I will present a solution to their conjecture in the so called replica symmetric regime. Our method is based on landscape complexity, or counting the number of critical points of the Hamiltonian.

Speaker:: Jiawen Zhang (Fudan University)
Title:: Quasi-local algebras for groupoids
Abstract:: The notion of quasi-locality was originally introduced by Roe to describe operators in (uniform) Roe algebras, and it attracted more and more attention recently. In this talk, I will introduce a generalisation of quasi-locality to the theory of groupoids, based on a joint work with Baojie Jiang and Jianguo Zhang. We show that when the groupoid is etale and amenable, then the quasi-local algebra coincides with the reduced groupoid C$^*$-algebra. This unifies the classic case of groups and metric spaces, and also gives new results for group actions and groupoid uniform Roe algebras.

Speaker:

Hao Zhang (Tsinghua University)

Title:

Conformal blocks and tensor categories of $C_2$-cofinite vertex operator algebras

Abstract:

Vertex operator algebras (VOAs) mathematically characterize 2d conformal field theories. For $C_2$-cofinite and rational VOAs (plus some minor assumptions), Yi-Zhi Huang proved that their representation categories are rigid modular tensor categories. For $C_2$-cofinite (not necessarily) VOAs, Huang-Lepowsky-Zhang proved that their representation categories are braided tensor categories, which mean that the rigidity and modularity are still open problems. Geometrically, it is mainly because for $C_2$-cofinite VOAs, their conformal blocks associated to compact Riemann surfaces of genera 1 are not clear yet.

In this talk, I will introduce a systematic approach towards higher genus conformal blocks for $C_2$-cofinite VOAs and explain how our approach generalizes Huang-Lepowsky-Zhang’s tensor category theory and higher level Zhu’s algebras. This is based on an ongoing project (arXiv: 2305.10180) joint with Bin Gui.

Speaker:: Haonan Zhang (University of South Carolina)
Title:: An inequality in Fourier analysis and its application to learning theory: classical and quantum
Abstract:: A fundamental problem from computational learning theory is to well-reconstruct an unknown function on the discrete hypercubes. One classical result of this problem for the random query model is the low-degree algorithm of Linial, Mansour, and Nisan in 1993. This method saw exponential improvement in 2022 by Eskenazis and Ivanisvili, who utilized a family of polynomial inequalities dating back to Littlewood’s work in 1930. More recently, quantum analogs of these polynomial inequalities were conjectured by Rouzé, Wirth, and Zhang (2022). This conjecture was subsequently resolved by Huang, Chen, and Preskill (2022), discovered during their study of learning problems in quantum dynamics. In this talk, I will discuss a reduction method that provides a simpler proof of this conjecture with better estimates. As an application, it allows us to recover the low-degree algorithm of Eskenazis and Ivanisvili in the quantum setting. Further exploration into qudit observables requires similar Fourier analysis inequalities on cyclic groups of general orders. Time permitting, we will discuss the challenges there and explain how to solve the problem using new ideas that lead to unexpected dimension-free Remez-type inequalities. This is based on joint work with Alexander Volberg (MSU) and Joseph Slote (Caltech).

Speaker:: Zishuo Zhao (Tsinghua University)
Title:: Relative entropy for quantum channels
Abstract:: We introduce an quantum entropy for bimodule quantum channels on finite von Neumann algebras, generalizing the remarkable Pimsner-Popa entropy. we derive an upper bound for this quantity as Araki’s relative entropy for quantum channels and relate it to the relative entropy for Fourier multipliers. Notably, the quantum entropy attains its maximum if there is a downward Jones basic construction. Applications to sandwiched Renyi relative entropies will be given.

Speaker:: Sen Zhu (Jilin University)
Title:: Irreducible approximation of Hilbert space operators
Abstract:: A reducing subspace for an operator $T$ on a complex Hilbert space $H$ is a closed subspace $M$ of $H$ such that $M$ is invariant under both $T$ and $T^*$. $T$ is said to be irreducible if it has no reducing subspaces other than $\{0\}$ and $H$. Halmos proved in 1968 that those irreducible operators on a complex separable Hilbert space $H$ constitute a dense subset of the algebra $B(H)$ of all bounded linear operators on $H$. We extend Halmos’ approximation result to certain special operator classes, including Toeplitz operators with continuous symbols, complex symmetric operators, skew symmetric operators and their finite-dimensional analogues.

Attachments: