Beijing Institute of Mathematical Sciences and Applications

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.

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 $r / R\rightarrow 0$ 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 $r / R \rightarrow 1$, 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.

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.

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.

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.

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.

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.

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

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 $U \oplus U \oplus \cdot\cdot\cdot \oplus U$. This is a joint work with Laurent Marcoux, Heydar Radjavi and Sascha Troscheit.

Let $\omega$ be a radial weight on the the unit ball BdBd 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.

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 $t \mapsto t^{1/p}, 0< t\le 1$, then there exists a noncommutative probability space $(N ,\sigma)$ such that $L_p (M)$ is isometric to a subspace of $E(N ,\sigma)$. In particular, this answers two questions raised by Randrianantoanina in 2006.

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.

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.

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.

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.

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.

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.

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.