Workshop on Quantum Analysis



August 17 - 22, 2020                   Alpensia Resort, Pyeongchang

Title/Abstract Home > Title/Abstract


계승혁 (서울대학교)

- Title: Positive maps in the theory of quantum information theory

- Abstract: Order structures have played crucial roles in the theory of operator algebras from the beginning of the theory, through Gelfand-Naimark-Segal construction in 1940s. Stinesprings(1955) and Stormer(1963) extended GNS construction to get the notions of completely positive maps and decomposable positive maps. These two notions of positivity now play important roles in the current quantum information theory, together with just positivity.

The main purpose of these talks is to introduce the above several kinds of positive maps together with central notions of quantum information theory, like separability/entanglement, Schmidt numbers, positive partial transpose. The main tool is the duality between mapping spaces and tensor products between matrix algebras. Especially, we introduce the notion of one-sided mapping cones with which we recover known characterizations of the above notions in a single frame.

We exhibit several kinds of nontrivial positive maps and PPT states in the relation of PPT square conjecture, which seems to be one of the hot research topics in the crossway between operator algebras and quantum information theory. We close this talk to explain multi-partite analogues of the above notions. This talk will have four sections with the following topics:

- positive linear maps in operator algebras
- entangled states in quantum information theory
- positive partial transpose square conjecture
- multi-partite analogues


박성철 (고등과학원)

- Title: Convergence of the Ising model transfer matrix on slit-strip geometry.

- Abstract: The transfer matrix and its diagonalisation have been central to the study of the planar Ising model on the square lattice, dating back to classical works of Onsager and Kaufman in the 1940s. We formulate the eigenstates and eigenvalues at criticality in terms of explicit functions, in terms of the recent discrete complex analytic approach pioneered by Mercat, Smirnov, and others. Consequently, we show that the fusion coefficients of the eigenstates in the slit-strip tend to the axiomatically derived vertex operator algebra structure coefficients. This talk is based on a joint work with T. Ameen, K. Kytölä, and D. Radnell.

- Title: Crossing estimates of the massive FK-Ising model.

- Abstract: Discrete complex analysis and its connection to the (FK-)Ising model on isoradial lattice (Mercat 2004, Chelkak-Smirnov 2012) have been instrumental in the rigorous verification of the conjecture that the critical planar Ising model is conformally invariant. We extend the discrete theory to a generalised analytic setting, where physical observables satisfy a discrete version of the Bers-Vekua system of equations. We prove convergence of the interface martingale and give estimates on crossing probabilities in the near-critical scaling limit of the FK-Ising model.


변성수 (서울대학교)

- Title: From conformal field theory to Schramm-Loewner evolution

- Abstract: Since Schramm introduced Schramm-Loewner evolution (SLE), the scaling limit of interface curve in planar critical models, SLE theory has successfully produced rigorous proofs of several remarkable conjectures in statistical physics. In this talk, I will introduce Gaussian free field based conformal field theory (CFT) and explain its implementations to construct martingale-observables for SLEs in simply/doubly connected domains. As an application, I will present the method of screening to derive explicit solutions to the parabolic partial differential equations for the annulus SLE partition functions.


이진엽 (고등과학원)

- Title: Time evolution of Bose-Einstein condensation in the mean-field limit
- Abstract: Bose-Einstein condensation (BEC) is one of the most famous phenomena, which cannot be explained by classical mechanics. Here, we discuss the time evolution of BEC in the mean-field limit. First, we review quantum mechanics briefly, and we understand the problem in a mathematically rigorous way. Then, we taste the idea of proof by using coherent state and the Fock space. Finally, some recent developments will be provided.


지운식 (충북대학교)

- Title: Implementation problems for CCRs and quantum white noise differential equations

- Abstract: We discuss the Fock representation of canonical commutation relation (CCR) and its transformation, and their implementation problems. From the implementation problems, we derive equivalent quantum white noise differential equations (QWNDEs), and then by solving the QWNDEs, we solve the implementation problems for CCRs.The implementation of CCR generalizes the Bogoliubov transformation. For our discussions, the Wick calculus for operators on Fock spaces plays an important role.


최진원 (숙명여자대학교)

- Title: Introduction to the positive partial transpose squared conjecture
- Abstract: The positive partial transpose (PPT) squared conjecture claims that the composition of any completely positive and completely copositive maps is entanglement breaking. We give an overview of the PPT squared conjecture.


한경훈 (수원대학교)

- Title: Separability criteria for multi-qubit X-states

- Abstract: The notion of entanglement is now considered as an indispensable resource in the current quantum physics and quantum information theory. In the multi-partite systems, there are various notions of separability/entanglement according to partitions of systems. A state is called an X-state if all the entries are zero except for diagonal and anti-diagonal entries. Multi-qubit X-states have a nice hereditary property: each X-parts of separable multi-qubit states are again separable. Thus, each separability criteria for X-states become entanglement criteria for general multi-qubit states.

Recently, several necessary and sufficient separability criteria for multi-qubit X-states have been found in the serial papers joint with Kye, Chen, Ha, Szalay. The criteria are simply algebraic, thus easy to check. Their proofs depend on the duality of convex cones, not explicit decomposition.