## Mini-Workshop on preserver problems and applications in Quantum information

Organizer: Jianlian Cui

July 14, 2013

Room A-304, Science Building

Schedule:

9.00-10.00am

Chi-kwong Li  (College of William and Mary, USA)

Title: Jordan products and preserver problems

AbstractWe discuss results and problems on Jordan products of operators, and related preserver problems.

10.00-11.00am

Nagi-ching Wong (National Sun Yat-sen University)

Title: Zero products and norm preserving orthogonally additive homogeneous polynomials on C*-algebras

Abstract: Let P:A\to B be a bounded orthogonally additive and zero product preserving $n$-homogeneous polynomial between C*-algebras. We show that, in the commutative case that A=C_0(X) and B=C_0(Y), there exist a bounded continuous function h in C(Y) and a map \varphi: Y\to X such that Pf=h\cdot (f\circ\varphi)^n. In the general case, we show that there is a central invertible multiplier h of B and a surjective Jordan homomorphism J: A\to B such that Pa = hJ(a)^n, provided that P(A)\supseteq B^+. Similar Banach-Stone type theorems also hold for orthogonally additive n-homogeneous polynomials which are n-isometries. We also discuss the structure of orthogonally additive and zero product preserving holomorphic functions on commutative C*-algebras.

11.00-12.00am

Raymond Sze (The Hong Kong Polytechnic University)

Title: Preserver problems arising in quantum information science

Abstract: The study of linear preserver problems has a long history in matrix and operator theory. It concerns the characterization of linear maps on matrices or operators with special properties. Recently, researchers are interested in linear preserver problems that are related to quantum information science. In this talk, some recent developments of this topic will be discussed.

2.30-3.30pm

Jinchuan Hou (Taiyuan University of Technology)

Title: Lie ring isomorphisms between nest algebras on Banach spaces

Abstract: Let N and M be nests on Banach spaces X and Y over the (real or complex) field F and let Alg N and Alg M be the associated nest algebras, respectively. It is shown that a map \Phi: Alg N\rightarrow Alg M is a Lie ring isomorphism (i.e., \Phi is additive, Lie multiplicative and bijective) if and only if \Phi has the form \Phi(A) = TAT^{-1} + h(A)I for all A in Alg N or \Phi(A)=-TA^*T^{-1}+h(A)I for all A in Alg N, where h is an additive functional vanishing on all commutators and T is an invertible bounded linear or conjugate linear operator when \dim X=\infty; T is a bijective \tau-linear transformation for some field automorphism \tau of F when \dim X<\infty.

3:30-4:00 pm, Tea Break

4.00-5.00pm

Yiu-tung Poon  (Iowa State University, USA)

Title: Linear maps preserving the higher numerical ranges of tensor product of matrices

Abstract: For a positive integer n, let M_n be the set of n \times n complex matrices. Suppose m,n\ge 2 are positive integers and k\in \{1,\ldots, mn-1\}. Denote by W_k(X) the k-numerical range of a matrix X\in M_{mn}. It is shown that a linear map \phi: M_{mn}\rightarrow M_{mn} satisfies W_k(\phi(A\otimes B)) = W_k(A\otimes B) for all A \in M_m and B \in M_n if and only if there is a unitary U in M_{mn} such that one of the following holds. For all A\in M_m, B\in M_n, \phi(A\otimes B)=U(\varphi(A\otimes B))U^*. mn = 2k and for all A\in M_m, B\in M_n, \phi(A\otimes B)=(\tr(A\otimes  B)/k)I_{mn}-U(\varphi(A\otimes B))U^*, where (1) \varphi is the identity map A \otimes B \mapsto A\otimes B or the transposition map A\otimes B \mapsto (A\otimes B)^t, or (2) \min\{m,n\} \le 2 and \varphi has the form A \otimes B \mapsto A \otimes B^t or A \otimes B \mapsto A^t \otimes B.

5.00-6.00pm

Yu Guo (Taiyuan University of Technology)

Title: Concurrence for infinite-dimensional quantum systems

Abstract: Concurrence is an important entanglement measure for states in finite-dimensional quantum systems that was explored intensively in the last decade. In this paper, we extend the concept of concurrence to infinite-dimensional bipartite systems and show that it is continuous and does not increase under local operation and classical communication (LOCC).

You are cordially invited to participate. Thank you very much for your attention.