Research at FAST

100 Department of Applied Mathematics Department of Applied Mathematics Qualification BSc (HKU) MPhil (HKU) PhD (HKU) ORCID ID 0000-0003-1567-2654 Higher Rank Numerical Range The higher rank numerical range (also called rank-k numerical range) is closely connected to the construction of quantum error correction code for a noisy quantum channel. Mathematically, the higher rank numerical range is a generalization of the classical numerical range. The numerical range of an operator is the set of quadratic forms on unit vectors determined by the operator. The theory of quadratic forms appears in many parts of mathematics and science, and has a long and distinguished history. Researchers are interested in the algebraic and geometric properties of generalized numerical ranges and their applications to many branches of applied mathematics and other fields of science and engineering. • Lau, C.K Li, Poon, Sze , J. Funct. Anal., 275:2497-2515, 2018 • Gau, Li, Poon, Sze , SIAM J. Matrix Analysis Appl, 32:23-43, 2011 • Li, Sze , Proc. Amer. Math. Soc., 136: 3013-3023, 2008 Dr SZE Nung Sing Raymond Associate Professor Research Overview Dr Sze’s current research interests are in the area of quantum information science, specifically the related mathematical problems in matrix and operator theory. Research topics involve quantum error correction and generalized numerical ranges. Other topics of interest include sensitivity analysis in nonnegative matrix theory and preserver problems. Award • Journal of Mathematical Analysis and Applications (JMAA) Ames Award, 2014 Quantum Error Correction The idea of quantum error correction is to protect quantum information from errors due to decoherence and other quantum noise during the transmission of information in quantum channels. One fundamental question of quantum error correction is the existence of quantum error correcting code for a noisy quantum system. In this study, some practical and effective implementation schemes including circuit diagrams are proposed which provide insight into new correcting codes and improve the design of existing codes. • Li, Nakahara, Poon, Sze , Quantum Inf. Process, 14:4039-4055, 2015 • Gungordu, Li, Nakahara, Poon, Sze , Phys. Rev. A, 89:042301, 2014 • Li, Nakahara, Poon, Sze , Tomita, Quantum Inf Comput., 12:149-158, 2012 • Li, Nakahara, Poon, Sze , Tomita, Phys. Rev. A 84:044301, 2011 The rank-2 numerical ranges of a 5 x 5 normal matrix (left), a 9 x 9 normal matrix (middle), and a 5 x 5 non-normal matrix (right).