The structure of states and maps in quantum theory

SUDHAVATHANI SIMON¹, S P RAJAGOPALAN² and R SIMON^3,*

¹Department of Computer Science, Women's Christian College, Chennai 600 006, India

²Mohamed Sathak A.J. College of Engineering, Old Mahabalipuram Road,

Egattur 603 103, India

³The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600 113, India

^*Corresponding author. E-mail: simon@imsc.res.in

Abstract. The structure of statistical state spaces in the classical and quantum

theories are compared in an interesting and novel manner. Quantum state

spaces and maps on them have rich convex structures arising from the superposition

principle and consequent entanglement. Communication channels (physical

processes) in the quantum scheme of things are in one-to-one

correspondence with completely positive maps. Positive maps which are not

completely positive do not correspond to physical processes. Nevertheless

they prove to be invaluable mathematical tools in establishing or

witnessing entanglement of mixed states. We consider some of the recent

developments in our understanding of the convex structure of

states and maps in quantum theory, particularly in the context of

quantum information theory.

Keywords. Positive maps; completely positive maps; indecomposable maps; entanglement witness.

PACS Nos 03.65.Ud; 03.67.-a; 89.70.+c