Partial Order on Classical and Quantum States

In this work we extend the work done by Bob Coecke and Keye Martin in their paper "Partial Order on Classical States and Quantum States (2003)". We review basic notions involving elementary domain theory, the set of probability measures on a finite set {a1, a2, ..., an}, which we identify with the standard (n-1)-simplex ?n and Shannon Entropy. We consider partial orders on ?n, which have the Entropy Reversal Property (ERP) : elements lower in the order have higher (Shannon) entropy or equivalently less information . The ERP property is important because of its applications in quantum information theory. We define a new partial order on ?n, called Stochastic Order , using the well-known concept of majorization order and show that it has the ERP property and is also a continuous domain. In contrast, the bayesian order on ?n defined by Coecke and Martin has the ERP property but is not continuous.

Autorentext

Arka Prava Bandyopadhyay, originally a student of Ramkrishna Mission, Narendrapur has an MS in Mathematics from India. He came to USA in 2010 and finished another MS in Computer science in just one year from Louisiana State University. Currently he is doing PhD in Mathematics in Courant Institute of Mathematical Sciences in New York University.