Quantum Information Processing and Quantum Communications

Extremal states of coupled finite-level quantum systems
Aula 326, Lunedì 9 febbraio ore 11.00
  K.R. PARTHASARATHY, Indian Statistical Institute of New Delhi

>VITAE: K.R.Parthasarathy is currently Professor Emeritus at the Indian Statistical Institute of New Delhi, and previously Visiting Fellow at USSR Acad.Sci., Steklov Mathematical Institute, Moscow (1962-63) and professor at the Universities of Manchester, Bombay and finally Professor Distinguished Scientist at the Indian Statistical Institute,Delhi. Fellow of numerous Science academies, he is one of most recognized experts of the mathematical statistical structure of Quantum Mechanics. Author of several books, among the the most popular “An Introduction to Quantum Stochastic Calculus”, Birkhauser Verlag.

  It is a famous theorem of Birkhoff and von Neumann that in the convex set of all joint distributions on {1,2,...,n}x{1,2,...,n} whose both the marginal distributions are uniform the extreme points are uniform distributions with support in {(1,s(1)),(2,s(2),...,(n,s(n))} where s is a permutation of {1,2,...,n}.
Here we investigate the quantized version of this problem when the set {1,2,...,n} is replaced by an n-dimensional complex Hilbert space and distributions are replaced by states. We shall present a simple criterion for the extremality of a joint state when the marginals are fixed and use it to obtain an upper bound to the dimension of the support of any extremal state. In particular, when n=2, the extremal states are maximally entangled. We shall present some interesting examples of extremal states and also some open problems.