Workshop on QUANTUM INFORMATION PROCESSING AND QUANTUM COMMUNICATIONS

Quantum Information Processing and Quantum Communications
Extremal states of coupled finitelevel
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 (196263) 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.


Abstract 


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 ndimensional 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. 

