[qinfo] szeminarium

[Vrana Peter]

Kovetkezo eloadas:

2015. aprilis 9. 9:00-10:00
Zimboras Zoltan: Representation theory and its application to quantum control


In this talk we present our new representation theoretic results and show how 
these can be used in quantum control theory.

We first study how tensor products of representations decompose when 
restricted from a compact Lie algebra to one of its subalgebras. In 
particular, the focus will be on tensor squares which are tensor products of a 
representation with itself. We show in a classification-free manner that the 
sum of multiplicities and the sum of squares of multiplicities in the 
corresponding decomposition of a tensor square into irreducible 
representations has to strictly grow when restricted from a compact semisimple 
Lie algebra to a proper subalgebra. The sum of squares of multiplicities is 
equal to the dimension of the commutant of all complex matrices commuting with
the tensor square representation. Hence, our results offer a test if a 
subalgebra of a compact semisimple Lie algebra is a proper one which uses only 
linear-algebra computations on sets of generators without calculating the 
relevant Lie closures.

In the second part of the talk, we show how the previous test can be naturally 
applied in the field of controlled quantum systems. We provide complete 
symmetry criteria deciding whether some effective target interaction(s) can be 
simulated by a set of given interactions. Several physical examples are 
illustrated, including entanglement invariants, the relation to unitary gate 
membership problems as well as the central spin model. 


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