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Symmetries of the finite Heisenberg group for composite systems. (English) Zbl 1198.81062
Summary: Symmetries of the finite Heisenberg group represent an important tool for the study of the deeper structure of finite-dimensional quantum mechanics. As it is well known, these symmetries are properly expressed in terms of a certain normalizer. This paper extends previous investigations to composite quantum systems consisting of two subsystems – qudits – with arbitrary dimensions $n$ and $m$. In this paper, we present detailed descriptions – in the group of inner automorphisms of $\text{GL}\left(nm,ℂ\right)$ – of the normalizer of the Abelian subgroup generated by tensor products of generalized Pauli matrices of orders $n$ and $m$. The symmetry group is then given by the quotient group of the normalizer.
##### MSC:
 81P45 Quantum information, communication, networks 81Q05 Closed and approximate solutions to quantum-mechanical equations 81R05 Representations of finite-dimensional groups and algebras in quantum theory 81R15 Operator algebra methods (quantum theory)