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Linear Algebra and its Applications

Subspace controllability of bipartite symmetric spin networks

Francesca Albertini, Domenico D’Alessandro

Published Sep 30, 2019• 4 citations
Publisher
DOI
10.1016/j.laa.2019.09.034

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We consider a class of spin networks where each spin in a certain set interacts, via Ising coupling, with a set of central spins, and the control acts simultaneously on all the spins. This is a common situation for instance in NV centers in diamonds, and we focus on the physical case of up to two central spins. Due to the permutation symmetries of the network, the system is not globally controllable but it displays invariant subspaces of the underlying Hilbert space. The system is said to be subspace controllable if it is controllable on each of these subspaces. We characterize the given invariant subspaces and the dynamical Lie algebra of this class of systems and prove subspace controllability in every case.
Controlling the dynamics of quantum systems has been a long-standing goal of quantum physics. To understand the controllability of a finite-dimensional quantum system is a fundamental problem in quantum control theory, but can be very challenging in many cases. In particular, results on controllability of open-loop coherently controlled closed quantum systems under symmetry constraints have been elusive. In this paper, we study the controllability of spin systems with some specific symmetry where spins are coupled via Ising interaction and manipulated collectively by a global control field. Based on Lie group and Lie algebra techniques, we find the conditions under which the system is universally controllable in its irreducible subspaces. We believe that the results obtained here will contribute to quantum controlling on symmetric quantum systems.
We propose a technique to investigate multipartite entanglement in the symmetric subspace. Our approach is to map an N -qubit symmetric state onto a bipartite symmetric state of higher local dimension. We show that this mapping preserves separability and allows to characterize the entanglement of the original multipartite state. In particular, we establish a connection between the border rank and the Schmidt rank, and derive lower bounds on entanglement measures. Finally, we reveal the existence of entangled symmetric subspaces, where all bipartite states are entangled.