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- W4385262318 abstract "We describe a framework for the controllability analysis of networks of $n$ quantum systems of an arbitrary dimension $d$, {it qudits}, with dynamics determined by Hamiltonians that are invariant under the permutation group $S_n$. Because of the symmetry, the underlying Hilbert space, ${cal H}=(mathbb{C}^d)^{otimes n}$, splits into invariant subspaces for the Lie algebra of $S_n$-invariant elements in $u(d^n)$, denoted here by $u^{S_n}(d^n)$. The dynamical Lie algebra ${cal L}$, which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra $u^{S_n}(d^n)$. If ${cal L}$ acts as $suleft( dim(V) right)$ on each of the invariant subspaces $V$, the system is called {it subspace controllable}. Our approach is based on recognizing that such a splitting of the Hilbert space ${cal H}$ coincides with the {it Clebsch-Gordan} splitting of $(mathbb{C}^d)^{otimes n}$ into {it irreducible representations} of $su(d)$. In this view, $u^{S_n}(d^n)$, is the direct sum of certain $su(n_j)$ for some $n_j$'s we shall specify, and its {it center} which is the Abelian (Lie) algebra generated by the {it Casimir operators}. Generalizing the situation previously considered in the literature, we consider dynamics with arbitrary local simultaneous control on the qudits and a symmetric two body interaction. Most of the results presented are for general $n$ and $d$ but we recast previous results on $n$ qubits in this new general framework and provide a complete treatment and proof of subspace controllability for the new case of $n=3$, $d=3$, that is, {it three qutrits}." @default.
- W4385262318 created "2023-07-26" @default.
- W4385262318 creator A5008152552 @default.
- W4385262318 date "2023-07-24" @default.
- W4385262318 modified "2023-10-18" @default.
- W4385262318 title "Subspace Controllability and Clebsch-Gordan Decomposition of Symmetric Quantum Networks" @default.
- W4385262318 doi "https://doi.org/10.48550/arxiv.2307.12908" @default.
- W4385262318 hasPublicationYear "2023" @default.
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