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SIAM Journal on Control and Optimization

Subspace Controllability and Clebsch–Gordan Decomposition of Symmetric Quantum Networks

Domenico D’Alessandro

Published Dec 13, 2024• 1 citation
Publisher

Abstract

Abstract. This paper describes a framework for the controllability analysis of networks of [Formula: see text] quantum systems of an arbitrary dimension [Formula: see text] , qudits, with dynamics determined by Hamiltonians that are invariant under the permutation group [Formula: see text]. Because of the symmetry, the underlying Hilbert space, [Formula: see text], splits into invariant subspaces for the Lie algebra of [Formula: see text]-invariant elements in [Formula: see text], denoted here by [Formula: see text]. The dynamical Lie algebra [Formula: see text], which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra [Formula: see text]. If [Formula: see text] acts as [Formula: see text] on each of the invariant subspaces [Formula: see text], the system is called subspace controllable. The described approach is based on recognizing that such a splitting of the Hilbert space [Formula: see text] coincides with the Clebsch–Gordan splitting of [Formula: see text] into irreducible representations of [Formula: see text]. In this view, [Formula: see text] is the direct sum of certain [Formula: see text] for some [Formula: see text]’s we shall specify and its center, which is the Abelian (Lie) algebra generated by the Casimir operators. Most of the results presented are for general [Formula: see text] and [Formula: see text], but the paper provides a complete treatment and proof of subspace controllability for the new case of a system with [Formula: see text], [Formula: see text], that is, three qutrits. The results are motivated by recent great interest in symmetric quantum states and systems (see, e.g., [K. Eckert et al., Ann. Phys., 299 (2001), pp. 88–127], [A. W. Harrow, “The Church of the Symmetric Subspace,” https://arxiv.org/abs/1308.6595 , 2013], [P. Migdal, J. Rodriguez-Laguna, and M. Lewenstein, Phys. Rev. A, 88 (2013), 021335]) both at the theoretical and experimental levels and by recent proposals in geometric quantum machine learning [M. Cerezo et al., Nature Comput. Sci., 2 (2022), pp. 567–576], [Q. T. Nguyen et al., PRX Quant., 5 (2024), 020328] to exploit symmetries in the data and in quantum circuits to improve the performance of learning protocols.

DOI
10.1137/23m1589128

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Abstract. This paper describes a framework for the controllability analysis of networks of [Formula: see text] quantum systems of an arbitrary dimension [Formula: see text] , qudits, with dynamics determined by Hamiltonians that are invariant under the permutation group [Formula: see text]. Because of the symmetry, the underlying Hilbert space, [Formula: see text], splits into invariant subspaces for the Lie algebra of [Formula: see text]-invariant elements in [Formula: see text], denoted here by [Formula: see text]. The dynamical Lie algebra [Formula: see text], which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra [Formula: see text]. If [Formula: see text] acts as [Formula: see text] on each of the invariant subspaces [Formula: see text], the system is called subspace controllable. The described approach is based on recognizing that such a splitting of the Hilbert space [Formula: see text] coincides with the Clebsch–Gordan splitting of [Formula: see text] into irreducible representations of [Formula: see text]. In this view, [Formula: see text] is the direct sum of certain [Formula: see text] for some [Formula: see text]’s we shall specify and its center, which is the Abelian (Lie) algebra generated by the Casimir operators. Most of the results presented are for general [Formula: see text] and [Formula: see text], but the paper provides a complete treatment and proof of subspace controllability for the new case of a system with [Formula: see text], [Formula: see text], that is, three qutrits. The results are motivated by recent great interest in symmetric quantum states and systems (see, e.g., [K. Eckert et al., Ann. Phys., 299 (2001), pp. 88–127], [A. W. Harrow, “The Church of the Symmetric Subspace,” https://arxiv.org/abs/1308.6595 , 2013], [P. Migdal, J. Rodriguez-Laguna, and M. Lewenstein, Phys. Rev. A, 88 (2013), 021335]) both at the theoretical and experimental levels and by recent proposals in geometric quantum machine learning [M. Cerezo et al., Nature Comput. Sci., 2 (2022), pp. 567–576], [Q. T. Nguyen et al., PRX Quant., 5 (2024), 020328] to exploit symmetries in the data and in quantum circuits to improve the performance of learning protocols.
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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.
In the analysis of controllability of finite dimensional quantum systems, subspace controllability refers to the situation where the underlying Hilbert space splits into the direct sum of invariant subspaces, and, on each of such invariant subspaces, it is possible to generate any arbitrary unitary operation using appropriate control functions. This is a typical situation in the presence of symmetries for the dynamics. We investigate whether and when if subspace controllability is verified, the addition of an extra Hamiltonian to the dynamics implies full controllability of the system. Under the natural (and necessary) condition that the new Hamiltonian connects all the invariant subspaces, we show that this is always the case, except for a very specific case we shall describe. Even in this specific case, a weaker notion of controllability, controllability of the state ( Pure State Controllability ) is verified.
We consider a class of spin networks where each spin in a certain set interacts via Ising coupling with a single central spin. This is a common situation for instance in NV centers in diamonds. 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.
We study, classify, and explore the mathematical properties of finite-dimensional Lie algebras occurring in the quantum dynamics of single-mode and self-interacting bosonic systems. These Lie algebras are contained in the real skew-hermitian Weyl algebra A^1\hat{A}_1A^1​, defined as the real subalgebra of the Weyl algebra A1A_1A1​ consisting of all skew-hermitian polynomials. A central aspect of our analysis is the choice of basis for A^1\hat{A}_1A^1​, which is composed of skew-symmetric combinations of two elements of the Weyl algebra called monomials, namely strings of creation and annihilation operators combined with their hermitian conjugate. Motivated by the quest for analytical solutions in quantum optimal control and dynamics, we aim at answering the following three fundamental questions: (i) What are the finite-dimensional Lie subalgebras in A^1\hat{A}_1A^1​ generated by monomials alone? (ii)~What are the finite-dimensional Lie subalgebras in A^1\hat{A}_1A^1​ that contain the free Hamiltonian? (iii) What are the non-abelian and finite-dimensional Lie subalgebras that can be faithfully realized in A^1\hat{A}_1A^1​? We answer the first question by providing all possible realizations of all finite-dimensional non-abelian Lie algebras that are generated by monomials alone. We answer the second question by proving that any non-abelian and finite-dimensional subalgebra of A^1\hat{A}_1A^1​ that contains a free Hamiltonian term must be a subalgebra of the Schrödinger algebra. We partially answer the third question by classifying all nilpotent and non-solvable Lie algebras that can be realized in A^1\hat{A}_1A^1​, and comment on the remaining cases. Finally, we also discuss the implications of our results for quantum control theory. Our work constitutes an important stepping stone to understanding quantum dynamics of bosonic systems in full generality.