Abstract The Ising spin model is a fundamental theoretical model in statistical physics which has received large attention in the context of quantum phase transitions (Sachdev S 2011 Quantum Phase Transitions (Cambridge University) Press) and, more recently, in quantum information (King et al 2022 Nat. Phys. 18 1324–8). In this paper, we present a controllability analysis of the quantum Ising periodic chain (1-dimensional) of n spin 1 2 particles (Pfeuty 1970 Ann. Phys 57 79–90) where the interpolating parameter between the two Hamiltonians plays the role of the control. The controllability is characterized by the dynamical Lie algebra generated by the two Hamiltonians (D’Alessandro 2021 Introduction to Quantum Control and Dynamics (CRC Press Taylor and Francis)). It is known (see, e.g. (Minami 2021 Nucl. Phys. B 973 115599; Roan 1991 Max-Plank-. Math. ; Von Gehlen and Rittenberg 1985 Nucl. Phys. B 257 351–70) that such a Lie algebra is an Onsager algebra . We present an alternative derivation of this result which is based on the algebra of Pauli matrices and Pauli strings . This derivation displays the explicit basis of the Lie algebra and of its simple ideals and center in the relevant representation of the space of n qubits ( C 2 ) ⊗ n . We discuss how these results match the ones of the seminal paper by Onsager (Onsager 1944 Phys. Rev. 65 3–4) which were obtained in the context of classical statistical mechanics. Applications to quantum information and control are also discussed.
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
Sn. Because of the symmetry, the underlying Hilbert space,
H=(Cd)⊗n, splits into invariant subspaces for the Lie algebra of
Sn-invariant elements in
u(dn), denoted here by
uSn(dn). The dynamical Lie algebra
L, which determines the controllability properties of the system, is a Lie subalgebra of such a Lie algebra
uSn(dn). If
L acts as
su(dim(V)) 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
H coincides with the {\it Clebsch-Gordan} splitting of
(Cd)⊗n into {\it irreducible representations} of
su(d). In this view,
uSn(dn), is the direct sum of certain
su(nj) for some
nj'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}.
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 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.
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.
Determining the unitary dynamics accessible from finite Hamiltonian resources is a central problem in Hamiltonian engineering and quantum control. Dynamical Lie algebras (DLAs) connect available control Hamiltonians with the reachable dynamics, but their use as a design tool for modifying Hamiltonian generator sets remains less developed. In this work, we develop a finite-dimensional DLA framework for three generator-set operations: composition, invariance, and reduction. For composition, we construct direct sums of component DLAs using spectral projectors on an auxiliary register. For invariance, we analyze when modifications of Pauli-string generating sets preserve the generated Lie algebra, and introduce algebraic diagnostics for added generators. For reduction, we consider compact reductive DLAs and show how projection onto selected simple ideals gives reduced generating sets whose Lie closures are the corresponding ideal sums. We illustrate these results with finite-dimensional examples and numerical checks, including direct-sum dimension addition, central-spin invariance diagnostics, and DLA-based ansatz reduction for block-local Hamiltonians. The results show how DLA structure can be used to diagnose controllability and guide Hamiltonian generator design under explicit algebraic assumptions.