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.
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.
In this paper, we present a controllability analysis of the quantum Ising periodic chain of n spin 1/2 particles where the interpolating parameter between the two Hamiltonians plays the role of the control. A fundamental result in the control theory of quantum systems states that the set of achievable evolutions is (dense in) the Lie group corresponding to the Lie algebra generated by the Hamiltonians of the system. Such a dynamical Lie algebra therefore characterizes all the state transitions available for a given system. For the Ising spin periodic chain we characterize such a dynamical Lie algebra and therefore the set of all reachable states. In particular, we prove that the dynamical Lie algebra is a (3n-1)-dimensional Lie sub-algebra of su(2^n) which is a direct sum of a two dimensional center and a (3n-3)-dimensional semisimple Lie subalgebra. This in turn is the direct sum of n-1 Lie algebras isomorphic to su(2) parametrized by the eigenvalues of a fixed matrix. We display the basis for each of these Lie subalgebras. Therefore the problem of control for the Ising spin periodic chain is, modulo the two dimensional center, a problem of simultaneous control of n-1 spin 1/2 particles. In the process of proving this result, we develop some tools which are of general interest for the controllability analysis of quantum systems with symmetry.
Much is understood about 1-dimensional spin chains in terms of entanglement properties, physical phases, and integrability. However, the Lie algebraic properties of the Hamiltonians describing these systems remain largely unexplored. In this work, we provide a classification of all Lie algebras generated by the terms of 2-local spin chain Hamiltonians, or so-called dynamical Lie algebras, on 1-dimensional linear and circular lattice structures. We find 17 unique dynamical Lie algebras. Our classification includes some well-known models such as the transverse-field Ising model and the Heisenberg chain, and we also find more exotic classes of Hamiltonians that appear new. In addition to the closed and open spin chains, we consider systems with a fully connected topology, which may be relevant for quantum machine learning approaches. We discuss the practical implications of our work in the context of variational quantum computing, quantum control and the spin chain literature.
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.
Controllability of dynamical systems is a fundamental problem for both classical control as well as quantum control.This paper focuses on investigating the controllability Lie algebra of single-spin and two spin quantum systems.Firstly,based on the equivalent controllability conditions of quantum systems,some general necessary and sufficient conditions related to the basis coefficients are obtained by computing Lie brackets of Hamiltonian operators for single-spin quantum systems.Secondly,the matrix basis of Lie algebra su(4) is generated by Cartan decomposition.Using the matrix basis and employing the controllability theorem of quantum system,a calculation method of multiple Lie brackets for the Hamiltonian operators and a controllability criterion of quantum systems are also put forward in this article.
Spin chains are correlated quantum models of great interest in quantum systems and materials exhibiting quasi-one-dimensional magnetic properties. Here, we review results on quantum problems associated with spin chains that are beyond the usual spinon paradigm. Alternatively, we use a representation valid in the thermodynamic limit, N→∞, in terms of the N spin-1/2 physical spins of the spin-1/2XXZ chain in its whole Hilbert space. It was originally introduced for the isotropic point in Carmelo et al. [Phys. Rev. B 92, 165133 (2015)], co-authored by David, and more recently extended to spin anisotropies Δ>1 in Carmelo et al. [Phys. Rev. Res. 5, 043058 (2023)] and J. M. P. Carmelo and P. D. Sacramento [Nucl. Phys. B 974, 115610 (2022); Nucl. Phys. B 997, 116385 (2023) (Corrigendum)]. The physical-spins representation accounts for the spin-1/2XXZ chain's continuous SUq(2) symmetry parameterized by q=Δ+Δ2-1∈]1,∞] and associated with q-spin Sq. Specifically, in this review we consider two quantum problems that are beyond the spinon representation: (a) Spin Bethe strings of length n that have no spinon representation, contribute to the dynamical properties of the spin-1/2XXZ chain with anisotropy Δ>1 and for n=1,2,3 were experimentally identified and realized in the zigzag materials SrCo2V2O8 and BaCo2V2O8; (b) The spin stiffness associated with ballistic spin transport at arbitrary finite temperature, which involves a huge number of energy eigenstates, many of which are generated in the thermodynamic limit from ground states by an infinite number of elementary processes. As found in Carmelo et al. [Phys. Rev. Res. 5, 043058 (2023)] and J. M. P. Carmelo and P. D. Sacramento [Nucl. Phys. B 974, 115610 (2022); Nucl. Phys. B 997, 116385 (2023) (Corrigendum)], the use of the continuous SUq(2) symmetry reveals that for anisotropy Δ>1 the Bethe strings of length n=1,2,3,… describe a number n of physical-spins Sq=0 singlet pairs that for n>1 are bound within a Sq=0 singlet configuration. Their contribution to the spin dynamical structure factor of both the spin-1/2XXZ chain in a longitudinal magnetic field and the spin chains in SrCo2V2O8 is one of the issues addressed in this paper. In addition, the SUq(2) symmetry imposes that only 2Sq out of the N physical spins are the spin carriers. We also review recent results of J. M. P. Carmelo and P. D. Sacramento ["Diffusive spin transport of the spin-1/2 XXZ chain in the Ising regime at zero magnetic field and finite temperature," (submitted) (2024)] concerning the vanishing of the contributions to finite-temperature ballistic spin transport at zero magnetic field. Within the physical-spins representation, this merely follows from the absolute value of the elementary spin currents carried by the M=2Sq spin carriers of all finite-Sq states that contribute to the spin stiffness being finite. Finally, we discuss deviations of the zigzag materials BaCo2V2O8 and SrCo2V2O8 from the one-dimensional physics described the spin-1/2XXZ chain due to selective interchain couplings.
Simple physical interactions between spin-1∕2 particles may result in quantum states that exhibit exotic correlations that are difficult to find if one simply explores state spaces of multipartite systems. In particular, we present a detailed investigation of the well-known Ising model of a chain (ring) of spin-1∕2 particles (qubits) in a transverse magnetic field. We present explicit expressions for eigenstates of the model Hamiltonian for arbitrary number of spin-1∕2 particles in the chain in the standard (computer) basis, and we investigate quantum entanglement between individual qubits. We analyze bipartite as well as multipartite entanglement in the ground state of the model. In particular, we show that bipartite entanglement between pairs of qubits of the Ising chain (measured in terms of a concurrence) as a function of the parameter $\ensuremath{\lambda}$ has a maximum around the point $\ensuremath{\lambda}=1$, and it monotonically decreases for large values of $\ensuremath{\lambda}$. We prove that in the limit $\ensuremath{\lambda}\ensuremath{\rightarrow}\ensuremath{\infty}$ this state is locally unitary equivalent to an N-partite Greenberger-Horn-Zeilinger state. We also analyze a very specific eigenstate of the Ising Hamiltonian with a zero eigenenergy (we denote this eigenstate as the X-state). This X-state exhibits the ``extreme'' entanglement in a sense that an arbitrary subset A of $k\ensuremath{\leqslant}n$ qubits in the Ising chain composed of N=2n+1 qubits is maximally entangled with the remaining qubits (set B) in the chain. In addition, we prove that by performing a local operation just on the subset B, one can transform the X-state into a direct product of k singlets shared by the parties A and B. This property of the X-state can be utilized for new secure multipartite communication protocols.