Subspace Of States Research Articles

Rapid stabilization of stochastic quantum systems in a unified framework

Rapid stabilization of general stochastic quantum systems is investigated based on the rapid stability of stochastic differential equations. We introduce a Lyapunov–LaSalle-like theorem for a class of nonlinear stochastic systems first, based on which a unified framework of rapidly stabilizing stochastic quantum systems is proposed. According to the proposed unified framework, we design the switching state feedback controls to achieve the rapid stabilization of single-qubit systems, two-qubit systems, and N-qubit systems. From the unified framework, the state space is divided into two state subspaces, and the target state is located in one state subspace, while the other system equilibria are located in the other state subspace. Under the designed state feedback controls, the system state can only transit through the boundary between the two state subspaces no more than two times, and the target state is globally asymptotically stable in probability. In particular, the system state can converge exponentially in (all or part of) the state subspace where the target state is located. Moreover, the effectiveness and rapidity of the designed state feedback controls are shown in numerical simulations by stabilizing GHZ states for a three-qubit system.

Shallow water waves generated by a floating object: A control theoretical perspective

<p style='text-indent:20px;'>We consider a control system describing the interaction of water waves with a partially immersed rigid body constraint to move only in the vertical direction. The fluid is modeled by the shallow water equations. The control signal is a vertical force acting on the floating body. We first derive the full governing equations of the fluid-body system in a water tank and reformulate them as an initial boundary value problem of a first-order evolution system. We then linearize the equations around the equilibrium state and we study its well-posedness. Finally we focus on the reachability and stabilizability of the linear system. Our main result asserts that, provided that the floating body is situated in the middle of the tank, any symmetric waves with appropriate regularity can be obtained from the equilibrium state by an appropriate control force. This implies, in particular, that we can project this system on the subspace of states with appropriate symmetry properties to obtain a reduced system which is approximately controllable and strongly stabilizable. Note that, in general, this system is not controllable (even approximately).</p>

Non-local property of single-photon states: Illustration with spontaneously emitted photon from a Hydrogen atom

We discuss a new proof of the non-locality of single-photons using a concrete physical observable such as the local energy density. As an illustration of this non-locality, we compute the mean value of the local energy observable for the spontaneous emission from a Hydrogen atom. We find that, in the subspace of single-photon states, the mean value of the local energy density observable is non-zero everywhere and decreases as a power of the distance.

Complete hierarchy of linear systems for certifying quantum entanglement of subspaces

We introduce a hierarchy of linear systems for showing that a given subspace of pure quantum states is entangled (i.e., contains no product states). This hierarchy outperforms known methods already at the first level, and it is complete in the sense that every entangled subspace is shown to be so at some finite level of the hierarchy. It straightforwardly generalizes to the case of higher Schmidt rank, as well as the multipartite cases of completely and genuinely entangled subspaces. These hierarchies work extremely well in practice even in very large quantum systems, as they can be implemented via elementary linear algebra techniques rather than the semidefinite programming techniques that are required by previously known hierarchies.

Theory of triangulene two-dimensional crystals

Abstract Equilateral triangle-shaped graphene nanoislands with a lateral dimension of n benzene rings are known as [n] triangulenes. Individual [n] triangulenes are open-shell molecules, with single-particle electronic spectra that host n − 1 half-filled zero modes and a many-body ground state with spin S = ( n − 1 ) / 2 . The on-surface synthesis of triangulenes has been demonstrated for n = 3 , 4 , 5 , 7 and the observation of a Haldane symmetry-protected topological phase has been reported in chains of [3]triangulenes. Here, we provide a unified theory for the electronic properties of a family of two-dimensional honeycomb lattices whose unit cell contains a pair of triangulenes with dimensions n a , n b . Combining density functional theory and tight-binding calculations, we find a wealth of half-filled narrow bands, including a graphene-like spectrum (for n a = n b = 2 ), spin-1 Dirac electrons (for n a = 2 , n b = 3 ), p x , y -orbital physics (for n a = n b = 3 ), as well as a gapped system with flat valence and conduction bands (for n a = n b = 4 ). All these results are rationalized with a class of effective Hamiltonians acting on the subspace of the zero-energy states that generalize the graphene honeycomb model to the case of fermions with an internal pseudospin degree of freedom with C 3 symmetry.

Modelling of an interacting series process for rector cooling system using system identification method

This research presents the modelling of cooling system plant in the TRIGA PUSPATI reactor (RTP). The RTP cooling system involves the interacting series process which are reactor model and heat exchanger model. The behaviour process is complex due to the nonlinearity, uncertainty, and time variation of the system. Due to the nonlinearity and interacting series process of system, the numerical algorithm for subspace state space system identification (N4SID) was proposed in the modelling of RTP cooling system. However, the model was converted to the transfer function form. The output response from the simulation model of RTP cooling subsystems were analysed by comparing the model with the real operational data. The responses of the model were within the respective range of a good model data. The residual analysis for each output was also calculated. The comparison and residual analysis were done to validate the model accuracy to determine the behaviour of the model. Overall, the model accuracy assessment shows that all models are still fit in the good model region (<10%) and has a good representation of a real plant.

Model reduction of convection-dominated partial differential equations via optimization-based implicit feature tracking

This work introduces a new approach to reduce the computational cost of solving partial differential equations (PDEs) with convection-dominated solutions: model reduction with implicit feature tracking. Traditional model reduction techniques use an affine subspace to reduce the dimensionality of the solution manifold and, as a result, yield limited reduction and require extensive training due to the slowly decaying Kolmogorov n-width of convection-dominated problems. The proposed approach circumvents the slowly decaying n-width limitation by using a nonlinear approximation manifold systematically defined by composing a low-dimensional affine space with a space of bijections of the underlying domain. Central to the implicit feature tracking approach is a residual minimization problem over the reduced nonlinear manifold that simultaneously determines the reduced coordinates in the affine space and the domain mapping that minimize the residual of the unreduced PDE discretization. This is analogous to standard minimum-residual reduced-order models, except instead of only minimizing the residual over the affine subspace of PDE states, our method enriches the optimization space to also include admissible domain mappings. The nonlinear trial manifold is constructed by using the proposed residual minimization formulation to determine domain mappings that cause parametrized features to align in a reference domain for a set of training parameters. Because the feature is stationary in the reference domain, i.e., the convective nature of solution removed, the snapshots are effectively compressed to define an affine subspace. The space of domain mappings, originally constructed using high-order finite elements, is also compressed in a way that ensures the boundaries of the original domain are maintained. Several numerical experiments are provided, including transonic and supersonic, inviscid, compressible flows, to demonstrate the potential of the method to yield accurate approximations to convection-dominated problems with limited training.

Quantum phase measurement of two-qubit states in an open waveguide

We present a new method for quantum state tomography within a single-excitation subspace of two-qubit states in an open waveguide. The system under investigation consists of three qubits in an open waveguide, separated by a distance comparable to the wavelength of the electromagnetic field. We show that the modulation of the frequency of the central ancillary qubit allows us to obtain unambiguous information about the initial phase difference $\varphi_1-\varphi_3$ of the edge qubits via the measurement of the evolution of their probability amplitudes.

FROM DYSON–SCHWINGER EQUATIONS TO QUANTUM ENTANGLEMENT

We apply combinatorial Dyson–Schwinger equations and their Feynman graphon representations to study quantum entanglement in a gauge field theory $${\varPhi }$$ in terms of cut-distance regions of Feynman diagrams in the topological renormalization Hopf algebra $$H^{\text {cut}}_{\text {FG}}({\varPhi })$$ and lattices of intermediate structures. Feynman diagrams in $$H_{\text {FG}}({\varPhi })$$ are applied to describe states in $${\varPhi }$$ where we build the Fisher information metric on finite dimensional linear subspaces of states in terms of homomorphism densities of Feynman graphons which are continuous functionals on the topological space $$\mathcal {S}^{{\varPhi },M \subseteq [0,\infty )}_{\text {graphon}}([0,1])$$ . We associate Hopf subalgebras of $$H_{\text {FG}}({\varPhi })$$ generated by quantum motions to separated regions of space-time to address some new correlations. These correlations are encoded by assigning a statistical manifold to the space of 1PI Green’s functions of $${\varPhi }$$ . These correlations are applied to build lattices of Hopf subalgebras, Lie subgroups, and Tannakian subcategories, derived from towers of combinatorial Dyson–Schwinger equations, which contribute to separated but correlated cut-distance topological regions. This lattice setting is applied to formulate a new tower of renormalization groups which encodes quantum entanglement of space-time separated particles under different energy scales.

Advanced Constrained Model Predictive Control of Vapor Pressure Deficit in Agricultural Greenhouses

The agricultural greenhouse is an open system to the external environment. The challenge is to track the reference trajectory of relevant climatic parameters and regulate the internal climate, despite the strong meteorological disturbances. This study focuses on the modelling and control of Vapor Pressure Deficit (VPD) coupled with temperature and hygrometry in greenhouses. The importance of this parameter is related to plant growth and its sensitivity to external climate disturbances. The VPD is a primordial parameter for preventing plant diseases caused by an inappropriate interior climate. This work provides a methodology for efficient control of VPD with a Multi-setting parameterization of the controller according to the needs of the plant and the actuators’ power available in the greenhouse. For this purpose, the VPD is indirectly evaluated through the ambient air temperature and humidity using the psychometric chart embedded in a fuzzy inference algorithm. For dynamic modelling of VPD, a discrete state-space model is obtained by considering an agricultural greenhouse as a black-box system. Numerical algorithms for Subspace State Space System Identification (N4SID) of Matlab is applied to parametric identification based on experimental measurements. A Constrained Model Predictive Control (CMPC) method is applied for VPD control design, considering physical and operational constraints on Input/output variables. Here, the CMPC strategy is improved by information on the greenhouse’s outside climatic conditions. Promising numerical simulation results show the feasibility of the proposed modelling and control strategy. They show good performances, despite the climatic exchange phenomenon between inside and outside the greenhouse and strong external weather disturbances. The inside VPD tracks nearly reference trajectories with a small overshooting as desired.

Coherence limitations in the optical control of the singlet-triplet qubit in a quantum dot molecule

We analyze the optically driven dynamics of a qubit implemented on a singlet-triplet subspace of two-electron states in a self-assembled quantum dot molecule. We study two possible control schemes based on the coupling to an excited (four-particle) state either by two spectrally separated laser pulses or by a single spectrally broad pulse. We quantitatively characterize the imperfections of the qubit operation resulting from non-adiabatic evolution and from limited spectral selectivity in a real system, as compared to the ideal adiabatic Raman transfer of occupation in the $\Lambda$-system. Next, we study the effects of decoherence induced by the coupling to the phonons of the surrounding crystal lattice and by radiative recombination. As a result, we are able to identify the optimization trade-offs between different sources of errors and indicate the most favorable conditions for quantum control of the singlet-triplet qubit in the two optical control schemes.

Local photons

The classical free-space solutions of Maxwell’s equations for light propagation in one dimension include wave packets of any shape that travel at the speed of light. This includes highly-localised wave packets that remain localised at all times. Motivated by this observation, this paper builds on recent work by Southall et al. [J. Mod. Opt. 68, 647 (2021)] and shows that a local description of the quantised electromagnetic field, which supports such solutions and which must overcome several no-go theorems, is indeed possible. Starting from the assumption that the basic building blocks of photonic wave packets are so-called bosons localised in position (blips), we identify the relevant Schrödinger equation and construct Lorentz-covariant electric and magnetic field observables. In addition we show that our approach simplifies to the standard description of quantum electrodynamics when restricted to a subspace of states.

Toponomic quantum computation

Holonomic quantum computation makes use of non-abelian geometric phases, associated to the evolution of a subspace of quantum states, to encode logical gates. We identify a special class of subspaces, for which a sequence of rotations results in a non-abelian holonomy of a topological nature, so that it is invariant under any SO(3)-perturbation. Making use of a Majorana-like stellar representation for subspaces, we give explicit examples of topological-holonomic (or toponomic) NOT and CNOT gates.

Tensor network approach to electromagnetic duality in (3+1)d topological gauge models

Given the Hamiltonian realisation of a topological (3+1)d gauge theory with finite group G, we consider a family of tensor network representations of its ground state subspace. This family is indexed by gapped boundary conditions encoded into module 2-categories over the input spherical fusion 2-category. Individual tensors are characterised by symmetry conditions with respect to non-local operators acting on entanglement degrees of freedom. In the case of Dirichlet and Neumann boundary conditions, we show that the symmetry operators form the fusion 2-categories 2VecG of G-graded 2-vector spaces and 2Rep(G) of 2-representations of G, respectively. In virtue of the Morita equivalence between 2VecG and 2Rep(G) — which we explicitly establish — the topological order can be realised as the Drinfel’d centre of either 2-category of operators; this is a realisation of the electromagnetic duality of the theory. Specialising to the case G = ℤ2, we recover tensor network representations that were recently introduced, as well as the relation between the electromagnetic duality of a pure ℤ2 gauge theory and the Kramers-Wannier duality of a boundary Ising model.

Analysis of unitarity in conformal quantum gravity

We perform a canonical quantization of Weyl’s conformal gravity by means of the covariant operator formalism and investigate the unitarity of the resulting quantum theory. After reducing the originally fourth-order theory to second-order in time derivatives via the introduction of an auxiliary tensor field, we identify the full Fock space of quantum states under a Becchi–Rouet–Stora–Tyutin (BRST) construction that includes Faddeev–Popov ghost fields corresponding to Weyl transformations. This second-order formulation allows the formal tools of operator-based quantum field theory to be applied to quadratic gravity for the first time. Using the Kugo–Ojima quartet mechanism, we identify the physical subspace of quantum states and find that the subspace containing the transverse spin-2 states comes equipped with an indefinite inner product metric and a one-particle Hamiltonian that possesses only a single eigenstate. We construct the Lehmann–Symanzik–Zimmermann (LSZ) reduction formula for the S-matrix in this spin-2 subspace and find that unitarity is violated in scattering events. The explicit way in which this violation occurs represents a new view on the ghost-problem in quadratic theories of quantum gravity.

Families of symmetries and the hydrogen atom

We study a new type of symmetry for the hydrogen atom involving algebraic families of groups parametrized by the energy value in the time-independent Schrödinger equation. We construct an algebraic family of Harish-Chandra modules from the solutions of the Schrödinger equation, and we characterize this family. We show that the subspaces of physical states may be obtained from our algebraic family using a Jantzen filtration, and we relate our algebraic methods with spectral theory and scattering theory using the limiting absorption principle.

Experimental mutual coherence from separable coherent qubits

Quantum coherence is a fundamental resource in modern quantum physics with important applications in quantum technologies. In composite quantum systems, various forms of coherence emerge and play an important role, such as the global coherence, coherence of local subsystems, and the recently introduced mutual coherence. We investigate states that maximize the mutual coherence in various subspaces of the overall two-qubit Hilbert space and discover a nontrivial asymmetric optimal state in the three-dimensional subspace. We experimentally generate this optimal state from two factorized photonic qubits by a strictly incoherent probabilistic quantum operation that projects the input state onto the desired three-dimensional subspace. For comparison, we also experimentally test the preparation of states with maximal mutual coherence by unitary transformations of input product states. These proof-of-principle tests demonstrate the initial steps of control of mutual quantum coherence in qubit systems.

The gauge structure of double field theory follows from Yang-Mills theory

We show that to cubic order double field theory is encoded in Yang-Mills theory. To this end we use algebraic structures from string field theory as follows: The ${L}_{\ensuremath{\infty}}$-algebra of Yang-Mills theory is the tensor product $\mathcal{K}\ensuremath{\bigotimes}\mathfrak{g}$ of the Lie algebra $\mathfrak{g}$ of the gauge group and a ``kinematic algebra'' $\mathcal{K}$ that is a ${C}_{\ensuremath{\infty}}$-algebra. This structure induces a cubic truncation of an ${L}_{\ensuremath{\infty}}$-algebra on the subspace of level-matched states of the tensor product $\mathcal{K}\ensuremath{\bigotimes}\overline{\mathcal{K}}$ of two copies of the kinematic algebra. This ${L}_{\ensuremath{\infty}}$-algebra encodes double field theory. More precisely, this construction relies on a particular form of the Yang-Mills ${L}_{\ensuremath{\infty}}$-algebra following from string field theory or from the quantization of a suitable worldline theory.

Unsupervised detection of decoupled subspaces: Many-body scars and beyond

Highly excited eigenstates of quantum many-body systems are typically featureless thermal states. Some systems, however, possess a small number of special, low-entanglement eigenstates known as quantum scars. We introduce a quantum-inspired machine-learning platform based on a quantum variational autoencoder (QVAE) that detects families of scar states in spectra of many-body systems. Unlike a classical autoencoder, QVAE performs a parametrized unitary operation, allowing us to compress a single eigenstate into a smaller number of qubits. We demonstrate that the autoencoder trained on a scar state is able to detect the whole family of scar states sharing common features with the input state. We identify families of quantum many-body scars in the PXP model beyond the ${\mathbb{Z}}_{2}$ and ${\mathbb{Z}}_{3}$ families and find dynamically decoupled subspaces in the Hilbert space of disordered, interacting spin-ladder model. The possibility of an automatic detection of subspaces of scar states opens new pathways in studies of models with a weak breakdown of ergodicity and fragmented Hilbert spaces.

Hierarchic Controllability Analysis in High-Dynamic Guidance for Autonomous Vehicle Landing

Controllability analysis is significant for guidance law optimization and control system synthesis during atmospheric landing of hypersonic flight vehicles. This article concerns the problem of quantifying the controllability and developing a novel controllability measure, i.e., the degree of controllability (DOC), for linear (or linearized) time-varying systems, and conducts the hierarchically quantitative controllability analysis in high-dynamic guidance for autonomous vehicle landing. In particular, a novel DOC criterion is derived via analyzing the characteristics of singular values and the condition number of the controllability Gramian. The proposed DOC criterion can offer a deep insight into the controllability of both the system and each state subspace, and thus, it possesses a clear physical meaning in practice. Furthermore, a certain normalization and modification method is designed to make the criterion more applicable in aerospace engineering. To validate the derived criterion, the scenario of 3-DOF Mars entry guidance is simulated based on a reference-trajectory tracking scheme with active disturbance rejection control algorithm. According to the simulation results, the optimality of different reference trajectories can be evaluated and compared via the DOC criterion. Therefore, the DOC criterion can function as an indicator to analyze the influence of trajectory parameters on the landing process from the perspective of controllability analysis.