Two-dimensional Subspace Research Articles

Nonadiabatic geometric quantum computation of W-state codes using invariant-based reverse engineering

In this paper, we propose a protocol to realize nonadiabatic geometric quantum computation (NGQC)of W-state codes in a spin system using invariant-based reverse engineering. The Heisenberg XY interaction of spin qubits provides a two-dimensional computational subspace spanned by a pair of W states. By applying a time-dependent magnetic on the spin qubits, we realize the effective Pauli operations for the computational subspace. Assisted by the invariant-based reverse engineering, the waveform of the control field is designed and the evolution paths for the NGQC is found. The performance of the protocol under the influence of experimental imperfections is estimated by the numerical simulations with available parameters. The results demonstrate that the protocol is robust against systematic error, random noise and decoherence. Therefore, the protocol may be promising to implement fast and robust manipulation of W states in spin systems.

Bounds on quantum adiabaticity in driven many-body systems from generalized orthogonality catastrophe and quantum speed limit

We provide two inequalities for estimating adiabatic fidelity in terms of two other more handily calculated quantities, i.e., generalized orthogonality catastrophe and quantum speed limit. As a result of considering a two-dimensional subspace spanned by the initial ground state and its orthogonal complement, our method leads to stronger bounds on adiabatic fidelity than those previously obtained. One of the two inequalities is nearly sharp when the system size is large, as illustrated using a driven Rice-Mele model, which represents a broad class of quantum many-body systems whose overlap of different instantaneous ground states exhibits orthogonality catastrophe.

Theoretical and Numerical Study of Self-Organizing Processes in a Closed System Classical Oscillator and Random Environment

A self-organizing joint system classical oscillator–random environment is considered within the framework of a complex probabilistic process that satisfies a Langevin-type stochastic differential equation. Various types of randomness generated by the environment are considered. In the limit of statistical equilibrium (SEq), second-order partial differential equations (PDE) are derived that describe the distribution of classical environmental fields. The mathematical expectation of the oscillator trajectory is constructed in the form of a functional-integral representation, which, in the SEq limit, is compactified into a two-dimensional integral representation with an integrand: the solution of the second-order complex PDE. It is proved that the complex PDE in the general case is reduced to two independent PDEs of the second order with spatially deviating arguments. The geometric and topological features of the two-dimensional subspace on which these equations arise are studied in detail. An algorithm for parallel modeling of the problem has been developed.

Multipartite subspaces containing no locally inaccessible information

One notion of non-locality in quantum theory is the fact that information may be encoded in a composite system in such a way that it is not accessible through local measurements, even with the assistance of classical communication. Thus, contrary to the classical case, there exists information in quantum many body systems which cannot be accessed locally. We show however that, remarkably, two-dimensional subspaces do not have this property: any physically allowed measurement on information encoded in any two-dimensional subspace, regardless of entanglement or multi-partite structure, may be performed locally. Further, this requires only local measurement and feed-forward of classical information, readily achievable in many experimental platforms. As an application to quantum secret sharing we suggest a twist on a well known quantum information splitting protocol, which ensures that no receiving party ever has access to the full state sent, but parties must work together to perform measurements on the state. These results may have practical applications to the measurement of encoded qubits in e.g. quantum secret sharing, quantum error correction, and reveal a fundamental property unique to two-dimensional subspaces.

Invariant measures for Hamiltonian flows and diffusion in infinitely dimensional phase spaces

We study Hamiltonian flows in a real separable Hilbert space endowed with a symplectic structure. Measures on the Hilbert space that are invariant with respect to the group of symplectomorphisms preserving two-dimensional symplectic subspaces are investigated. This construction gives the opportunity to present a random Hamiltonian flow in phase space by means of a random unitary group in the space of functions that are quadratically integrable by invariant measure. The properties of mean values of random shift operators are studied.

Bi-Criteria Radio Spectrum Sharing With Subspace-Based Pareto Tracing

Radio spectrum is a scarce resource. To meet demands, new wireless technologies must operate in shared spectrum over unlicensed bands (coexist). We consider coexistence of Long-Term Evolution (LTE) License-Assisted Access (LAA) with incumbent Wi-Fi systems. Our scenario consists of multiple LAA and Wi-Fi links sharing an unlicensed band; we aim to simultaneously optimize performance of both coexistence systems. To do this, we present a technique to continuously estimate the Pareto frontier of parameter sets (traces) which approximately maximize all convex combinations of network throughputs over network parameters. We use a dimensionality reduction approach known as active subspaces to determine that this near-optimal parameter set is primarily composed of two physically relevant parameters. A choice of two-dimensional subspace enables visualizations augmenting explainability and the reduced-dimension convex problem results in approximations which dominate random grid search.

Relationship between the Mandelbrot Algorithm and the Platonic Solids

This paper focuses on the dynamics of the eight tridimensional principal slices of the tricomplex Mandelbrot set: the Tetrabrot, the Arrowheadbrot, the Mousebrot, the Turtlebrot, the Hourglassbrot, the Metabrot, the Airbrot (octahedron), and the Firebrot (tetrahedron). In particular, we establish a geometrical classification of these 3D slices using the properties of some specific sets that correspond to projections of the bicomplex Mandelbrot set on various two-dimensional vector subspaces, and we prove that the Firebrot is a regular tetrahedron. Finally, we construct the so-called “Stella octangula” as a tricomplex dynamical system composed of the union of the Firebrot and its dual, and after defining the idempotent 3D slices of M3, we show that one of them corresponds to a third Platonic solid: the cube.

Generalized orbital angular momentum symmetry in parametric amplification

We investigate interesting symmetry properties verified by the down-converted beams produced in optical parametric amplification with structured light. We show that the Poincar\'e sphere symmetry, previously demonstrated for first-order spatial modes, translates to a multiple Poincar\'e sphere structure for higher orders. Each one of these multiple spheres is associated with a two-dimensional subspace defined by a different value of the orbital angular momentum. Therefore, the symmetry verified by first order modes is reproduced independently in each subspace. This effect can be useful for parallel control of independently correlated beams.

Comparing the performance of a nested to a continuous evolution strategy with covariance matrix adaption for optimization of drilling EDM

Electrical discharge machining (EDM) is a complex manufacturing process where the correlation of the individual process parameters is not fully known. When introducing new materials or complex, individual geometries in EDM, a satisfactory vector of parameters for the process must be found. This challenge is often encountered in the aerospace industry as well as in mold and tool making. One previously successful method to tackle this challenge is the stochastic optimization procedure Evolutionary Strategy (ES). Utilizing an appropriately chosen objective function, the search for a suitable vector of input parameters may be formulated as a mathematical minimization problem over the parameter space. The ES with a covariance matrix adaption (CMA) was utilized to sample from this parameter space in a stochastic manner. Examining the impact of the CMA within an ES is a promising way to gain better insight into the performance of ES. For this purpose, a vector of input parameters was optimized for a drilling EDM process with a comma and a nested comma ES with CMA. It was found that the comma ES led to a reduction in erosion duration tero of 38 % compared to the initially chosen parameters and the nested comma ES led to a reduction in erosion duration tero of 27 % compared to the same initial parameters. The additional information stored in the covariance matrix enables the ES to find a local optimum of the parameter vector faster and more consistently. This fact is verified by use of visualizations of the covariance matrix on a two-dimensional subspace. From these findings it can be concluded, that for the application of the ES to the optimization of EDM processes the CMA should be performed continuously over all generations as opposed to resetting this matrix after a number of generations.

Limit cycles in mass-conserving deficiency-one mass-action systems

We present some simple mass-action systems with limit cycles that fall under the scope of the Deficiency-One Theorem. All the constructed examples are mass-conserving and their stoichiometric subspace is two-dimensional. Using the continuation software MATCONT, we depict the limit cycles in all stoichiometric classes at once. The networks are trimolecular and tetramolecular, and some exhibit two or even three limit cycles. Finally, we show that the associated mass-action system of a bimolecular reaction network with two-dimensional stoichiometric subspace does not admit a limit cycle.

Subspace Decomposition Based Adaptive Density Peak Clustering for Radar Signals Sorting

Radar signal sorting (RSS) plays an important role in the electronic support measurement system. However, the existing clustering-based RSS methods depend heavily on prior knowledge to achieve excellent performance, which may bring severe challenges to RSS in actual scenarios. This letter proposes a novel subspace decomposition based adaptive density peak clustering (SD-ADPC) method to address the problems of low accuracy and high computational cost in RSS. First, the original complex radar signal data is directly decomposed into two-dimensional (2D) subspace by t-distributed random neighborhood embedding (t-SNE). Then, based on the outlier detection of the products between the peak density and the distance of the data points, ADPC is used to adaptively determine the optimal clustering centers of the original data in 2D subspace. Finally, the reminding data is assigned to its nearest cluster with Euclidean distance in one step. The experimental results of the simulated RSS dataset and the open baselines show that our proposed method does not require any knowledge and can achieve better or competitive performance in terms of accuracy and computational cost, compared to existing state-of-the-art methods.

On the Symmetric Lamination Convex and Quasiconvex Hull for the Coplanar $n$-Well Problem in Two Dimensions

We study some particular cases of the $n$ -well problem in two-dimensional linear elasticity. Assuming that every well in $\mathcal{U}\subset \mathbb{R}^{2\times 2}_{\text{sym}}$ belong to the same two-dimensional affine subspace, we characterize the symmetric lamination convex hull $L^{e}(\mathcal{U})$ for any number of wells in terms of the symmetric lamination convex hull of all three-well subsets contained in $\mathcal{U}$ . For a family of four-well sets where two pairs of wells are rank-one compatible, we show that the symmetric lamination convex and quasiconvex hulls coincide, but are strictly contained in its convex hull $C(\mathcal{U})$ . We extend this result to some particular configurations of $n$ wells. Most of the proofs are constructive, and we also present explicit examples.

Angle-micro-Doppler frequency image of underwater target multi-highlight combining with sparse reconstruction

Aiming at the problem of low angle resolution and difficult extraction of time-varying features in underwater target multi-highlight acoustic imaging, an acoustic imaging method of multi-highlight angle-micro-Doppler frequency for underwater target combining with sparse reconstruction is proposed by introducing sparse decomposition and time–frequency analysis into array signal processing. According to the difference between the underwater target highlight reflected wave modulation form caused by the micro-Doppler effect and time–frequency distribution, a kind of MicroDopplerlet time–frequency atom matching the complex modulation of the highlight reflection wave is designed. The multi-component signal processing is transformed into multiple single component signal processing problems by separating the highlight reflected wave from the echo by MP decomposition. In addition, a two-dimensional subspace processing method based on MP (MP-MUSIC) and a time–frequency distribution based on MP (MP-WVD) are constructed for sparsely reconstructed reflected wave of one single highlight, which are used to extract azimuth angle, pitch angle, micro-Doppler frequency information and acoustic image respectively. The proposed method has the advantages of noise robustness, high resolution and the ability to solve the underdetermined problem. The simulation results show that the proposed method can distinguish the multi-highlight structure of underwater target, and estimate the highlight angle, micro-Doppler frequency and micro-motion parameters of the target components correctly.

Algebras of Symmetry Operators of the Klein–Gordon–Fock Equation for Groups Acting Transitively on Two-Dimensional Subspaces of a Space-Time Manifold

Algebras of Symmetry Operators of the Klein–Gordon–Fock Equation for Groups Acting Transitively on Two-Dimensional Subspaces of a Space-Time Manifold

Four dimensional second-order topological insulator based on a synthetic plasmonic metasurface.

It is possible to explore higher dimensional topological properties in lower dimensional structures by introducing additional synthetic dimensions. In this Letter, we construct a four-dimensional (4D) second-order topological insulator using gradient nanoparticle arrays arranged in a periodic lattice. The nanoparticle array has spatially varying inter-particle distance along x and y directions, which can be regarded as two synthetic dimensions. Different from higher-order topological insulators in classical wave systems, the higher-order topological states in this 4D system are protected by a pair of first Chern numbers in two-dimensional (2D) subspaces instead of by the quantized 2D Zak phases. It is shown that there exist (4-1)- and (4-2)-dimensional boundary states for both transverse and longitudinal collective resonant modes, which provides new, to the best of our knowledge, mechanisms for light confinement and control in such a plasmonic superlattice.

Analytical computation of quasinormal modes of slowly rotating black holes in dynamical Chern-Simons gravity

Using gravitational wave observations to search for deviations from general relativity (GR) in the strong-gravity regime has become an important research direction. One aspect of the strong gravity modifications to GR is parity violation. Chern Simons (CS) gravity is one of the most frequently studied parity-violating models of strong gravity. CS gravity is indistinguishable from GR for all conformally flat space-times and for space-times that possess a maximally symmetric two-dimensional subspace. Also, it is known that the Kerr black hole is not a solution for CS gravity. At the same time, the only rotating solution available in the literature for dynamical CS (DCS) gravity is the slow-rotating case most accurately known to quadratic order in spin. In this work, for the slow-rotating case (accurate to first order in spin), we derive the linear perturbation equations governing the metric and the DCS field accurate to linear order in spin and quadratic order in the CS coupling parameter ($\ensuremath{\alpha}$) and obtain the quasinormal mode (QNM) frequencies. After confirming the recent results of Wagle et al. arXiv:2103.09913, we find an additional contribution to the eigenfrequency correction at the leading perturbative order of ${\ensuremath{\alpha}}^{2}$. Unlike Wagle et al., we also find corrections to frequencies in the polar sector. We compute these extra corrections by evaluating the expectation values of the perturbative potential on unperturbed QNM wave functions along a contour deformed into the complex-$r$ plane. For $\ensuremath{\alpha}=0.1{M}^{2}$, we obtain the ratio of the imaginary parts of the DCS correction to the GR correction in the first QNM frequency (in the polar sector) to be 0.263 implying significant change. For the (2, 2) mode, the DCS corrections make imaginary part of the first QNM of the fundamental mode less negative, thereby decreasing the decay rate. Our results, along with future gravitational wave observations, can be used as a test for DCS gravity and to further constrain the CS coupling parameters.

Robust semi-device-independent certification of all pure bipartite maximally entangled states via quantum steering

The idea of self-testing is to render guarantees concerning the inner workings of a device based on the measurement statistics. It is one of the most formidable quantum certification and benchmarking schemes. Recently it was shown by Coladangelo et. al. (Nat Commun 8, 15485 (2017)) that all pure bipartite entangled states can be self tested in the device independent scenario by employing subspace methods introduced by Yang et. al. (Phys. Rev. A 87, 050102(R)). Here, we have adapted their method to show that any bipartite pure entangled state can be certified in the semi-device independent scenario through Quantum Steering. Analogous to the tilted CHSH inequality, we use a steering inequality called Tilted Steering Inequality for certifying any pure two-qubit entangled state. Further, we use this inequality to certify any bipartite pure entangled state by certifying two-dimensional sub-spaces of the qudit state by observing the structure of the set of assemblages obtained on the trusted side after measurements are made on the un-trusted side. As a feature of quantum state certification via steering, we use the notion of Assemblage based robust state certification to provide robustness bounds for the certification result in the case of pure maximally entangled states of any local dimension.

High-energy EFT probes with fully differential Drell-Yan measurements

We study the potential of fully-differential measurements of high-energy dilepton cross-sections at the LHC to probe heavy new physics encapsulated in dimension-6 interaction operators. The assessment is performed in the seven-dimensional parameter space of operators that induce energy-growing corrections to the Standard Model partonic cross-sections at the interference level, and in the two-dimensional subspace associated with the W and Y parameters. A considerable sensitivity improvement is found relative to single-differential measurements, owing to the possibility of probing at the interference level more directions in the seven-dimensional parameter space. The reduction of parton distribution function uncertainties in the fully-differential fit is also found to play a significant role. The results are interpreted in the minimal Z′ new-physics model, providing a concrete illustration of the advantages of the fully-differential analysis. We find that high-energy dilepton measurements can extend the Z′ exclusion and discovery potential well beyond the reach of direct searches in a large region of the parameter space.

Thermal one-point functions and single-valued polylogarithms

I point out that the thermal one-point functions of a pair of relevant operators in massive free QFTs, in odd dimensions and in the presence of an imaginary chemical potential for a U(1) global charge, are given by certain classes of single-valued polylogarithms. This result is verified by a direct calculation using the thermal OPE. The complex argument of the polylogarithms parametrizes a two-dimensional subspace of relevant deformations of generalised free CFTs, while the rank of the polylogarithms is related to the dimension d. This may be compared with the well-known representation of single-valued polylogarithms as multiloop Feynman amplitudes. As an example, the thermal one-point function of the U(1) charge in d-dimensions generalises the thermal average of the twist operator in a pair of harmonic oscillators and is given by the well-known conformal ladder graphs in four dimensions.

A Generalized Decomposition Model of Dual Three-Phase Permanent Magnet Synchronous Machines Considering Asymmetric Impedances and Compensation Capability

This article first proposes a generalized decomposition model for dual three-phase (3-ph) permanent magnet synchronous machines (PMSMs) with 0°, 30°, and 60° angle displacements between two sets of windings, allowing machines controlled in two-dimensional orthogonal αβ and xy subspaces. This model can decouple mutual inductances coupling phases from two sets, respectively. However, there often exist asymmetric impedances, due to inherent asymmetric machine parameters, cables with unequal lengths, manufacture tolerances, etc., in dual 3-ph PMSMs inevitably causing unbalanced phase currents and deteriorating the decoupling performance of the generalized machine model. Therefore, the generalized decomposition model incorporating asymmetric impedances is further developed, in which additional terms incurred by asymmetric impedances lead to the cross-coupling of αβ subspace and xy subspace. Through this model, the compensation capability of asymmetries is derived at a given dc link voltage. Then, a compensation strategy is illustrated to suppress unbalanced phase currents together with current harmonics caused by nonlinearities. Finally, experimental results testify the current balancing performance and simulation results further validate the machine model and the compensation capability.