Unitary Invariance Research Articles

The Gauge Picture of Quantum Dynamics

Although local Hamiltonians exhibit local time dynamics, this locality is not explicit in the Schrödinger picture in the sense that the wavefunction amplitudes do not obey a local equation of motion. We show that geometric locality can be achieved explicitly in the equations of motion by "gauging" the global unitary invariance of quantum mechanics into a local gauge invariance. That is, expectation values ⟨ψ|A|ψ⟩ are invariant under a global unitary transformation acting on the wavefunction |ψ⟩→U|ψ⟩ and operators A→UAU†, and we show that it is possible to gauge this global invariance into a local gauge invariance. To do this, we replace the wavefunction with a collection of local wavefunctions |ψJ⟩, one for each patch of space J. The collection of spatial patches is chosen to cover the space; e.g. we could choose the patches to be single qubits or nearest-neighbor sites on a lattice. Local wavefunctions associated with neighboring pairs of spatial patches I and J are related to each other by dynamical unitary transformations UIJ. The local wavefunctions are local in the sense that their dynamics are local. That is, the equations of motion for the local wavefunctions |ψJ⟩ and connections UIJ are explicitly local in space and only depend on nearby Hamiltonian terms. (The local wavefunctions are many-body wavefunctions and have the same Hilbert space dimension as the usual wavefunction.) We call this picture of quantum dynamics the gauge picture since it exhibits a local gauge invariance. The local dynamics of a single spatial patch is related to the interaction picture, where the interaction Hamiltonian consists of only nearby Hamiltonian terms. We can also generalize the explicit locality to include locality in local charge and energy densities.

Locally unitarily invariantizable NEPv and convergence analysis of SCF

We consider a class of eigenvector-dependent nonlinear eigenvalue problems (NEPv) without the unitary invariance property. Those NEPv commonly arise as the first-order optimality conditions of a particular type of optimization problems over the Stiefel manifold, and previously, special cases have been studied in the literature. Two necessary conditions, a definiteness condition and a rank-preserving condition, on an eigenbasis matrix of the NEPv that is a global optimizer of the associated optimization problem are revealed, where the definiteness condition has been known for the special cases previously investigated. We show that, locally close to the eigenbasis matrix satisfying both necessary conditions, the NEPv can be reformulated as a unitarily invariant NEPv, the so-called aligned NEPv, through a basis alignment operation — in other words, the NEPv is locally unitarily invariantizable. Numerically, the NEPv is naturally solved by a self-consistent field (SCF)-type iteration. By exploiting the differentiability of the coefficient matrix of the aligned NEPv, we establish a closed-form local convergence rate for the SCF-type iteration and analyze its level-shifted variant. Numerical experiments confirm our theoretical results.

Quantum conditional entropy based on local quantum Bernoulli noises

Quantum noise has always been an important issue in the field of quantum information transmission. As the localization of quantum noise, local quantum Bernoulli noises (LQBNs) have attracted extensive attention in recent years. In this paper, the quantum conditional entropy based on LQBNs is deeply discussed, and a new definition of quantum conditional entropy is given. Then we find that this quantum conditional entropy has some basic properties such as concavity, conditional reduced entropy and unitary invariance. This research is of great significance to the field of quantum information, which can help us to understand the influence of local quantum noises on quantum information transmission.

High Accuracy and Low Complexity Lidar Place Recognition Using Unitary Invariant Frobenius Norm

Simultaneous Localization and Mapping (SLAM) is used in solving the problems of localization, navigation, and map construction for autonomous vehicles moving in unknown environments. Place recognition is an inevitable subject in SLAM, and the current lidar-based methods have been popularized for their rising environmental robustness. Currently, the place recognition method with lidar has attracted much attention due to its high environmental robustness. However, many 3D laser point cloud methods for place recognition descriptors construction with the local or global point-cloud data omitting some intrinsic properties of the point clouds. In this article, we propose a place recognition method based on the unitary invariance of the Frobenius norm for utilizing different attributes of ground points and non-ground points. Specifically, the interpretable filtering framework and a dynamic threshold adjustment strategy are raised according to different environments and sensors with intensity and geometric information. As commissioning, the comparative experiments are conducted with CHDloop datasets and public KITTI datasets of different scales and access types. Compared to the original Scan Context (SC) and Intensity Scan Context (ISC) methods, our proposed method achieves higher efficiency while maintaining a improved recall rate and precision. This novel method has been integrated into the existing LiDAR SLAM and formed a new framework FSC_ALOAM that reduces drifts in point-cloud mapping. The entire process improves the accordance of maps at the identical position in auto-driving.

Realism-based nonlocality: Invariance under local unitary operations and asymptotic decay for thermal correlated states

The realism-based nonlocality (RBN) is a recently introduced measure that differs from the well-known Bell’s nonlocality. For bipartite states, the RBN concerns how much an element of reality associated with a given observable is affected upon local measurements on the other subsystem. Here, we present an analytical proof for the unitary invariance of the RBN and that it presents a monotonous behavior upon the action of unital and non-unital local quantum noise. We illustrate our results by employing the two-qubit Werner state and thermal quantum correlated states. We show how the RBN is limited by the initial equilibrium temperature and, especially, that it decays asymptotically with it. These results also corroborate the hierarchy relationship between the quantifiers of RBN and global quantum discord, showing that RBN can capture undetectable nonlocal aspects even for non-discordant states.

Physical Implementability of Linear Maps and Its Application in Error Mitigation

Completely positive and trace-preserving maps characterize physically implementable quantum operations. On the other hand, general linear maps, such as positive but not completely positive maps, which can not be physically implemented, are fundamental ingredients in quantum information, both in theoretical and practical perspectives. This raises the question of how well one can simulate or approximate the action of a general linear map by physically implementable operations. In this work, we introduce a systematic framework to resolve this task using the quasiprobability decomposition technique. We decompose a target linear map into a linear combination of physically implementable operations and introduce the physical implementability measure as the least amount of negative portion that the quasiprobability must pertain, which directly quantifies the cost of simulating a given map using physically implementable quantum operations. We show this measure is efficiently computable by semidefinite programs and prove several properties of this measure, such as faithfulness, additivity, and unitary invariance. We derive lower and upper bounds in terms of the Choi operator's trace norm and obtain analytic expressions for several linear maps of practical interests. Furthermore, we endow this measure with an operational meaning within the quantum error mitigation scenario: it establishes the lower bound of the sampling cost achievable via the quasiprobability decomposition technique. In particular, for parallel quantum noises, we show that global error mitigation has no advantage over local error mitigation.

Local tail statistics of heavy-tailed random matrix ensembles with unitary invariance

We study heavy-tailed Hermitian random matrices that are unitarily invariant. The invariance implies that the eigenvalue and eigenvector statistics are decoupled. The motivating question has been whether a freely stable random matrix has stable eigenvalue statistics for the largest eigenvalues in the tail. We investigate this question through the use of both numerical and analytical means, the latter of which makes use of the supersymmetry method. A surprising behaviour is uncovered in that a freely stable random matrix does not necessarily yield stable statistics and if it does then it might exhibit Poisson or Poisson-like statistics. The Poisson statistics have been already observed for heavy-tailed Wigner matrices. We conclude with two conjectures on this peculiar behaviour.

Deep random walk of unitary invariance for large-scale data representation

Data representation aims at learning an efficient low-dimensional representation, which is always a challenging task in machine learning and computer vision. It can largely improve the performance of specific learning tasks. Unsupervised methods are extensively applied to data representation, which considers the internal connection among data. Most of existing unsupervised models usually use a specific norm to favor certain distributions of the input data, leading to an unsustainable encouraging performance for given learning tasks. In this paper, we propose an efficient data representation method to address large-scale feature representation problems, where the deep random walk of unitary invariance is exploited for learning discriminative features. First, the data representation is formulated as deep random walk problems, where unitarily invariant norms are employed to capture diverse beneficial perspectives hidden in the data. It is embedded into a state transition matrix model, where an arbitrary number of transition steps is available for an accurate affinity evaluation. Second, data representation problems are then transformed as high-order matrix factorization tasks with unitary invariance. Third, a closed-form solution is proved for the formulated data representation problem, which may provide a new perspective for solving high-order matrix factorization problems. Finally, extensive comparative experiments are conducted in publicly available real-world data sets. In addition, experimental results demonstrate that the proposed method achieves better performance than other compared state-of-the-art approaches in terms of data clustering.

Implications of the unitary invariance and symmetry restrictions on the development of proper approximate one-body reduced-density-matrix functionals

In many of the approximate functionals in one-body reduced density matrix (1RDM) functional theory, the approximate two-body reduced density matrix (2RDM) in the natural orbital representation only depends on the natural occupation numbers. In Phys. Rev. A 92, 012520 (2015) Wang and Knowles initialised a discussion of to what extent this simplification is valid, by introducing two different H$_4$ geometries with identical natural occupation numbers, but different 2RDMs. Gritsenko has argued that this feature is due symmetry [Phys. Rev. A 97, 026501 (2018)]. This work aims to contribute to the discussion on the following points: 1) one should rather speak of symmetry-restricted variants of the universal functional, than saying that the universal functional is symmetry dependent; 2) the unitary invariance of degenerate NOs can lead to large deviations in the 2RDM elements, especially the phase of the NOs; 3) symmetry-restricted functionals are constructed for the H$_4$ geometries considered by Wang and Knowles, whose structure could serve as guide in the construction of approximate 1RDM functionals.

Quantifying the quantumness of ensembles via generalized α-z-relative rényi entropy

Quantifying the quantumness of ensembles is a vital and practical task in quantum information theory. In this paper, we quantify the quantumness of quantum ensembles based on a quantifier of the coherence of ensembles via generalized α-z-relative Renyi entropy. It is shown that this measure satisfies the intuitive and desirable properties which are required by a measure of quantumness, such as positivity, unitary invariance, monotonicity under any commutativity preserving operations and so on. The relationship between our measure and some existing measures for quantumness of quantum ensembles is discussed. Detailed examples are presented to some important ensembles to illustrate the quantumness given by our measure.

New upper bounds for noncentral chi-square cdf

Some new upper bounds for noncentral chi-square cumulative density function are derived from the basic symmetries of the multidimensional standard Gaussian distribution: unitary invariance, components independence in both polar and Cartesian coordinate systems. The proposed new bounds have analytically simple form compared to analogues available in the literature: they are based on combination of exponents, direct and inverse trigonometric functions, including hyperbolic ones, and the cdf of the one dimensional standard Gaussian law. These new bounds may be useful both in theory and in applications: for proving inequalities related to noncentral chi-square cumulative density function, and for bounding powers of Pearson’s chi-squared tests.

An Overview of Self-Consistent Field Calculations Within Finite Basis Sets.

A uniform derivation of the self-consistent field equations in a finite basis set is presented. Both restricted and unrestricted Hartree–Fock (HF) theory as well as various density functional approximations are considered. The unitary invariance of the HF and density functional models is discussed, paving the way for the use of localized molecular orbitals. The self-consistent field equations are derived in a non-orthogonal basis set, and their solution is discussed also in the presence of linear dependencies in the basis. It is argued why iterative diagonalization of the Kohn–Sham–Fock matrix leads to the minimization of the total energy. Alternative methods for the solution of the self-consistent field equations via direct minimization as well as stability analysis are briefly discussed. Explicit expressions are given for the contributions to the Kohn–Sham–Fock matrix up to meta-GGA functionals. Range-separated hybrids and non-local correlation functionals are summarily reviewed.

On the predictive potential of kernel principal components

We give a probabilistic analysis of a phenomenon in statistics which, until recently, has not received a convincing explanation. This phenomenon is that the leading principal components tend to possess more predictive power for a response variable than lower-ranking ones despite the procedure being unsupervised. Our result, in its most general form, shows that the phenomenon goes far beyond the context of linear regression and classical principal components --- if an arbitrary distribution for the predictor $X$ and an arbitrary conditional distribution for $Y \vert X$ are chosen then any measureable function $g(Y)$, subject to a mild condition, tends to be more correlated with the higher-ranking kernel principal components than with the lower-ranking ones. The ``arbitrariness'' is formulated in terms of unitary invariance then the tendency is explicitly quantified by exploring how unitary invariance relates to the Cauchy distribution. The most general results, for technical reasons, are shown for the case where the kernel space is finite dimensional. The occurency of this tendency in real world databases is also investigated to show that our results are consistent with observation.

On unitary invariance of some classes of operators in Hilbert spaces

On unitary invariance of some classes of operators in Hilbert spaces

Knit Product of Finite Groups and Sampling

A finite sampling theory associated with a unitary representation of a finite non-abelian group \({\mathbf {G}}\) on a Hilbert space is established. The non-abelian group \({\mathbf {G}}\) is a knit product \({\mathbf {N}}\bowtie {\mathbf {H}}\) of two finite subgroups \({\mathbf {N}}\) and \({\mathbf {H}}\) where at least \({\mathbf {N}}\) or \({\mathbf {H}}\) is abelian. Sampling formulas where the samples are indexed by either \({\mathbf {N}}\) or \({\mathbf {H}}\) are obtained. Using suitable expressions for the involved samples, the problem is reduced to obtain dual frames in the Hilbert space \(\ell ^2({\mathbf {G}})\) having a unitary invariance property; this is done by using matrix analysis techniques. An example involving dihedral groups illustrates the obtained sampling results.

REMP: A hybrid perturbation theory providing improved electronic wavefunctions and properties.

We propose a new perturbation theoretical approach to the electron correlation energy by choosing the zeroth order Hamiltonian as a linear combination of the corresponding "Retaining the Excitation degree" (RE) and the Møller-Plesset (MP) operators. In order to fulfill Kato cusp conditions, the RE and MP contributions are chosen to sum up to one. 15% ± 5% MP contribution is deduced to be in an optimal range from a fit of the first order REMP wavefunction to near full configuration interaction reference data. For closed shell systems, the same range of MP weights shows best performance for equilibrium bond distances and vibrational wavenumbers of diatomic molecules, the reaction energies in the spin component scaled MP2 fit set, the transition energies of the BHPERI test set, and the parameterized coupled cluster with singles and doubles (pCCSD) fit set. For these properties, REMP outperforms all other tested perturbation theories at second order and shows equal performance as the best coupled pair approaches or pCCSD methods as well as the best double hybrid density functionals. Furthermore, REMP is shown to fulfill all required fundamental boundary conditions of proper wavefunction based quantum chemical methods (unitary invariance and size consistency).

Unitary symmetry constraints on tensorial group field theory renormalization group flow

Renormalization group methods are an essential ingredient in the study of nonperturbative problems of quantum field theory. This paper deal with the symmetry constraints on the renormalization group flow for quartic melonic tensorial group field theories. Using the unitary invariance of the interactions, we provide a set of Ward–Takahashi identities which leads to relations between correlation functions. There are numerous reasons to consider such Ward identities in the functional renormalization group. Their compatibility along the flow provides a non-trivial constraint on the reliability of the approximation schemes used in the non-perturbative regime, especially on the truncation and the choice of the regulator. We establish the so-called structure equations in the melonic sector and in the symmetric phase. As an example we consider the TGFT model without gauge constraint. The Wetterich flow equation is given and the way to improve the truncation on the effective action is also scrutinized.

Quantifying the quantumness of ensembles via unitary similarity invariant norms

The quantification of the quantumness of a quantum ensemble has theoretical and practical significance in quantum information theory. We propose herein a class of measures of the quantumness of quantum ensembles using the unitary similarity invariant norms of the commutators of the constituent density operators of an ensemble. Rigorous proof shows that they share desirable properties for a measure of quantumness, such as positivity, unitary invariance, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Several specific examples illustrate the applications of these measures of quantumness in studying quantum information.

Numerical radius inequalities via convexity

The main goal of this article is to present numerical radius inequalities for matrices based on convexity of certain numerical radius functions. This simple approach extends some unitarily invariant norms inequalities, such as Heinz and Young inequalities, to the context of numerical radius. However, due to weak unitary invariance of the numerical radius, these extended inequalities will be weaker than the corresponding unitarily invariant norms versions.

Quantifying the quantumness of ensembles

We quantify the quantumness of quantum ensembles in terms of noncommutativity between pairs of the square root of the constituent states. We prove that this measure satisfies desirable and intuitive properties required naturally for a measure of quantumness, such as positivity, unitary invariance, subadditivity, concavity under probabilistic union, convexity under state decomposition, decreasing under coarse graining, and increasing under fine graining. Some applications and implications are indicated.