Trace Norm Research Articles

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.

{0,1}-Brauer Configuration Algebras and Their Applications in Graph Energy Theory

The energy E(G) of a graph G is the sum of the absolute values of its adjacency matrix. In contrast, the trace norm of a digraph Q, which is the sum of the singular values of the corresponding adjacency matrix, is the oriented version of the energy of a graph. It is worth pointing out that one of the main problems in this theory consists of determining appropriated bounds of these types of energies for significant classes of graphs, digraphs and matrices, provided that, in general, finding out their exact values is a problem of great difficulty. In this paper, the trace norm of a {0,1}-Brauer configuration is introduced. It is estimated and computed by associating suitable families of graphs and posets to Brauer configuration algebras.

A nonlocal enhanced Low-Rank tensor approximation framework for 3D Magnetic Resonance image denoising

Magnetic Resonance Imaging (MRI) has become an increasingly essential tool in clinical detection and diagnosis for its ability to disclose the distinctive information of human anatomic structures in vivo. However, the image intensity in magnitude MRI images is frequently corrupted by Rician noise which is inherent to the acquisition process. This study aimed to propose a denoising model for 3D MR image restoration based on low-rank tensor approximation with a Logarithmic-Sum regularization framework. Nonlocal similarity of images was exploited with grouping and block matching techniques, and tensor decomposition approaches were utilized taking consideration of multi-dimensional structural dependence conservation. The low-rank tensor approximation problem was regulated by introducing a Logarithmic-Sum trace norm, which thresholded the singular value spectrum adaptively with different thresholds. Noise removal was processed through the low-rank approximation, and the denoised tensor blocks were aggregated to restore the clear images. Numerical experiments were conducted under comprehensive noise conditions for 3D MR volumetric datasets. Through denoising comparison, the results demonstrated that the proposed algorithm achieves state-of-the-art performance for Rician noise removal with excellent detail preservation..

Optimal convex approximation of qubit states and geometry of completely represented states

We investigate the optimal convex approximation, optimally approximating a desired and unavailable qubit state by the convex mixing of a given set of available states. When the available states are the eigenvectors of three Pauli matrices, we present the complete exact solution for the optimal convex approximation of an arbitrary qubit state based on the fidelity distance. By the comparison of optimal states based on fidelity and trace norm, the advantages and disadvantages of the optimal convex approximation are identified. Several examples are provided to support this. We also analyze the geometrical properties of the target states which can be completely represented by a set of available states. Using the feature of convex combination, we derive the maximum range of completely represented states and clearly illustrate that any qubit state can be optimally prepared by at most three available states in known useable states.

Low-Rank Matrix Recovery via Efficient Schatten p-Norm Minimization

As an emerging machine learning and information retrieval technique, the matrix completion has been successfully applied to solve many scientific applications, such as collaborative prediction in information retrieval, video completion in computer vision, \emph{etc}. The matrix completion is to recover a low-rank matrix with a fraction of its entries arbitrarily corrupted. Instead of solving the popularly used trace norm or nuclear norm based objective, we directly minimize the original formulations of trace norm and rank norm. We propose a novel Schatten $p$-Norm optimization framework that unifies different norm formulations. An efficient algorithm is derived to solve the new objective and followed by the rigorous theoretical proof on the convergence. The previous main solution strategy for this problem requires computing singular value decompositions - a task that requires increasingly cost as matrix sizes and rank increase. Our algorithm has closed form solution in each iteration, hence it converges fast. As a consequence, our algorithm has the capacity of solving large-scale matrix completion problems. Empirical studies on the recommendation system data sets demonstrate the promising performance of our new optimization framework and efficient algorithm.

Common Positive Solution of Two Nonlinear Matrix Equations Using Fixed Point Results

We discuss a pair of nonlinear matrix equations (NMEs) of the form X=R1+∑i=1kAi*F(X)Ai, X=R2+∑i=1kBi*G(X)Bi, where R1,R2∈P(n), Ai,Bi∈M(n), i=1,⋯,k, and the operators F,G:P(n)→P(n) are continuous in the trace norm. We go through the necessary criteria for a common positive definite solution of the given NME to exist. We develop the concept of a joint Suzuki-implicit type pair of mappings to meet the requirement and achieve certain existence findings under weaker assumptions. Some concrete instances are provided to show the validity of our findings. An example is provided that contains a randomly generated matrix as well as convergence and error analysis. Furthermore, we offer graphical representations of average CPU time analysis for various initializations.

Approximate tensor decompositions: Disappearance of many separations

It is well known that tensor decompositions show separations, that is, constraints on local terms (such as positivity) may entail an arbitrarily high cost in their representation. Here, we show that many of these separations disappear in the approximate case. Specifically, for every approximation error ɛ and norm, we define the approximate rank as the minimum rank of an element in the ɛ-ball with respect to that norm. For positive semidefinite matrices, we show that the separations between rank, purification rank, and separable rank disappear for a large class of Schatten p-norms. For non-negative tensors, we show that the separations between rank, positive semidefinite rank, and non-negative rank disappear for all ℓp-norms with p > 1. For the trace norm (p = 1), we obtain upper bounds that depend on the ambient dimension. We also provide a deterministic algorithm to obtain the approximate decomposition attaining our bounds. Our main tool is an approximate version of the Carathéodory theorem. Our results imply that many separations are not robust under small perturbations of the tensor, with implications in quantum many-body systems and communication complexity.

Variational quantum algorithms for trace norms and their applications

The trace norm of matrices plays an important role in quantum information and quantum computing. How to quantify it in today’s noisy intermediate scale quantum (NISQ) devices is a crucial task for information processing. In this paper, we present three variational quantum algorithms on NISQ devices to estimate the trace norms corresponding to different situations. Compared with the previous methods, our means greatly reduce the requirement for quantum resources. Numerical experiments are provided to illustrate the effectiveness of our algorithms.

Distribution of kinks in an Ising ferromagnet after annealing and the generalized Kibble-Zurek mechanism

We consider the annealing dynamics of a one-dimensional Ising ferromagnet induced by a temperature quench in finite time. In the limit of slow cooling, the asymptotic two-point correlator is analytically found under Glauber dynamics, and the distribution of the number of kinks in the final state is shown to be consistent with a Poissonian distribution. The mean kink number, the variance, and the third centered moment take the same value and obey a universal power-law scaling with the quench time in which the temperature is varied. The universal power-law scaling of cumulants is corroborated by numerical simulations based on Glauber dynamics for moderate cooling times away from the asymptotic limit, when the kink-number distribution takes a binomial form. We analyze the relation of these results to physics beyond the Kibble-Zurek mechanism for critical dynamics, using the kink number distribution to assess adiabaticity and its breakdown. We consider linear, nonlinear, and exponential cooling schedules, among which the latter provides the most efficient shortcuts to cooling in a given quench time. The non-thermal behavior of the final state is established by considering the trace norm distance to a canonical Gibbs state.

Some Fixed Point Results on Relational Quasi Partial Metric Spaces and Application to Non-Linear Matrix Equations

We introduce a qϱ-implicit contractive condition by an implicit relation on relational quasi partial metric spaces and establish new (unique) fixed point results and periodic point results based on it. We justify the results by two suitable examples and compare with them related work. We discuss sufficient conditions ensuring the existence of a unique positive definite solution of the non-linear matrix equation U=B+∑i=1mAi*G(U)Ai, where B is an n×n Hermitian positive definite matrix, A1, A2, … Am are n×n matrices, and G is a non-linear self-mapping of the set of all Hermitian matrices which is continuous in the trace norm. Two examples (with randomly generated matrices and complex matrices, respectively) are given, together with convergence and error analysis, as well as average CPU time analysis and visualization of solution in surface plot.

Algorithms for nonnegative C2(R2) interpolation

Let E⊂R2 be a finite set, and let f:E→[0,∞). In this paper, we address the algorithmic aspects of nonnegative C2 interpolation in the plane. Specifically, we provide an efficient algorithm to compute a nonnegative C2(R2) extension of f with norm within a universal constant factor of the least possible. We also provide an efficient algorithm to approximate the trace norm.

Tensor completion-based trajectory imputation approach in air traffic control

The flight trajectory in air traffic control systems usually misses some updating positions because of unexpected errors. In this paper, a tensor completion-based approach is proposed to recover missing positions from a whole trajectory dataset. Considering the trajectory dependencies among different operations, the flight trajectories with the same flight number are organized as a three-dimensional tensor. A trace norm minimizing based tensor completion method is performed on the trajectory tensor to achieve the imputation task, in which the Block Coordinate Descent algorithm is applied to optimize the tensor model. Unlike other data-driven algorithms, the proposed approach captures the global information (route similarity and transition patterns) from the whole tensor, which is further applied to estimate the missing values in a training-free manner. Several experiments are designed to validate the proposed approach, including the padding methods, the dataset size, and the imputation performance on different missing patterns and rates. Experimental results on real-world flight trajectories show that the proposed approach can (1) estimate missing positions with high accuracy even on a small dataset, (2) recover missing positions even if the random missing rate up to 90%, (3) overcome the situation of the flight chain missing and block missing, which are the barriers of existing methods. The proposed approach serves as a post-processing procedure of air traffic data and can further provide high-quality data to other air traffic studies.

Multitask Feature Learning Meets Robust Tensor Decomposition for EEG Classification.

In this article, we study a tensor-based multitask learning (MTL) method for classification. Taking into account the fact that in many real-world applications, the given training samples are limited and can be inherently arranged into multidimensional arrays (tensors), we are motivated by the advantages of MTL, where the shared structural information among related tasks can be leveraged to produce better generalization performance. We propose a regularized tensor-based MTL method for joint feature selection and classification. For feature selection, we employ the Fisher discriminant criterion to both select discriminative features and control the within-class nonstationarity. For classification, we take both shared and task-specific structural information into consideration. We decompose the regression tensor for each task into a linear combination of a shared tensor and a task-specific tensor and propose a composite tensor norm. Specifically, we use the scaled latent trace norm for regularizing the shared tensor and the l1 -norm for task-specific tensor. Further, we give a computationally efficient optimization algorithm based on the alternating direction method of multipliers (ADMMs) to tackle the joint learning of discriminative features and multitask classification. The experimental results on real electroencephalography (EEG) datasets demonstrate the superiority of our method over the state-of-the-art techniques.

Hyperspectral Restoration and Fusion With Multispectral Imagery via Low-Rank Tensor-Approximation

Tensor-based fusion that couples the high spatial resolution of a multispectral image (MSI) to the high spectral resolution of a hyperspectral image (HSI) is considered. The fusion problem is first formulated mathematically as a convex optimization of a tensor trace norm imposing low-rank spatially as well as spectrally, with an alternating-directions optimization featuring linearization providing the solution. Although prior tensor-based fusion approaches typically resort to tensor decomposition, the proposed algorithm exploits ideas from the field of tensor completion to directly impose a low-rank property spatially and spectrally while avoiding the computationally complex patch clustering and dictionary learning common to competing fusion techniques. Additionally, small modifications to the basic optimization permit a fusion process robust to missing hyperspectral values such as those that can result from dead stripes in real hyperspectral sensors. The experimental evaluations on both synthetic imagery as well as real imagery demonstrate that the resulting low-rank tensor-approximation (LRTA) fusion algorithm preserves both spatial details and texture, yielding significantly improved image quality when compared to other state-of-the-art fusion methods as well as effective restoration under conditions of missing stripes within the HSI.

Bounds on amplitude-damping-channel discrimination

Amplitude-damping (AD) channels are good models for many physical scenarios, and so the development of protocols to discriminate between them is an important task in quantum information science. It is therefore important to bound the performance of such protocols. Since adaptivity has been shown to improve the performance of discrimination protocols, bounds on the distinguishability of AD channels must take this into account. In this paper, we use both channel simulation and a bound based on the diamond norm to significantly tighten the upper bound on the trace norm between the possible outputs of binary channel discrimination protocols acting on AD channels (and hence the lower bound on the error probability of such protocols). The diamond norm between any two AD channels is found analytically, giving the optimal error probability for a one-shot discrimination protocol. We also present a tighter lower bound on the achievable trace norm between protocol outputs (and a corresponding upper bound on the achievable error probability). The upper and lower bounds are compared with existing bounds and then applied to quantum hacking and biological quantum sensing scenarios.

Convergence Rates for the Quantum Central Limit Theorem

Various quantum analogues of the central limit theorem, which is one of the cornerstones of probability theory, are known in the literature. One such analogue, due to Cushen and Hudson, is of particular relevance for quantum optics. It implies that the state in any single output arm of an n-splitter, which is fed with n copies of a centred state rho with finite second moments, converges to the Gaussian state with the same first and second moments as rho . Here we exploit the phase space formalism to carry out a refined analysis of the rate of convergence in this quantum central limit theorem. For instance, we prove that the convergence takes place at a rate mathcal {O}left( n^{-1/2}right) in the Hilbert–Schmidt norm whenever the third moments of rho are finite. Trace norm or relative entropy bounds can be obtained by leveraging the energy boundedness of the state. Via analytical and numerical examples we show that our results are tight in many respects. An extension of our proof techniques to the non-i.i.d. setting is used to analyse a new model of a lossy optical fibre, where a given m-mode state enters a cascade of n beam splitters of equal transmissivities lambda ^{1/n} fed with an arbitrary (but fixed) environment state. Assuming that the latter has finite third moments, and ignoring unitaries, we show that the effective channel converges in diamond norm to a simple thermal attenuator, with a rate mathcal {O}Big (n^{-frac{1}{2(m+1)}}Big ). This allows us to establish bounds on the classical and quantum capacities of the cascade channel. Along the way, we derive several results that may be of independent interest. For example, we prove that any quantum characteristic function chi _rho is uniformly bounded by some eta _rho <1 outside of any neighbourhood of the origin; also, eta _rho can be made to depend only on the energy of the state rho .

Hypoenergetic and nonhypoenergetic digraphs

The energy of a graph G, E(G), is the sum of absolute values of the eigenvalues of its adjacency matrix. This concept was extended by Nikiforov to arbitrary complex matrices. Recall that the trace norm of a digraph D is defined as, N(D)=∑i=1nσi, where σ1≥⋯≥σn are the singular values of the adjacency matrix of D. In this paper we would like to present some lower and upper bounds for N(D). For any digraph D it is proved that N(D)≥rank(D) and the equality holds if and only if D is a disjoint union of directed cycles and directed paths. Finally, we present a lower bound on σ1 and N(D) in terms of the size of digraph D.

Lower bounds for the spectral norm of digraphs

Let D be a digraph with n vertices and σ1(D)≥σ2(D)≥⋯≥σn(D) the singular values of the adjacency matrix A of D. The spectral norm of D is σ1(D) and the trace norm of D is ‖D‖⁎=∑i=1nσi(D). In this paper we find lower bounds for the spectral norm of a digraph in terms of the structure of the digraph. Moreover, we introduce the concept of almost regular digraphs (extension of the well known almost regular graphs), and show that the lower bounds are attained precisely in almost regular digraphs. When we apply this theory to graphs, we recover well known lower bounds for the spectral radius of graphs. Also, we give a new upper bound for the trace norm of a digraph. Moreover, we determine the digraphs for which this bound is sharp: sink-source complete bipartite digraphs or symmetric balanced incomplete block designs (BIBD).

Partial Multi-Label Learning With Noisy Label Identification.

Partial multi-label learning (PML) deals with problems where each instance is assigned with a candidate label set, which contains multiple relevant labels and some noisy labels. Recent studies usually solve PML problems with the disambiguation strategy, which recovers ground-truth labels from the candidate label set by simply assuming that the noisy labels are generated randomly. In real applications, however, noisy labels are usually caused by some ambiguous contents of the example. Based on this observation, we propose a partial multi-label learning approach to simultaneously recover the ground-truth information and identify the noisy labels. The two objectives are formalized in a unified framework with trace norm and l1 norm regularizers. Under the supervision of the observed noise-corrupted label matrix, the multi-label classifier and noisy label identifier are jointly optimized by incorporating the label correlation exploitation and feature-induced noise model. Furthermore, by mapping each bag to a feature vector, we extend PML-NI method into multi-instance multi-label learning by identifying noisy labels based on ambiguous instances. A theoretical analysis of generalization bound and extensive experiments on multiple data sets from various real-world tasks demonstrate the effectiveness of the proposed approach.

Sparse Phase Retrieval Via PhaseLiftOff

The aim of sparse phase retrieval is to recover a $k$-sparse signal $\mathbf{x}_0\in \mathbb{C}^{d}$ from quadratic measurements $|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|^2$ where $\mathbf{a}_i\in \mathbb{C}^d, i=1,\ldots,m$. Noting $|\langle \mathbf{a}_i,\mathbf{x}_0\rangle|^2={\text{Tr}}(A_iX_0)$ with $A_i=\mathbf{a}_i\mathbf{a}_i^*\in \mathbb{C}^{d\times d}, X_0=\mathbf{x}_0\mathbf{x}_0^*\in \mathbb{C}^{d\times d}$, one can recast sparse phase retrieval as a problem of recovering a rank-one sparse matrix from linear measurements. Yin and Xin introduced PhaseLiftOff which presents a proxy of rank-one condition via the difference of trace and Frobenius norm. By adding sparsity penalty to PhaseLiftOff, in this paper, we present a novel model to recover sparse signals from quadratic measurements. Theoretical analysis shows that the solution to our model provides the stable recovery of $\mathbf{x}_0$ under almost optimal sampling complexity $m=O(k\log(d/k))$. The computation of our model is carried out by the difference of convex function algorithm (DCA). Numerical experiments demonstrate that our algorithm outperforms other state-of-the-art algorithms used for solving sparse phase retrieval.