In this paper, some useful necessary and sufficient conditions for the unique solution of the generalized absolute value equation (GAVE) $Ax-B|x|=b$ with $A, Bin mathbb{R}^{ntimes n}$ from the optimization field are first presented, which cover the f
undamental theorem for the unique solution of the linear system $Ax=b$ with $Ain mathbb{R}^{ntimes n}$. Not only that, some new sufficient conditions for the unique solution of the GAVE are obtained, which are weaker than the previous published works.
The entanglement witness is an important and experimentally applicable tool for entanglement detection. In this paper, we provide a nonlinear improvement of any entanglement witness for $2otimes d$ quantum systems. Compared with any existing entangle
ment witness, the improved separability criterion only needs two more measurements on local observables. Detailed examples are employed to illustrate the efficiency of the nonlinear improvement for general, optimal and non-decomposable entanglement witnesses.
In this work, we present a quantum neighborhood preserving embedding and a quantum local discriminant embedding for dimensionality reduction and classification. We demonstrate that these two algorithms have an exponential speedup over their respectiv
ely classical counterparts. Along the way, we propose a variational quantum generalized eigenvalue solver that finds the generalized eigenvalues and eigenstates of a matrix pencil $(mathcal{G},mathcal{S})$. As a proof-of-principle, we implement our algorithm to solve $2^5times2^5$ generalized eigenvalue problems. Finally, our results offer two optional outputs with quantum or classical form, which can be directly applied in another quantum or classical machine learning process.
We study quantum anomaly detection with density estimation and multivariate Gaussian distribution. Both algorithms are constructed using the standard gate-based model of quantum computing. Compared with the corresponding classical algorithms, the res
ource complexities of our quantum algorithm are logarithmic in the dimensionality of quantum states and the number of training quantum states. We also present a quantum procedure for efficiently estimating the determinant of any Hermitian operators $mathcal{A}inmathcal{R}^{Ntimes N}$ with time complexity $O(polylog N)$ which forms an important subroutine in our quantum anomaly detection with multivariate Gaussian distribution. Finally, our results also include the modified quantum kernel principal component analysis (PCA) and the quantum one-class support vector machine (SVM) for detecting classical data.
The ultrafine entanglement witness, introduced in [F. Shahandeh, M. Ringbauer, J.C. Loredo, and T.C. Ralph, Phys. Rev. Lett. textbf{118}, 110502 (2017)], can seamlessly and easily improve any standard entanglement witness. In this paper, by combining
the constraint and the test operators, we rotate the hyperplane determined by the test operator and improve further the original ultrafine entanglement witness. In particular, we present a series of new ultrafine entanglement witnesses, which not only can detect entangled states that the original ultrafine entanglement witnesses cannot detect, but also have the merits that the original ultrafine entanglement witnesses have.
The entanglement detection via local measurements can be experimentally implemented. Based on mutually unbiased measurements and general symmetric informationally complete positive-operator-valued measures, we present separability criteria for bipart
ite quantum states, which, by theoretical analysis, are stronger than the related existing criteria via these measurements. Two detailed examples are supplemented to show the efficiency of the presented separability criteria.
The correlation matrices or tensors in the Bloch representation of density matrices are encoded with entanglement properties. In this paper, based on the Bloch representation of density matrices, we give some new separability criteria for bipartite a
nd multipartite quantum states. Theoretical analysis and some examples show that the proposed criteria can be more efficient than the previous related criteria.
By combining a parameterized Hermitian matrix, the realignment matrix of the bipartite density matrix $rho$ and the vectorization of its reduced density matrices, we present a family of separability criteria, which are stronger than the computable cr
oss norm or realignment (CCNR) criterion. With linear contraction methods, the proposed criteria can be used to detect the multipartite entangled states that are biseparable under any bipartite partitions. Moreover, we show by examples that the presented multipartite separability criteria can be more efficient than the corresponding multipartite realignment criterion based on CCNR, multipartite correlation tensor criterion and multipartite covariance matrix criterion.
In this paper, based on a matrix norm, we first present a ball of separable unnormalized states around the identity matrix for the bipartite quantum system, which is larger than the separable ball in Frobenius norm. Then the proposed ball is used to
get not only simple sufficient conditions for the separability of pseudopure states and the states with strong positive partial transposes, but also a separable ball centered at the identity matrix for the multipartite quantum system.
Based on the mutually unbiased bases, the mutually unbiased measurements and the general symmetric informationally complete positive-operator-valued measures, we propose three separability criteria for $d$-dimensional bipartite quantum systems, which
are more powerful than the corresponding ones introduced in [C. Spengler, M. Huber, S. Brierley, T. Adaktylos, and B.C. Hiesmayr, Phys. Rev. A textbf{86}, 022311 (2012); B. Chen, T. Ma, and S.M. Fei, Phys. Rev. A textbf{89}, 064302 (2014); B. Chen, T. Li, and S.M. Fei, arXiv:1406.7820v1 [quant-ph] (2014)]. Some states such as Werner states and Bell-diagonal states are used to further illustrate the efficiency of the presented criteria.
