arXiv subjects

Jianlian Cui

Publications and source records attributed to Jianlian Cui.

6 recordsLinked to original sources

arXiv2018

A Quantum Algorithm for Solving Linear Differential Equations: Theory and Experiment

We present and experimentally realize a quantum algorithm for efficiently solving the following problem: given an $N\times N$ matrix $\mathcal{M}$, an $N$-dimensional vector $\textbf{\emph{b}}$, and an initial vector $\textbf{\emph{x}}(0)$, obtain a target vector $\textbf{\emph{x}}(t)$ as a function of time $t$ according to the constraint $d\textbf{\emph{x}}(t)/dt=\mathcal{M}\textbf{\emph{x}}(t)+\textbf{\emph{b}}$. We show that our algorithm exhibits an exponential speedup over its classical counterpart in certain circumstances. In addition, we demonstrate our quantum algorithm for a $4\times4$ linear differential equation using a 4-qubit nuclear magnetic resonance quantum information processor. Our algorithm provides a key technique for solving many important problems which rely on the solutions to linear differential equations.

quant-ph↗

arXiv2017

Unitary similarity invariant function preservers of skew products of operators

Let ${\mathcal B}(H)$ denote the Banach algebra of all bounded linear operators on a complex Hilbert space $H$ with $\dim H\geq 3$, and let $\mathcal A$ and $\mathcal B$ be subsets of ${\mathcal B}(H)$ which contain all rank one operators. Suppose $F(\cdot )$ is a unitary invariant norm, the pseudo spectra, the pseudo spectral radius, the $C$-numerical range, or the $C$-numerical radius for some finite rank operator $C$. The structure is determined for surjective maps $Φ:{\mathcal A}\rightarrow \mathcal B$ satisfying $F(A^*B)=F(Φ(A)^*Φ(B))$ for all $A, B \in {\mathcal A}$. To establish the proofs, some general results are obtained for functions $F:{\mathcal F}_1(H) \cup \{0\} \rightarrow [0, +\infty)$, where ${\mathcal F}_1(H)$ is the set of rank one operators in ${\mathcal B}(H)$, satisfying (a) $F(μUAU^*)=F(A)$ for a complex unit $μ$, $A\in {\mathcal F}_1(H)$ and unitary $U \in {\mathcal B}(H)$ (b) for any rank one operator $X\in {\mathcal F}_1(H)$ the map $t\mapsto F(tX)$ on $[0, \infty)$ is strictly increasing, and (c) the set $\{F(X): X \in {\mathcal F}_1(H) \hbox{ and } \|X\| = 1\}$ attains its maximum and minimum.

math.FA↗

arXiv2015

Product of positive semi-definite matrices

It is known that every complex square matrix with nonnegative determinant is the product of positive semi-definite matrices. There are characterizations of matrices that require two or five positive semi-definite matrices in the product. However, the characterizations of matrices that require three or four positive semi-definite matrices in the product are lacking. In this paper, we give a complete characterization of these two types of matrices. With these results, we give an algorithm to determine whether a square matrix can be expressed as the product of $k$ positive semi-definite matrices but not fewer, for $k = 1,2,3,4,5$.

math.FA↗

arXiv2014

Preservers of Unitary Similarity Functions on Lie Products of Matrices

Denote by $M_n$ the set of $n\times n$ complex matrices. Let $f: M_n \rightarrow [0,\infty)$ be a continuous map such that $f(μUAU^*)= f(A)$ for any complex unit $μ$, $A \in M_n$ and unitary $U \in M_n$, $f(X)=0$ if and only if $X=0$ and the induced map $t \mapsto f(tX)$ is monotonic increasing on $[0,\infty)$ for any rank 1 nilpotent $X \in M_n$. Characterizations are given for surjective maps $ϕ$ on $M_n$ satisfying $f(AB-BA) = f(ϕ(A)ϕ(B)-ϕ(B)ϕ(A))$. The general theorem are then used to deduce results on special cases when the function is the pseudo spectrum and the pseudo spectral radius, that answers a question of Molnar raised at the 2014 CMS summer meeting.

math.FA↗

arXiv2013

Quantifying entanglement of arbitrary-dimensional multipartite pure states in terms of the singular values of coefficient matrices

The entanglement quantification and classification of multipartite quantum states are two important research fields in quantum information. In this work, we study the entanglement of arbitrary-dimensional multipartite pure states by looking at the averaged partial entropies of various bipartite partitions of the system, namely, the so-called Manhattan distance ($l_1$ norm) of averaged partial entropies (MAPE), and it is proved to be an entanglement measure for pure states. We connected the MAPE with the coefficient matrices, which are important tools in entanglement classification and reexpressed the MAPE for arbitrary-dimensional multipartite pure states by the nonzero singular values of the coefficient matrices. The entanglement properties of the $n$-qubit Dicke states, arbitrary-dimensional Greenberger-Horne-Zeilinger states, and $D_3^n$ states are investigated in terms of the MAPE, and the relation between the rank of the coefficient matrix and the degree of entanglement is demonstrated for symmetric states by two examples.

quant-ph↗

arXiv2013

Classification of arbitrary-dimensional multipartite pure states under stochastic local operations and classical communication using the rank of coefficient matrix

We study multipartite entanglement under stochastic local operations and classical communication (SLOCC) and propose the entanglement classification under SLOCC for arbitrary-dimensional multipartite ($n$-qudit) pure states via the rank of coefficient matrix, together with the permutation of qudits. The ranks of the coefficient matrices have been proved to be entanglement monotones. The entanglement classification of the $2 \otimes 2 \otimes 2 \otimes 4$ system is discussed in terms of the generalized method, and 22 different SLOCC families are found.

quant-ph↗