Linear preservers and quantum information science

Let $m,n\ge 2$ be positive integers, $M_m$ the set of $m\times m$ complex matrices and $M_n$ the set of $n\times n$ complex matrices. Regard $M_{mn}$ as the tensor space $M_m\otimes M_n$. Suppose $|\cdot|$ is the Ky Fan $k$-norm with $1 \le k \le mn$, or the Schatten $p$-norm with $1 \le p \le \infty$ ($p\ne 2$) on $M_{mn}$. It is shown that a linear map $\phi: M_{mn} \rightarrow M_{mn}$ satisfying $$|A\otimes B| = |\phi(A\otimes B)|$$ for all $A \in M_m$ and $B \in M_n$ if and only if there are unitary $U, V \in M_{mn}$ such that $\phi$ has the form $A\otimes B \mapsto U(\varphi_1(A) \otimes \varphi_2(B))V$, where $\varphi_i(X)$ is either the identity map $X \mapsto X$ or the transposition map $X \mapsto X^t$. The results are extended to tensor space $M_{n_1} \otimes ... \otimes M_{n_m}$ of higher level. The connection of the problem to quantum information science is mentioned.Comment: 13 page

Physical transformations between quantum states

Given two sets of quantum states {A_1, ..., A_k} and {B_1, ..., B_k}, represented as sets of density matrices, necessary and sufficient conditions are obtained for the existence of a physical transformation T, represented as a trace-preserving completely positive map, such that T(A_i) = B_i for i = 1, ..., k. General completely positive maps without the trace-preserving requirement, and unital completely positive maps transforming the states are also considered

Human Microbe-Disease Association Prediction With Graph Regularized Non-Negative Matrix Factorization

A microbe is a microscopic organism which may exists in its single-celled form or in a colony of cells. In recent years, accumulating researchers have been engaged in the field of uncovering microbe-disease associations since microbes are found to be closely related to the prevention, diagnosis, and treatment of many complex human diseases. As an effective supplement to the traditional experiment, more and more computational models based on various algorithms have been proposed for microbe-disease association prediction to improve efficiency and cost savings. In this work, we developed a novel predictive model of Graph Regularized Non-negative Matrix Factorization for Human Microbe-Disease Association prediction (GRNMFHMDA). Initially, microbe similarity and disease similarity were constructed on the basis of the symptom-based disease similarity and Gaussian interaction profile kernel similarity for microbes and diseases. Subsequently, it is worth noting that we utilized a preprocessing step in which unknown microbe-disease pairs were assigned associated likelihood scores to avoid the possible negative impact on the prediction performance. Finally, we implemented a graph regularized non-negative matrix factorization framework to identify potential associations for all diseases simultaneously. To assess the performance of our model, cross validations including global leave-one-out cross validation (LOOCV) and local LOOCV were implemented. The AUCs of 0.8715 (global LOOCV) and 0.7898 (local LOOCV) proved the reliable performance of our computational model. In addition, we carried out two types of case studies on three different human diseases to further analyze the prediction performance of GRNMFHMDA, in which most of the top 10 predicted disease-related microbes were verified by database HMDAD or experimental literatures

Conductivity measurement of ionic liquids confined in the nanopores of metal–organic frameworks: a case study for [BMIM][TFSI] in HKUST-1

Nanoporous materials like metal–organic frameworks (MOFs) attract considerable attention as porous host for electrolytes like ionic liquids (ILs). The conductivity and mobility of the ions in the pores are among the key properties and their experimental quantification is of paramount importance. Here, three different approaches for the quantification of the ion conductivity of IL@MOF via electrochemical impedance spectroscopy (EIS) are compared: the material in the form of IL-impregnated MOF powders pressed into pellets between two planar electrodes, MOF films grown on substrates with deposited electrodes loaded with IL by impregnation, and the IL-loaded MOF films where excess IL is removed. Contact-angle measurements and EIS data show that the excess IL on the outer MOF surface of the film or pellet results in apparent conductivities, larger than the intrinsic conductivity of the IL@MOF. Removing the excess IL enables the experimental quantification of the intrinsic IL@MOF conductivity

On the second-order zero differential spectra of some power functions over finite fields

Boukerrou et al. (IACR Trans. Symmetric Cryptol. 2020(1), 331-362) introduced the notion of Feistel Boomerang Connectivity Table (FBCT), the Feistel counterpart of the Boomerang Connectivity Table (BCT), and the Feistel boomerang uniformity (which is the same as the second-order zero differential uniformity in even characteristic). FBCT is a crucial table for the analysis of the resistance of block ciphers to power attacks such as differential and boomerang attacks. It is worth noting that the coefficients of FBCT are related to the second-order zero differential spectra of functions. In this paper, by carrying out certain finer manipulations of solving specific equations over the finite field $\mathbb{F}_{p^n}$, we explicitly determine the second-order zero differential spectra of some power functions with low differential uniformity, and show that our considered functions also have low second-order zero differential uniformity. Our study pushes further former investigations on second-order zero differential uniformity and Feistel boomerang differential uniformity for a power function $F$