An entanglement measure for n-qubits

Recently, Coffman, Kundu, and Wootters introduced the residual entanglement for three qubits to quantify the three-qubit entanglement in Phys. Rev. A 61, 052306 (2000). In Phys. Rev. A 65, 032304 (2007), we defined the residual entanglement for $n$ qubits, whose values are between 0 and 1. In this paper, we want to show that the residual entanglement for $n$ qubits is a natural measure of entanglement by demonstrating the following properties. (1). It is SL-invariant, especially LU-invariant. (2). It is an entanglement monotone. (3). It is invariant under permutations of the qubits. (4). It vanishes or is multiplicative for product states.Comment: 16 pages, no figure

Analytical modeling of surface roughness in precision grinding of particle reinforced metal matrix composites considering nanomechanical response of material

Grinding is usually applied for particle reinforced metal matrix composites (PRMMCs) to achieve high ground surface quality. However, the surface quality especially surface roughness is difficult to predict theoretically due to different mechanical properties of two or more phases inside the PRMMCs. In this study, an analytical model of the surface roughness of ground PRMMCs is developed based on an undeformed chip thickness model with Rayleigh probability distribution by considering the different removal mechanism of metal matrix and reinforcement particles in grinding. GT35, a typical kind of steel based metal matrix composite reinforced with TiC particles is investigated as an example. Nanoindentation experiments are employed for the investigation of nanomechanical properties and cracking behavior of GT35 and the nanoindentation results are integrated in the model. Single factor surface grinding experiments of GT35 are also carried out to understand the material removal mechanism of GT35 and validate this novel surface roughness prediction model. The predicted surface roughness from this model shows good agreement with the experimental results

SLOCC invariant and semi-invariants for SLOCC classification of four-qubits

We show there are at least 28 distinct true SLOCC entanglement classes for four-qubits by means of SLOCC invariant and semi-invariants and derive the number of the degenerated SLOCC classes for n-qubits.Comment: 22 pages, no figures, 9 tables, submit the paper to a journa

The Simple Criteria of SLOCC Equivalence Classes

We put forward an alternative approach to the SLOCC classification of entanglement states of three-qubit and four-qubit systems. By directly solving matrix equations, we obtain the relations satisfied by the amplitudes of states. The relations are readily tested since in them only addition, subtraction and multiplication occur.Comment: The original version was submitted to PRA in Feb. 2005, the paper No. is AA10020. 14 pages for the present version. No figure

No-cloning of nonorthogonal states does not require inner product preserving

The no-cloning theorem says there is no quantum copy machine which can copy any one-qubit state. Inner product preserving was always used to prove the no-cloning of nonorthogonal states. In this paper we show that the no-cloning of nonorthogonal states does not require inner product preserving and discuss the minimal properties which a linear operator possesses to copy two different states at the same device. In this paper, we obtain the following necessary and sufficient condition. For any two different states ∣ψ〉 = a∣0〉+b∣1〉∣ψ〉=a∣0〉+b∣1〉 and ∣ϕ〉 = c∣0〉+d∣1〉∣ϕ〉=c∣0〉+d∣1〉, assume that a linear operator LL can copy them, that is, L(∣ψ,0〉) = ∣ψ,ψ〉L(∣ψ,0〉)=∣ψ,ψ〉 and L(∣ϕ,0〉) = ∣ϕ,ϕ〉L(∣ϕ,0〉)=∣ϕ,ϕ〉. Then the two states are orthogonal if and only if L(∣0,0〉)L(∣0,0〉) and L(∣1,0〉)L(∣1,0〉) are unit length states. Thus we only need linearity and that L(∣0,0〉)L(∣0,0〉) and L(∣1,0〉)L(∣1,0〉) are unit length states to prove the no-cloning of nonorthogonal states. It implies that inner product preserving is not necessary for the no-cloning of nonorthogonal states.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/87751/2/082102_1.pd

Fundamental Cycles and Graph Embeddings

In this paper we present a new Good Characterization of maximum genus of a graph which makes a common generalization of the works of Xuong, Liu, and Fu et al. Based on this, we find a new polynomially bounded algorithm to find the maximum genus of a graph

Fixed-point Quantum Search for Different Phase Shifts

Grover recently presented the fixed-point search algorithm. In this letter, we study the fixed-point search algorithm obtained by replacing equal phase shifts of $\pi /3$ by different phase shifts.Comment: 8 page

The Spatial Associations between Crime and Economy in Chicago 2015-2020

The severity of the crime is often the most intuitive reflection of whether a region is safe and the top factor for the public when evaluating a region. Economist\u27s list of the safest cities in seven major North American cities, Chicago was ranked at six, just above Dallas. Chicago scored the lowest in personal security, which is closely tied to the crime. Against the backdrop of higher unemployment and prices, this study is interested in how property-based crimes are related to the economic decline in Chicago geographically. The study used the heterogeneity analysis tool Geodetector to investigate the correlation between property-related crimes and selected economic indicators; examine the interactions of economic indicators on the selected crimes and estimate the likelihood of occurrence of the crimes base on classifications of the economic indicators. The results show that robbery, burglary, and Motor Vehicle Theft are less likely to occur in areas with stable economic environments such as high income, low housing vacancy rates, and low unemployment rates than in other areas, but not in Larceny. The spatial distribution of Larceny is not highly correlated with the distribution of either Median Household Income, Vacant Rate, or Unemployment Rate, but is correlated with the Median Household Income & housing vacancy rate and housing vacancy rate & unemployment rate. but is highly correlated with the spatial distribution of the combination of Median Household Income & housing vacancy rate and housing vacancy rate & unemployment rate. Higher burglary crime risk is likely associated with higher unemployment rates whereas robbery and Motor Vehicle Theft have a greater association with the unemployment rate and housing vacancy than with the other factors. The rates of vacant homes and unemployment are higher from 2015 to 2020. There is some level of association between higher crime risk and low income, high vacancy, and high unemployment rates in general, the increase in unemployment rates in 2020 is the dominant factor associated with high crime risk. Geodetector is not only able to identify the spatial associations between crimes and individual EIs but also how the interactions between EIs would affect crime risk at different levels