Quantum information entropies of the eigenstates for the Pschl-Teller-like potential

Shannon entropy for lower position and momentum eigenstates of Ptschl-Teller-like potential is evaluated. Based on the entropy densities demonstrated graphically, we note that the wave through of the position information entropy density p （x） moves right when the potential parameter V1 increases and its amplitude decreases. However, its wave through moves left with the increase in the potential parameter 丨V2丨. Concerning the momentum information entropy density p（p）, we observe that its amplitude increases with increasing potential parameter V1, but its amplitude decreases with increasing丨V2丨. The Bialynicki-Birula-Mycielski （BBM） inequality has also been tested for a number of states. Moreover, there exist eigenstates that exhibit squeezing in the momentum information entropy. Finally, we note that position information entropy increases with V1, but decreases with 丨V2丨, However, the variation of momentum information entropy is contrary to that of the position information entropy.

Tetrapartite entanglement measures of W-class in noninertial frames

We present the entanglement measures of a tetrapartite W-class entangled system in a noninertial frame, where the transformation between Minkowski and Rindler coordinates is applied.Two cases are considered.First, when one qubit has uniform acceleration whilst the other three remain stationary.Second, when two qubits have nonuniform accelerations and the others stay inertial.The 1–1 tangle, 1–3 tangle, and whole entanglement measurements π4 and Π4, are studied and illustrated with graphics through their dependence on the acceleration parameter rd for the first case and rc and rd for the second case.It is found that the negativities(1–1 tangle and 1–3 tangle) and π-tangle decrease when the acceleration parameter rd or in the second case rc and rd increase, remaining a nonzero entanglement in the majority of the results.This means that the system will be always entangled except for special cases.It is shown that only the 1–1 tangle for the first case vanishes at infinite accelerations, but for the second case the 1–1 tangle disappears completely when r > 0.472473.An analytical expression for the von Neumann information entropy of the system is found and we note that it increases with the acceleration parameter.

Molecular-scale controllable conversion of biopolymers into hard carbons towards lithium and sodium ion batteries: A review

Hard carbons are widely investigated as potential anodes for lithium and sodium ion batteries owing to their internally well-tailored textures(closed pores and defects) and large microcrystalline interlayer spacing. The renewable biomass is a green and economically attractive carbon source to produce hard carbons. However, the chemical and structural complexity of biomass has plagued the understanding of evolution mechanism from organic precursors to hard carbons and the structure-property relationship.This makes it difficult to finely tune the microstructure of biomass-derived hard carbons, thus greatly restricting their high-performance applications. Most recently, the optimal utilization and controllable conversion of biomass-derived biopolymers(such as starch, cellulose and lignin) at the molecular level have become a burgeoning area of research to develop hard carbons for advanced batteries.Considering the principal source of carbonaceous materials is from biomass pyrolysis, we firstly overview the chemical structures and pyrolysis behaviors of three main biopolymers. Then, the controllable preparation of hard carbons using various physicochemical properties of biopolymers at the molecular level is systematically discussed. Furthermore, we highlight present challenges and further opportunities in this field. The Review will guide future research works on the design of sustainable hard carbons and the optimization of battery performance.

Tetrapartite entanglement measures of generalized GHZ state in the noninertial frames

Using a single-mode approximation, we carry out the entanglement measures, e.g., the negativity and von Neumann entropy when a tetrapartite generalized GHZ state is treated in a noninertial frame, but only uniform acceleration is considered for simplicity. In terms of explicit negativity calculated, we notice that the difference between the algebraic average π_(4) and geometric average Π_(4) is very small with the increasing accelerated observers and they are totally equal when all four qubits are accelerated simultaneously. The entanglement properties are discussed from one accelerated observer to all four accelerated observers. It is shown that the entanglement still exists even if the acceleration parameter r goes to infinity. It is interesting to discover that all 1-1 tangles are equal to zero, but 1-3 and 2-2 tangles always decrease when the acceleration parameter γ increases. We also study the von Neumann entropy and find that it increases with the number of the accelerated observers. In addition, we find that the von Neumann entropy S_(ABCDI), S_(ABCIDI), S_(ABICIDI) and S_(AIBICIDI) always decrease with the controllable angle θ, while the entropies S_(3-3 non), S_(3-2 non), S_(3-1 non) and S_(3-0 non) first increase with the angle θ and then decrease with it.

Tripartite Entanglement Measures of Generalized GHZ State in Uniform Acceleration

Using the single-mode approximation,we study entanglement measures including two independent quantities;i.e.,negativity and von Neumann entropy for a tripartite generalized Greenberger-Horne-Zeilinger(GHZ)state in noninertial frames.Based on the calculated negativity,we study the whole entanglement measures named as the algebraic average π3-tangle and geometric average Π3-tangle.We find that the difference between them is very small or disappears with the increase of the number of accelerated qubits.The entanglement properties are discussed from one accelerated observer and others remaining stationary to all three accelerated observers.The results show that there will always exist entanglement,even if acceleration r arrives to infinity.The degree of entanglement for all 1-1 tangles are alwa.ys equal to zero,but 1-2 tangles always decrease with the acceleration parameter r.We notice that the von Neumann entropy increases with the number of the accelerated observers and SκΙζΙ(κ,ζ∈(A,B,C)) first increases and then decreases with the acceleration parameter r.This implies that the subsystem ρκΙζΙ is first more disorder and then the disorder will be reduced as the acceleration parameter r increases.Moreover,it is found that the von Neumann entropies SABCI,SABICI and SAIBICI always decrease with the controllable angle θ,while the entropies of the bipartite subsystems S2-2non(two accelerated qubits),S2-1non(one accelerated qubit) and S2-0non(without accelerated qubit) first increase with the angle θ and then decrease with it.

Shannon information entropies for rectangular multiple quantum well systems with constant total lengths

We first study the Shannon information entropies of constant total length multiple quantum well systems and then explore the effects of the number of wells and confining potential depth on position and momentum information entropy density as well as the corresponding Shannon entropy.We find that for small full width at half maximum（FWHM） of the position entropy density,the FWHM of the momentum entropy density is large and vice versa.By increasing the confined potential depth,the FWHM of the position entropy density decreases while the FWHM of the momentum entropy density increases.By increasing the potential depth,the frequency of the position entropy density oscillation within the quantum barrier decreases while that of the position entropy density oscillation within the quantum well increases.By increasing the number of wells,the frequency of the position entropy density oscillation decreases inside the barriers while it increases inside the quantum well.As an example,we might localize the ground state as well as the position entropy densities of the1 st,2 nd,and 6 th excited states for a four-well quantum system.Also,we verify the Bialynicki–Birula–Mycieslki（BBM）inequality.

Entanglement measures of a new type pseudo-pure state in accelerated frames

In this work we analyze the characteristics of quantum entangleinent of the Dirac field in noninertial reference frames in the context of a new type pseudo-pure state, which is composed of the Bell states. This will help us to understand the relationship between the relativity and quantum information theory. Some states will be changed from entangled states into separable ones around the critical value F = 1/4, but there is no such a critical value for the variable y related to acceleration a. We find that the negativity Nas (ρ^TA AB 1) increases with F but decreases with the variable y、while the variation of the negativity Nb1b11(ρ^TB1 B1B11)is opposite to that of the negativity N AB1(ρ^TA AB1). We also study the von Neumann entropies S(ρAB1) and S(ρB1B11)、We find that the S(ρAB1) increases with variable y but S(ρB1B11) is independent of it. However, both S(ρAB1) and S(ρB1B11) first decreases with F and then increases with it. The concurrences C(ρAB1) and C(ρB1B11) are also discussed. We find that the former decreases with y while the latter increases with y but both of them first increase with F and then decrease with it.

Quantum teleportation and information splitting via four-qubit cluster state and a Bell state

Quantum teleportation provides a ＂bodiless＂ way of transmitting the quantum state from one object to another, at a distant location, using a classical communication channel and a previously shared entangled state. In this paper, we present a tripartite scheme for probabilistic teleportation of an arbitrary single qubit state, without losing the information of the state being teleported, via a fourqubit cluster state of the form ｜Ф） 1234 = α｜10000） ＋β｜1010） ＋ γ｜0101） - η｜1111）, as the quantum channel, where the nonzero real numbers α, β, γ, and η satisfy the relation ｜α｜^2 ＋ ｜β｜^2 ＋ ｜γ｜^2 ＋ ｜η｜^2 = 1. With the introduction of an auxiliary qubit with state ｜0}, using a suitable unitary transformation and a positive-operator valued measure （POVM）, the receiver can recreate the state of the original qubit. An important advantage of the teleportation scheme demonstrated here is that, if the teleportation fails, it can be repeated without teleporting copies of the unknown quantum state, if the concerned parties share another pair of entangled qubit. We also present a protocol for quantum information splitting of an arbitrary two-particle system via the aforementioned cluster state and a Bell-state as the quantum channel. Problems related to security attacks were examined for both the cases and it was found that this protocol is secure. This protocol is highly efficient and easy to implement.

Tetrapartite entanglem ent features of W-C lass state in uniform acceleration

Using the single-mode approximatiou,we first calculate entanglement measures such as negativity(1-3 and 11 tangles)and von Neumann entropy for a tetrapartite W-Class system in noninertial fiarne and then anal^e the whole entanglement measures,the residualπ4 and geometricП4 average of tangles.Notice that the difference betweenπ4 andП4 is very small or disappears with the increasing accelerated observers.The entanglement properties are compared among the different cases from oue accelerated observer to four accelerated observers.The results show that there still exists entanglement for the complete system even when accelemtion r tends to infinity.The degree of entanglement is disappeared for the 1-1 tangle case when the acceleration r>0.472473.We reexamine the Unriih effect in noninertiai frames.It is shown that the entanglement system in which only one qubit is accelerated is more robust than those eutangled systems in whicli two or three or four qubits are accelerated.It is also found that the von Neumann entropy S of the total system always increases with the increasing accelerated observers,but the Sκξand Sκζδwith two and three involved noniuertial qubits first increases and then decreases with the acceleration parameter r,but they are equal to constants 1 and 0.811278 respectively for zero involved iioniiiertial qubit.

Semi-exact Solutions of Konwent Potential

In this work we study the quantum system with the symmetric Konwent potential and show how to find its exact solutions. We find that the solutions are given by the confluent Heun function. The eigenvalues have to be calculated numerically because series expansion method does not work due to the variable z ≥ 1. The properties of the wave functions depending on the potential parameter A are illustrated for given potential parameters V_0 and a. The wave functions are shrunk towards the origin with the increasing |A|. In particular, the amplitude of wave function of the second excited state moves towards the origin when the positive parameter A decreases. We notice that the energy levels ε_i increase with the increasing potential parameter |A| ≥ 1, but the variation of the energy levels becomes complicated for |A| ∈(0, 1), which possesses a double well. It is seen that the energy levels ε_i increase with |A| for the parameter interval A ∈(-1, 0), while they decrease with |A| for the parameter interval A ∈(0, 1).