Optimization of Bell's Inequality Violation For Continuous Variable Systems

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半线性椭圆方程

µ − a + β . K•§ (0.12) – k˜‡
ë•©zµ [1] L. Caﬀarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53 (1984) 259–275. [2] K. Chou, C. Chu, On the best constant for a weighted Sobolev–Hardy inequality, J. Lond. Math. Soc. (2) 48 (1993) 137–151. [3] F. Catrina, Z. Wang, On the Caﬀarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of external functions, Comm. Pure Appl. Math. 54 (2) (2001) 229–258. [4] X.J. Huang, X.P. Wu, C.L. Tang, Multiple positive solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal. 74 (2011) 2602–2611 [5] L. Ding, C.L. Tang, Existence and multiplicity of positive solutions for a class of semilinear elliptic equations involving Hardy term and Hardy-Sobolev critical exponents, J. Math. Anal. Appl. 339 (2008) 1073–1083. [6] L. Huang, X.P. Wu, C.L. Tang, Existence and multiplicity of solutions for semilinear elliptic equations with critical weighted Hardy-Sobolev exponents, Nonlinear Anal. 71 (2009) 1916–1924. [7] P. H. Rabinowitz. Minimax methods in critical point theory with applications to diﬀerential equations, Conference Board of the Mathematical Sciences, vol. 65 Providence, RI: American Mathematical Society, 1986. [8] W. Rudin, Real and Complex Analysis, 3rd ed., McGraw-Hill, New York, 1987. [9] H. Brezis, L. Nirenberg, Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math. 36 (4) (1983) 437–477. [10] D. Kang, G. Li, S. Peng, Positive solutions and critical dimensions for the elliptic problems involving the Caﬀarelli-Kohn-Nirenberg inequalities, J. Jilin. Univ. Sci. 46 (2008) 423–427.

基于Bellman不等式的一类二阶微分方程的解的有界性

u ( t ) ≤ c 2 + ∫0 f ( s ) ds + ∫0 f ( s ) c 2 + ∫0 f ( u ) du exp ∫s f ( u ) du ds
t t s t
(
)
1 2
(23)
定理 7：若方程
y ′′ ( t ) + a 2 + φ ( t ) + ψ ( t ) y ( t ) = 0
t 1 2 a2 2 y′ (t ) + y ( t ) + ∫0 φ ( s ) y ( s ) y ′ ( s ) ds = c1 2 2
(25)
因而
y ′2 ( t ) ≤ 2c1 + 2 ∫ φ ( s ) y ( s ) y ′ ( s ) ds
t 0
(26)
由定理 3 以及引理 5 可得
u ≥ 0, v ≥ 0, c > 0
t
( 2′ )
则
u 2 ≤ c + ∫0 uvdt1
u ≤ c+
1 t vdt1 2 ∫0
利用该引理，证明了 y ′′ ( t ) + A ( t ) y ( t ) = 0 这类二阶微分方程的解的有界性，并给出了相关的结论。在 2014 年，Qusuay H. Alqifiary 和 Soon-Mo Jung 在文[2]中利用 Bellman 引理证明了一类二阶微分方程的 Hyers-Ulam 稳定性结论。而本文则是用 Bellman 引理 1 和引理 5，在欧阳亮[1]的基础上简化其证明过程，同时也减弱了文[1]中对二阶微分方程的系数 A ( t ) 的要求，即只需要文[1]中定理一中的条件(i)。除此之外，本文也与文[2]中的定理 2 中对 ψ ( t ) 的要求不同，即不需要当 t → ∞ 时，ψ ( t ) → 0 。从而对于证明微分方程的 Hyers-Ulam 稳定性起到了重要作用。

Application of particle swarm optimization to model the phase equilibrium of

derivative free optimization algorithms have emerged [5]. Particle swarm optimization (PSO) is a relatively recently population-based stochastic global optimization algorithm [6]. This technique is quite popular within the research community as a problem-design methodology and solver, because of its versatility and ability to ﬁnd optimal solutions in complex multimodal search spaces applied to non-differentiable cost functions [7]. As described by Eberhart and Kennedy, the PSO is inspired by the ability of ﬂocks of birds, schools of ﬁsh, and herds of animals to adapt to their environment, ﬁnd rich sources of food, and avoid predators by implementing an information sharing approach [8]. In PSO, a set of randomly generated solutions propagates in the design space toward the optimum solution over a number of iterations based on large amount of information about the design space that is assimilated and shared by all members of the swarm [9]. In this work, six binary gas–solid phase systems, and ﬁve binary vapor–liquid phase systems containing supercritical CO2 were evaluated using a PSO algorithm. The complete program was developed in Matlab, and was used to calculate the binary interaction parameters of complex mixtures by minimization of the difference between calculated and experimental data. 2. Particle swarm optimization

高三现代科技前沿探索英语阅读理解20题

高三现代科技前沿探索英语阅读理解20题1<背景文章>Artificial intelligence (AI) is rapidly transforming the field of healthcare. In recent years, AI has made significant progress in various aspects of medical care, bringing new opportunities and challenges.One of the major applications of AI in healthcare is in disease diagnosis. AI-powered systems can analyze large amounts of medical data, such as medical images and patient records, to detect diseases at an early stage. For example, deep learning algorithms can accurately identify tumors in medical images, helping doctors make more accurate diagnoses.Another area where AI is making a big impact is in drug discovery. By analyzing vast amounts of biological data, AI can help researchers identify potential drug targets and design new drugs more efficiently. This can significantly shorten the time and cost of drug development.AI also has the potential to improve patient care by providing personalized treatment plans. Based on a patient's genetic information, medical history, and other factors, AI can recommend the most appropriate treatment options.However, the application of AI in healthcare also faces some challenges. One of the main concerns is data privacy and security. Medicaldata is highly sensitive, and ensuring its protection is crucial. Another challenge is the lack of transparency in AI algorithms. Doctors and patients need to understand how AI makes decisions in order to trust its recommendations.In conclusion, while AI holds great promise for improving healthcare, it also poses significant challenges that need to be addressed.1. What is one of the major applications of AI in healthcare?A. Disease prevention.B. Disease diagnosis.C. Health maintenance.D. Medical education.答案：B。

高维Bell不等式及其最大违背的开题报告

高维Bell不等式及其最大违背的开题报告一、研究背景贝尔不等式是描述量子力学的本质区别于经典力学的重要工具之一。

20世纪60年代，约翰·贝尔提出了著名的贝尔不等式，在随后的几十年里，它成为了物理学家们探索量子力学与经典力学差异的基础。

然而，贝尔不等式只适用于描述两个物理系统之间的关系，并且只能适用于二元判定问题，即问题只有两个可能的答案。

因此，在研究多个物理系统之间的关系时，需要使用高维Bell不等式。

二、研究内容高维Bell不等式是一种用于描述多个量子系统之间关系的不等式。

它是通过推广贝尔不等式得到的，可以用于描述三个及以上的物理系统之间的关系。

高维Bell不等式和贝尔不等式一样，用于描述在类似Einstein-Podolsky-Rosen实验中，经典理论和量子力学之间的不同。

研究高维Bell不等式的过程需要用到数学工具，如Hilbert空间、线性算子、张量积等概念，以及离散数学中的图论、组合等知识。

通过这些工具，可以推导出不同维度下的Bell不等式及其最大违背。

三、研究意义高维Bell不等式及其最大违背的研究，对于深入理解量子力学与经典力学之间差异具有重要意义。

同时，高维Bell不等式对于实验检验量子力学的基本原理和理论的准确性具有重要意义。

研究结果还可以为量子信息科学、量子计算等领域的发展提供理论支持。

四、研究方法本文主要采用文献调研和数理推导相结合的方法，通过分析已有文献中的内容，运用数学工具对高维Bell不等式进行推导，探究其理论基础和数学性质，并分析其在实际应用中的意义。

五、研究进展至今，高维Bell不等式及其最大违背的研究仍然是一个活跃的领域。

目前已经有不少关于高维Bell不等式的研究成果，如高斯态的Bell不等式、含有n条臂的若干种类型的Bell不等式、三个系综的高维正态通信协议等等。

这些成果为高维Bell不等式的进一步探究提供了理论依据和实验验证的基础。

六、结论高维Bell不等式及其最大违背的研究，是探究量子世界和经典世界差异的重要研究方向。

Existence of Infinitely Many Solutions for a Quasilinear Elliptic Problem on Time Scales

Let f : T → R and t ∈ Tk (assume t is not left-scattered if t = sup T). We deﬁne f △(t) to be the number (provided it exists) such that given any ǫ > 0 there is a neighborhood U of t such that
arXiv:0705.3674v1 [math.AP] 24 May 2007
Existence of Inﬁnitely Many Solutions for a Quasilinear Elliptic Problem on Time Scales
Moulay Rchid Sidi Ammi sidiammi@mat.ua.pt
2 Preliminary results on time scales
We begin by recalling some basic concepts of time scales. Then, we prove some preliminary results that will be needed in the sequel.
Delﬁm F. M. Torres delfim@mat.ua.pt
Department of Mathematics
University of Aveiro 3810-193 Aveiro, Portugal
Abstract
We study a boundary-value quasilinear elliptic problem on a generic time scale. Making use of the ﬁxed-point index theory, suﬃcient conditions are given to obtain existence, multiplicity, and inﬁnite solvability of positive solutions.

量子物理的基础及其光学实验

量子物理的基础及其光学实验张开银　王树春　赵丽娟　黄　晖　张光寅　许京军(南开大学物理科学学院光子学中心,天津,300071)摘要:主要针对长期争论的量子力学的基本概念问题,即量子纠缠态,从实验方面对其做了回顾。

重点论述了近期光学实验在量子纠缠态方面的一系列新的进展,主要介绍了利用自发参量下转换产生两光子和三光子纠缠态的实验。

关键词:双缝干涉　量子纠缠　EPR Bell 态Optical experiments and the foundations of quantum physicsZhang Kaiyin ,Wang Shuchun ,Zhao Lijuan ,Huang Hui ,Zhang Guangyin ,Xu Jingjun(Photonics Research Center ,Institute of Physics ,Nankai University ,T ianjing ,300071)Abstract :This paper gives a review of experiments related to the foundations of quantum physics ,i.e.quantum entanglement.A series of late developments in the field of quantum entanglement ,especially tw o 2photon and three 2photon entanglement experiments which are based on the process of spontaneous parametric down con 2version ,is introduced.K ey w ords :double slit interference quantum entanglement EPR Bell state引言20世纪物理科学的革命是建立了相对论和量子力学,从根本上改变了人们对时间、空间、物质以及运动的概念。

多体纠缠态对贝尔不等式的不违背

多体纠缠态对贝尔不等式的不违背
量子纠缠在量子信息学中是一种切实的物理资源,在量子信息处理过程中起着至关重要的作用。非局域性是量子力学中一个十分深刻的概念,贝尔不等式在量子非局域性和纠缠性的研究中起着核心作用,并且在量子信息处理中有很多应用。
量子力学本质上是非局域的,在对非局域复合量子系统进行局域测量后,它与局域隐变量理论的不相容可以用贝尔不等式来揭示。局域实在理论限制了多粒子系统测量的统计关联性,它们以Bell 不等式的形式存在。
最后,得出了并非所有的纠缠纯态都违背Bell不等式的结论。第一章中我们介绍了量子信息的一些基础知识。
第二章中我们分别讨论并详细计算了3,4,5,L,N量子比特W态在局域坐标系（7） ?xj,y??j（8）下对Bell不等式的不违背,以及它们在局域坐标系（7） ?y?j,z?j （8）、（7） ?xj,z?j（8）下对Bell不等式的违背。第三章中我们对局域坐标系进行欧拉旋转,并详细计算和讨论了3量子比特、4量子比特W态在局域坐标系经过欧拉旋转后对Bell不等式的违背。
在局域实在理论中,测量结果是由粒子在观测之前和不受观测影响的隐藏性质决定的。即在局域实在理论中,在一个位置得到的结果不依赖于在类空分离时所做的任何测量。

关于一个改进的Hardy-Hilbert不等式

关于一个改进的Hardy-Hilbert不等式杨乔顺;龙萍;周昱【期刊名称】《大学数学》【年(卷),期】2012(028)001【摘要】通过建立权系数并利用改进了的H lder不等式,得到了一个新的改进的Hardy-Hilbert不等式.当p=2时,便得到了经典的Hilbert不等式的一个改进.%It is shown that a new improvement of Hardy-Hilbert inequality can be established by introducing a weight coefficient and by means of a sharpening of Holder＇ s inequality, for case p = 2 , a sharp result of the classical Hilbert inequality is obtained.【总页数】4页(P107-110)【作者】杨乔顺;龙萍;周昱【作者单位】吉首大学师范学院数计系,湖南吉首416000;吉首大学师范学院数计系,湖南吉首416000;吉首大学师范学院数计系,湖南吉首416000【正文语种】中文【中图分类】O178【相关文献】1.关于Hardy-Hilbert不等式的一个改进 [J], 陈小雨;高雪梅2.一个已加强的Hardy-Hilbert不等式的改进 [J], 王文杰;陈小雨;谭立3.Hardy-Hilbert 不等式一个新的改进 [J], 赵玉中;郭继峰4.关于一个Hardy-Hilbert型不等式的改进与推广 [J], 罗静;隆建军5.Hardy-Hilbert不等式一个新的改进 [J], 王祥;张太忠因版权原因，仅展示原文概要，查看原文内容请购买。

Optimization problem solving

专利名称：Optimization problem solving发明人：Trnka, Pavel,Pekar, Jaroslav申请号：EP11157244.2申请日：20110307公开号：EP2498189A1公开日：20120912专利内容由知识产权出版社提供专利附图：摘要：Optimization problem solving is described herein. For example, one or more method embodiments include assigning one of a number of predefined values to each of a number of shadow prices of the system, distributing the assigned predefined shadow price values to a number of sub-problems, wherein each sub-problem is associated withone of a number of subsystems of the system, performing an analysis, wherein the analysis includes determining a parametric solution and a region of validity for each of the number of sub-problems, determining an intersection of the regions of validity of all the parametric solutions, determining whether the optimization problem is solved from the parametric solutions of the number of sub-problems, determining one or more shadow price updates based on the parametric solutions, and distributing the updated shadow prices to sub-problems having a region of validity that does not include the updated shadow prices, and repeating the analysis using the updated shadow prices until the optimization problem is solved from the parametric solutions of the number of sub-problems.申请人：Honeywell spol s.r.o.地址：V Parku 2325/18 14800 Prague 4 CZ国籍：CZ代理机构：Buckley, Guy Julian更多信息请下载全文后查看。

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Theoretical Physics Institute, Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1 and
Departmentof Physics, Technion-Israel Institute of Technology, Haifa 32000, Israel
∞
Πx =
(|E O| + |O E|) dq
0
∞
Πy = i (|O E| − |E O|) dq.
(4)
0
The components of Π(i) = (Πx(i), Πy(i), Πz(i)) (i = 1, 2) satisfy the standard SU (2) algebra, and the square of each component is equal to the identity operator.
tional representatives for the operator Πz can be easily veriﬁed by considering their matrix elements, e.g., with respect to the number states.
(b) For the x and y components of Π(i) we choose
the Wigner function is not a prerequisite to BIQV. In this work we demonstrate the pres-
ence of conﬁgurational parameters controlling the extent of BIQV, which are not present
e.g. for delineating a dichotomic variable (even/odd parity in our case) as the veriﬁcation
involves diﬀerent observables (exact photon number in one case or simply even/odd number
pinned with a theory employing local hidden variables. They gained particular attention
with the demonstration, ﬁrst achieved by Banaszek and Wodkiewicz [4], that negativity of
continuous variable cases [9,10].
The state under study, |ζ (i.e. |T MSV ), involves two beams each headed in a diﬀerent
dir√ection and 1/ 2(q +√ ip)
delineated and a+ =
(a) The parity operator (the superscript i =1,2 refers to the channel - it is omitted where clarity allows),
∞
Πz = dq (|E E| − |O O|) ≡ IE − IO,
0
|E = √1 [|q + | − q ], |O = √1 [|q − | − q ],
∞
∞
IE = |2n 2n|, IO = |2n + 1 2n + 1|,
(3)
n=0
n=0
2
We refer to these two representations of IE, IO given in Eq.(2) and Eq.(3) as two conﬁgurational representations of the operators. The proof of the equivalence of the two conﬁgura-
BIQV for diﬀerent conﬁgurational parameterizations (one was obtained earlier in [5]) and
give the transformation between them. Our representative of Bells’ inequalities is the CHSH
|ζ = S(ζ)|00 ; S(ζ) ≡ exp[ζ(a+b+ − ab)],
(1)
where ζ is real. The CHSH inequality involves the so-called Bell operator B, [11], which, for our case, where parity is the dynamical variable, is given via the following observables:
of photons in another for the cases we consider below). BIQV for these states were observed
experimentally in several laboratories [8,7] within the general study of the problem for
inequality [6] and the observable is the parity rather than the spin. The novel conﬁgura-
tional dependence introduced herewith should be important when such states are employed
arXiv:quant-ph/0308063v2 21 Oct 2003
Optimization of Bell’s Inequality Violation For Continuous Variable Systems
G. Gour∗, F. C. Khanna†, A. Mann‡, M. Revzen§
in the Bohm-like entangled state (since that state involves only two levels per subsystem,
and therefore only orientational parameters are available for optimization of BIQV.) This
Abstract
Two mode squeezed vacuum states allow Bell’s inequality violation (BIQV) for all non-vanishing squeezing parameter (ζ). Maximal violation occurs at ζ → ∞ when the parity of either component averages to zero. For a given entangled two spin system BIQV is optimized via orientations of the operators entering the Bell operator (cf. S. L. Braunstein, A. Mann and M. Revzen: Phys. Rev. Lett. 68, 3259 (1992)). We show that for ﬁnite ζ in continuous variable systems (and in general whenever the dimensionality of the subsystems is greater than 2 ) additional parameters are present for optimizing BIQV. Thus the expectation value of the Bell operator depends, in addition to the orientation parameters, on conﬁguration parameters. Optimization of these conﬁgurational parameters leads to a unique maximal BIQV that depends only on ζ. The conﬁgurational parameter variation is used to show that BIQV relation to entanglement is, even for pure state, not monotonic.
by√having the operators of the ﬁrst 1/ 2(q − ip) and the second channel
channel by b =
de√signatand
b+ = 1/ 2(q′ − ip′). The TMSV state, |ζ , is given by,
Two mode squeezed vacuum states (TMSV) generated via pulsed non-degenerate para-
metric ampliﬁers are of interest following the pioneering study of Grangier et al. [1]. Such
particular. The latter’s relevance to the Bell inequality problem was stressed originally by