Tripartite Quantum State Sharing
利用冯· 诺依曼熵获得最大纠缠态的形式
利用冯诺依曼熵获得最大纠缠态的形式朱孟正;赵春然;李洪俊;张东杰【摘要】纠缠在量子信息处理中有许多重要的应用,正如Bell态对量子通信的实施是必不可少的.考虑如何得到Bell态,本文提出了一种用冯· 诺依曼熵求解二体或三体系统中最大纠缠态表示形式的方法.计算二体或三体系统的量子态的冯· 诺依曼熵,并将约化密度算符与用Bloch矢量表示的密度算符进行比较.根据密度算符具有正的、厄密性的特点,得到了最大纠缠态解析式,如Bell态和GHZ态.【期刊名称】《吉林师范大学学报（自然科学版）》【年(卷),期】2018(039)002【总页数】5页(P78-82)【关键词】冯·诺依曼熵;纠缠态;密度算符【作者】朱孟正;赵春然;李洪俊;张东杰【作者单位】淮北师范大学物理与电子信息学院,安徽淮北235000;淮北师范大学信息学院,安徽淮北235000;淮北师范大学物理与电子信息学院,安徽淮北235000;淮北师范大学物理与电子信息学院,安徽淮北235000;淮北师范大学物理与电子信息学院,安徽淮北235000【正文语种】中文【中图分类】O413.10 IntroductionThe key feature of quantum mechanics that lies behind quantum information theory is quantum entanglement.Quantum entanglement refers to correlations between the results of measurements made on distinct subsystems of a composite system that can not be explained in terms of standard statistical correlations between classical properties inherent in each subsystem.For the bipartite quantum systems，a correlation between two subsystems is simply the statement that if a measurement of one subsystem yields the result A then a measurement on the second subsystem will yield the result B with some probability.Perfect correlation occurs when the second result is certain，given the outcome of the first[1-2].It has become clear that entanglement is a new quantum resource for tasks that cannot be performed by means of classical resources.It can be manipulated，broadcast，controlled and distributed.Remarkably，entanglement is a resource which，though it does not carry information itself，can help in such tasks as the reduction of classical communication complexity，entanglement-assisted orientation in space，quantum estimation of a damping constant，frequency standards improvement，and clock synchronization.Entanglement plays a fundamental role in quantum communication between parties separated by macroscopic distances[3-5].For these tasks，the maximally entangled state is an indispensable quantum resource.The Bell states are the “canonical” maximally entangled states in the bipartite systems[6].Have we ever considered how the Bell states are given? This article presents a method for solving the representation of the maximally entangled states intwo-particle systems using von Neumann entropy.Then we generalize this approach to the three-body entanglement problem[7].1 Von Neumann entropy and the maximally entangled stateFor any pure state of two parties，for instance，a pure state of two qubits can be written as|ψ〉AB=c0|00〉+c1|01〉+c2|10〉+c3|11〉.(1)A unique measure of bipartite entanglement for pure states is given by the partial von Neumann entropy.The von Neumann entropy of a state is defined asS(ρ)=-Tr(ρlogρ)，(2)where the symbol ρ is the density operator for the system and Tr(…) denotes the trace operation.We can obtain the density operator associated with the quantum state in Eq.(1)，(3)The entanglement of the partly entangled pure state in Eq.(3) can be naturally parametrized by its entropy of entanglement[8]，defined as the von Neumann entropy of either ρA or ρB，S(ρAB)=S(ρA)=S(ρB).(4)We choose the standard basis to calculate the partial trace.For the densityoperator in Eq.(3)，we can obtain(5)where TrB refers to the partial trace over mode B.Analogously，(6)We diagonalise ρA or ρB.When the reduced density operator ρ is written in this diagonal form，our von Neumann entropy in Eq.(2) becomes(7)where the symbols ρn are the associated(non-negative) eigenvalues of the reduced density operator ρ diagonalised，which sum to unity，that is to the diagonalization，the reduced density operator ρA and ρB can be written as(8)where an d ζ≡4|c1c2-c0c3|2.Obviously，0≤ζ≤1.By making use of the diagonal representation of the reduced density operator in Eq.(8)，we write von Neumann entropy in the form：S(ρAB)=S(ρA)=S(ρB)=-ρ+logρ+-ρ-logρ-.(9)Any two-by-two matrix can be written as a weighted sum of the four Pauli operators.This means，in turn，that any operator associated with our qubit can also be expressed in terms of these operators.In particular，we can write the density operator in the form：(10)where I is the identity operator，r=(u，ν，w) is a Bloch vector，and σ=(σx，σy，σz) is the vector operator[9].Here Eq.(10)，the factor 1/2 ensures that Tr(ρ)=1.The density operator ρ is a positive Hermitian operator.The Hermiticity of ρ ensures that u，ν，and w are real.For the two-state system，the density operator of Eq.(10) can be written in the diagonal form：ρ=ρ+|ρ+〉〈ρ+|+ρ-|ρ-〉〈ρ-| .(11)where the states |ρ+〉and |ρ-〉are the eigenvectors of ρ correspondingto the eigenvalues Neumann entropy can be written as the equation(9)，but ρ± are described by the variable r.We have delineated Figure 1 about the von Neumann entropy as a function of the variable r.The positivity of the density operator ρ requires that u2+ν2+w2≤1.It is worth noting that the eigenvectors of ρ，namely |ρ+〉and |ρ-〉，are also the eigenvec tors of the operator r·σ corresponding to the eigenvalues ±r.If the vector’s tip of r lies on the surface of the Bloch sphere(r=1)，the diagonal density operator in Eq.(11) reduces the pure state |ρ+〉〈ρ+|.Ifr<1，the Bloch vector describes a point inside the Bloch sphere and corresponds to a mixed state.The farther the point is from the surface，the higher the degree of mixing of the mixed state is.This is to say，the resultis a more mixed state with a greater entropy.When the entropy of the subsystem reduced states are maximal，such states are called maximally entangled.The maximally entangled state of two subsystems associatedwith Eq.(1) requires r=0 from Fig.1.Fig.1 The von Neumann entropy versus the variable rAccording to the condition r=0，we can obtain the real parameteru=ν=w=0 due to the Hermiticity of ρ.Compared Eq.(5) and Eq.(6)with Eq.(10)，we can also write the following relationship：|c0|2+|c1|2-|c2|2-|c3|2=0，|c0|2-|c1|2+|c2|2-|c3|2=0，|c0|2+|c1|2+|c2|2+|c3|2=1，|c1c2-c0c3|=1/2(12)in the condition of the maximally entangled state.We choose the real coefficients c0，c1，c2，and c3 for simplicity.The individual equation in Eqs.(12) is not linearly independent of the other，for example，the last equation.By solving Eqs.(12)，we can obtain the coefficients of Eq.(1) for the maximally entangled state as follows.If and c1=c2=0；if andc0=c3=0.This result exactly corresponds to the Bell states for the maximally entangled subsystems.The four Bell states are conventionally written in the form(13)They are known as the four maximally entangled two-qubit Bell states.The von Neumann entropy of this density operator of the Bell states is positive and maximal.For the quantum state of two qubits，the Bell states of have a special prominence.The reasons for this include their simplicity and the fact that they have been realized in a number of diverse experiments.Just as Bell states are essential for the implementation of quantumcommunication with perfect fidelity，the importance of such a state in the distribution of bipartite entanglement is obvious.Certainly，multipartite maximally entangled states also play many crucial roles in quantum computation and quantum communication[10].Then we this approach is also generalized to solve the maximally entangled states of the three-body entanglement.For any pure state of tripartite，for instance，a pure state of three qubits can be written as|ψ〉ABC=c0|000〉+c1|001〉+c2|010〉+c3|011〉+c4|100〉+c5|101〉+c6|110〉+c7|111〉.(14)We can obtain the density operator associated with the quantum tripartite state in Eq.(14)，ρABC=|ψ〉ABC〈ψ|.(15)For the tripartite density operator，we choose the standard basis to calculate the partial trace in order to obtain(16)whereρA11≡|c0|2+|c1|2+|c2|2+|c3|2，ρB11≡|c0|2+|c1|2+|c4|2+|c5|2，ρC11≡|c0|2+|c2|2+|c4|2+|c6|2，Compared Eqs.(16) with Eq.(10) in the condition of r=0，we can obtain the coefficients of Eq.(14) in order to write the maximally entangled state of tripartite system as follows(17)Analogously，for simplicity we have chosen the real coefficients ci(i=0，1，2，…，7).They are known as the eight maximally entangled Greenberger-Horne-Zeilinger(GHZ)states[11].The GHZ state is a certain type of entangled state that involves at least three subsystems.The GHZ states are used in several protocols in quantum communication and cryptography，for example，in secret sharing.Thus the GHZ-state and the W-state represent two very different kinds of tripartite entanglement.In a certain sense，the W-state is “less entangled” than the GHZ-state.Therefore，the W-state does not belong to our solution.2 ConclusionThe degree which state in a quantum system consisting of two “particles” is entangled is measured by the von Neumann entropy of either of the two reduced density operators of the state.When the entropy of the subsystem reduced states are maximal，such states are called maximally entangled.We calculate the von Neumann entropy of the pipartite or tripartite systems and compare the reduced density operator with the density operator in the form of the Bloch vertor.According to the character that the density operator ρ is a positive Hermitian operator，we obtain the maximally entangled states such as the Bell states and the GHZ states. References【相关文献】[1]GERRY C C，KNIGHT P L.Introductory quantum optics[M].Cambridge：Cambridge University Press，2005.[2]PAN J W，CHEN Z B，LU C Y，et al.Multiphoton entanglement andinterferometry[J].Rev Mod Phys，2012，84(2)：777-838.[3]HORODECKI R，HORODECKI P L，HORODECKI M L，et al.Quantumentanglement[J].Rev Mod Phys，2009，81(2)：865-942.[4]SCHMID C，KIESEL N，WEBER U K，et al.Quantum teleportation and entanglement swapping with linear optics logic gates[J].New J Phys，2009，11：33008-1-33008-10. [5]ZHANG X L，WANG M L，YANG L L.Hawk-dovegame model of quantum under asymmetric information[J].Journal of Jilin Normal University(Natural Science Edition)，2011，32(4)：8-12.[6]EISERT J，CRAMER M，PLENIO M B.Colloquium：Area laws for the entanglement entropy[J].Rev Mod Phys，2010，82(1)：277-306.[7]AMICO L，FAZIO R，OSTERLOH A，et al.Entanglement in many-body systems[J].Rev Mod Phys，2008，80(2)：517-576.[8]BENNETT C H，BERNSTEIN H J，POPESCU S，et al.Concentrating partial entanglement by local operations[J].Phys Rev A，1996，53：2046-2052.[9]BARNETT S M.Quantum information[M].New York：Oxford University Press，2009.[10]DÜR W.Multipartit e entanglement that is robust against disposal of particles[J].Phys Rev A，2001，63(2)：020303-1-020303-4.[11]GREENBERGER D M，HORNE M A，SHIMONY A，et al.Bell’s theorem without inequalities[J].Am J Phys，1990，58(12)：1131-1143.。
量子信息和量子纠缠理论
Multipartite Schmidt-correlated State
Fully separable
PPT
Fully separable (maximally entangled)
~ 1 (N)
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S. Albeverio, S.M. Fei, Phys. Lett. A 276(2000)8 S. Albeverio, S.M. Fei and W.L. Yang, Comm. Theor. Phys. 38 (2002) 301
S. Albeverio, S.M. Fei and W.L. Yang, Phys. Rev. A 66 (2002) 012301 M. Horodecki, P. Horodecki and R. Horodecki, Phys. Rev. A 60, 1888 (1999)
Separable! Separable！
Separability of mixed states: no general criteria a) Peres (PPT) criterion:
Peres PRL 77, 1413 (1996)
2x2, 2x3:
PPT
Separable
Horodeckis, Phys. Lett. A 223,1 (1996)
Caltech (Kimble et al)
/~qoptics/teleport.html
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Wigner functions before &after
设备无关量子密钥分发协议
基于测量设备无关量子密钥分发协议虽然从理论上证明了QKD方案是一种绝对安全的量子密钥分发方案。
但是，由于测量设备和量子信号源的非完美性，量子密钥分配系统在实际应用中并不能保证传输信息的绝对安全。
例如光子探测器就容易受到"时移攻击"、"强光致盲攻击"等各种类型的攻击。
另外窃听者还可以利用量子信号源的非完美性进行攻击，例如窃听者可以利用光源的非完美性进行"光子数分流攻击"。
为了解决上述问题，人们提出了几种可能的方案，其中就包括使用诱骗态进行的量子密钥分发方案和基于设备无关量子密钥分发（DI-QKD）方案。
而最后一种方案有其自身独特的优点：不需要掌握QKD设备的运转状态，可以通过贝尔不等式来判断是否存在窃听者。
由于DI-QKD很难用于实际，后来，由Lo等人又提出了基于测量设备的无关量子密钥分发协议（MDI-QKD）。
该协议的优点十分突出，首先，该协议有很高的安全性，而且该协议的实现非常容易；其次，这个协议在传输距离上相对于传统量子密钥分发系统也有很大的优势，即使在MDI-QKD系统中使用普通二极管发出的激光光源，它的通信距离也几乎是传统量子密钥分发系统的两倍。
为了更好的理解基于测量设备的无关量子密钥分发协议，在此以该协议为基础介绍一个很简单MDI-QKD通信系统。
该通信系统与BB84协议使用相同的四种偏振态，即为偏振态，Alice和Bob都制备这四种偏振态，并随机从四种偏振态中选择一种发送给第三方（或者是EVE），这里我们无法判断第三方是否是窃听者，可以认为他是不受信任的。
然后由第三方将从两者接收到的信息结合起来并进行贝尔态的测量，即将输入信号转换为贝尔态。
像这种测量在实际环境下都是可以实现的，而且，Alice和Bob可以应用诱骗态技术来分析接收到的多光子的误码率。
使用了诱骗态技术的MDI-QKD偏振编码方案原理图如下图所示：图4.4 MDI-QKD协议的基本原理图正如图中所示，该系统采用弱相干激光脉冲作为光源，发送方Alice和接收方Bob通过偏振调制器对发射的弱相干激光脉冲进行偏振编码，随后，在强度调制器中制备诱骗态，然后，光束进入一个分束器中发生纠缠，最后经过偏振分束器中到达光子探测器。
中科大首次成功实现量子隐形传态首跨百公里卫星助力全球量子通信网络可行
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端装备 制造 、 软件服务外包 等三 合肥等城市 示范运营 ，并批量应 用于大连达沃斯 、上海世博会 、合 纯电动豪华旅 游客车。
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纠缠 ； 国际上首 次实验 实现 了拓 扑量 子纠错 ， 在 取得 星 的广域 量子 通信 和大尺 度 的量子力 学基 础原 理 检 了可扩 展容错性 量子计算 领域 的重要 突破 ，同时 。 潘 验 的实现奠定 了坚实 的技术基础 。
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多组态自洽场量子计算
多组态自洽场量子计算全文共四篇示例，供读者参考第一篇示例：多组态自洽场量子计算是一种重要的计算方法，它结合了量子力学的原理和计算机技术，专门用于处理原子和分子的电子结构问题。
在化学领域和材料科学领域，多组态自洽场量子计算被广泛应用，为研究者提供了强大的工具，使他们能够预测分子的性质和反应过程。
多组态自洽场量子计算的基本原理是基于量子力学的薛定谔方程。
在分子和固体中，电子的运动受到原子核和其他电子的相互作用的影响。
为了描述这种复杂的相互作用，我们需要解薛定谔方程，以确定体系的基态波函数和能量。
而多组态自洽场方法正是通过迭代求解薛定谔方程，得到最优的体系能量和波函数。
在多组态自洽场计算中，我们通常将原子和分子的电子波函数表示为多个自旋轨道的线性组合。
每个自旋轨道都对应一个电子的动力学运动。
我们通过将这些自旋轨道的线性组合代入薛定谔方程，并利用变分法求解得到最优的自旋轨道系数，从而获得体系的基态波函数和能量。
在实际的计算中，我们需要进行多次迭代，直至体系的能量收敛为止。
这涉及到解一系列复杂的线性代数方程，对计算资源和算法性能提出了较高的要求。
而随着计算机技术的发展，我们现在能够使用高性能计算平台进行大规模的多组态自洽场计算，加快计算速度和提高计算精度。
多组态自洽场量子计算在化学和材料科学研究中具有重要的应用价值。
通过计算分子和材料的电子结构，我们可以了解它们的化学性质、光学性质和电子输运性质。
这为设计新型功能材料和优化化学反应过程提供了重要的指导。
第二篇示例：多组态自洽场量子计算是一种基于量子力学原理的计算方法，用于模拟多体量子系统的性质。
在原子、分子和凝聚态物理领域，多组态自洽场量子计算被广泛应用于研究不同材料的电子结构、性质和反应动力学等方面。
本文将对多组态自洽场量子计算的基本原理、发展历程以及在科学研究中的应用进行介绍和探讨。
1. 多体量子系统的描述和自洽场方法在量子力学中，描述多体量子系统的方法之一是采用多体波函数的形式，即由多个粒子的波函数构成的复合波函数。
量子力学英语词汇
1、microscopic world 微观世界2、macroscopic world 宏观世界3、quantum theory 量子[理]论4、quantum mechanics 量子力学5、wave mechanics 波动力学6、matrix mechanics 矩阵力学7、Planck constant 普朗克常数8、wave-particle duality 波粒二象性9、state 态10、state function 态函数11、state vector 态矢量12、superposition principle of state 态叠加原理13、orthogonal states 正交态14、antisymmetrical state 正交定理15、stationary state 对称态16、antisymmetrical state 反对称态17、stationary state 定态18、ground state 基态19、excited state 受激态20、binding state 束缚态21、unbound state 非束缚态22、degenerate state 简并态23、degenerate system 简并系24、non-deenerate state 非简并态25、non-degenerate system 非简并系26、de Broglie wave 德布罗意波27、wave function 波函数28、time-dependent wave function 含时波函数29、wave packet 波包30、probability 几率31、probability amplitude 几率幅32、probability density 几率密度33、quantum ensemble 量子系综34、wave equation 波动方程35、Schrodinger equation 薛定谔方程36、Potential well 势阱37、Potential barrien 势垒38、potential barrier penetration 势垒贯穿39、tunnel effect 隧道效应40、linear harmonic oscillator线性谐振子41、zero proint energy 零点能42、central field 辏力场43、Coulomb field 库仑场44、δ-function δ-函数45、operator 算符46、commuting operators 对易算符47、anticommuting operators 反对易算符48、complex conjugate operator 复共轭算符49、Hermitian conjugate operator 厄米共轭算符50、Hermitian operator 厄米算符51、momentum operator 动量算符52、energy operator 能量算符53、Hamiltonian operator 哈密顿算符54、angular momentum operator 角动量算符55、spin operator 自旋算符56、eigen value 本征值57、secular equation 久期方程58、observable 可观察量59、orthogonality 正交性60、completeness 完全性61、closure property 封闭性62、normalization 归一化63、orthonormalized functions 正交归一化函数64、quantum number 量子数65、principal quantum number 主量子数66、radial quantum number 径向量子数67、angular quantum number 角量子数68、magnetic quantum number 磁量子数69、uncertainty relation 测不准关系70、principle of complementarity 并协原理71、quantum Poisson bracket 量子泊松括号72、representation 表象73、coordinate representation 坐标表象74、momentum representation 动量表象75、energy representation 能量表象76、Schrodinger representation 薛定谔表象77、Heisenberg representation 海森伯表象78、interaction representation 相互作用表象79、occupation number representation 粒子数表象80、Dirac symbol 狄拉克符号81、ket vector 右矢量82、bra vector 左矢量83、basis vector 基矢量84、basis ket 基右矢85、basis bra 基左矢86、orthogonal kets 正交右矢87、orthogonal bras 正交左矢88、symmetrical kets 对称右矢89、antisymmetrical kets 反对称右矢90、Hilbert space 希耳伯空间91、perturbation theory 微扰理论92、stationary perturbation theory 定态微扰论93、time-dependent perturbation theory 含时微扰论94、Wentzel-Kramers-Brillouin method W. K. B.近似法95、elastic scattering 弹性散射96、inelastic scattering 非弹性散射97、scattering cross-section 散射截面98、partial wave method 分波法99、Born approximation 玻恩近似法100、centre-of-mass coordinates 质心坐标系101、laboratory coordinates 实验室坐标系102、transition 跃迁103、dipole transition 偶极子跃迁104、selection rule 选择定则105、spin 自旋106、electron spin 电子自旋107、spin quantum number 自旋量子数108、spin wave function 自旋波函数109、coupling 耦合110、vector-coupling coefficient 矢量耦合系数111、many-partic le system 多子体系112、exchange forece 交换力113、exchange energy 交换能114、Heitler-London approximation 海特勒-伦敦近似法115、Hartree-Fock equation 哈特里-福克方程116、self-consistent field 自洽场117、Thomas-Fermi equation 托马斯-费米方程118、second quantization 二次量子化119、identical particles全同粒子120、Pauli matrices 泡利矩阵121、Pauli equation 泡利方程122、Pauli’s exclusion principle泡利不相容原理123、Relativistic wave equation 相对论性波动方程124、Klein-Gordon equation 克莱因-戈登方程125、Dirac equation 狄拉克方程126、Dirac hole theory 狄拉克空穴理论127、negative energy state 负能态128、negative probability 负几率129、microscopic causality 微观因果性本征矢量eigenvector本征态eigenstate本征值eigenvalue本征值方程eigenvalue equation本征子空间eigensubspace (可以理解为本征矢空间)变分法variatinial method标量scalar算符operator表象representation表象变换transformation of representation表象理论theory of representation波函数wave function波恩近似Born approximation玻色子boson费米子fermion不确定关系uncertainty relation狄拉克方程Dirac equation狄拉克记号Dirac symbol定态stationary state定态微扰法time-independent perturbation定态薛定谔方程time-independent Schro(此处上面有两点)dinger equation 动量表象momentum representation角动量表象angular mommentum representation占有数表象occupation number representation坐标（位置）表象position representation角动量算符angular mommentum operator角动量耦合coupling of angular mommentum对称性symmetry对易关系commutator厄米算符hermitian operator厄米多项式Hermite polynomial分量component光的发射emission of light光的吸收absorption of light受激发射excited emission自发发射spontaneous emission轨道角动量orbital angular momentum自旋角动量spin angular momentum轨道磁矩orbital magnetic moment归一化normalization哈密顿hamiltonion黑体辐射black body radiation康普顿散射Compton scattering基矢basis vector基态ground state基右矢basis ket ‘右矢’ket基左矢basis bra简并度degenerancy精细结构fine structure径向方程radial equation久期方程secular equation量子化quantization矩阵matrix模module模方square of module内积inner product逆算符inverse operator欧拉角Eular angles泡利矩阵Pauli matrix平均值expectation value （期望值）泡利不相容原理Pauli exclusion principle氢原子hydrogen atom球鞋函数spherical harmonics全同粒子identical partic les塞曼效应Zeeman effect上升下降算符raising and lowering operator 消灭算符destruction operator产生算符creation operator矢量空间vector space守恒定律conservation law守恒量conservation quantity投影projection投影算符projection operator微扰法pertubation method希尔伯特空间Hilbert space线性算符linear operator线性无关linear independence谐振子harmonic oscillator选择定则selection rule幺正变换unitary transformation幺正算符unitary operator宇称parity跃迁transition运动方程equation of motion正交归一性orthonormalization正交性orthogonality转动rotation自旋磁矩spin magnetic monent(以上是量子力学中的主要英语词汇，有些未涉及到的可以自由组合。
未来,的量子通行英语作文
未来,的量子通行英语作文Quantum Computing: Unlocking the Future's Potential.In the realm of technological advancements, quantum computing stands as a transformative force, poised to revolutionize industries and redefine the very fabric of our society. This nascent field harnesses the principles of quantum mechanics to manipulate quantum bits, or qubits, unleashing computational capabilities far beyond the reach of traditional computers. As quantum technology continues to evolve at an unprecedented pace, it holds immense promise for shaping the future across a myriad of domains.Scientific Discovery and Innovation.Quantum computers possess the potential to accelerate scientific research and fuel groundbreaking discoveries. Their unrivaled computational power could aid in unraveling complex scientific phenomena, such as the intricacies of quantum chemistry and the behavior of subatomic particles.By simulating complex systems with unmatched precision, quantum computers can pave the way for novel materials, advanced drug development, and groundbreaking medical treatments.Pharmaceuticals and Healthcare.The healthcare industry stands to witness transformative advancements with the advent of quantum computing. The ability to simulate molecular interactions and pharmaceutical compounds with unprecedented accuracy can accelerate drug discovery and optimize treatment regimens. Quantum-powered algorithms can analyze vast datasets of patient data, identifying patterns and correlations that escape traditional analysis, leading to personalized therapies and improved patient outcomes.Financial Modeling and Optimization.Quantum computing is poised to revolutionize the financial sector, enabling complex financial modeling and risk analysis in ways that are currently infeasible. Thesesystems can process massive amounts of data in real-time, providing insights into market trends, forecasting financial fluctuations, and optimizing investment strategies. Quantum algorithms can also enhance portfolio optimization, leading to more informed decision-making and improved financial performance.Materials Science and Engineering.The transformative power of quantum computing extends to materials science and engineering. Quantum simulations can elucidate the intricate properties of materials at the atomic and molecular level, enabling the development of lightweight, durable, and highly efficient materials. This advancement holds implications for industries ranging from aerospace to manufacturing, paving the way for innovations in next-generation vehicles, aircraft, and infrastructure.Artificial Intelligence and Machine Learning.Quantum computing has the potential to fuel the next era of artificial intelligence and machine learning. Byharnessing the power of qubits, quantum algorithms can accelerate the training of machine learning models, enabling them to process larger datasets and solve more complex problems. This computational surge can empower AI-driven systems to perform tasks that are currently beyond their grasp, such as natural language processing, speech recognition, and image analysis.Cryptography and Cybersecurity.The advent of quantum computing poses bothopportunities and challenges for cryptography and cybersecurity. While quantum algorithms can be harnessed to enhance encryption protocols, they also have the potential to break existing encryption standards. This necessitates the development of quantum-resistant cryptography, ensuring the continued security of sensitive information in the face of advancing computational capabilities.Ethical Considerations and Societal Impact.As the field of quantum computing continues to evolve,it is crucial to address the ethical and societal implications of this transformative technology. The immense computational power of these systems raises concerns about privacy, security, and the potential for misuse.Establishing clear ethical guidelines and regulations is paramount to ensure that quantum computing is developed and deployed for the benefit of society, while mitigating any potential risks.Conclusion.Quantum computing holds the potential to reshape the future across a vast array of industries and scientific disciplines. Its ability to accelerate scientific discovery, fuel innovation, and solve complex problems that are currently intractable opens boundless possibilities for human progress. However, as we harness the power of this transformative technology, it is essential to proceed with both excitement and caution, considering the ethical and societal implications and ensuring that quantum computingis used for the betterment of humanity.。
真正量子纠缠的检测与度量
摘要量子信息是量子力学中量子系统状态所带有的物理信息，量子纠缠态是量子信息理论中最主要的物理资源.本文主要研究了真正多体量子纠缠的检测与度量.首先介绍了量子纠缠与量子信息的关系及国内外的发展背景与现状.其次是基础知识，介绍了一些数学与物理的基本概念，如矩阵分解、密度矩阵、量子纠缠与真正多体量子纠缠的定义、一些经典的量子纠缠判据、用Bloch表示构造关联张量以及concurrence的定义和下界等.然后文中用密度矩阵的Bloch表示、密度矩阵部分转置、重排研究了真正多体纠缠的判定以及真正多体并发度的估计，通过分析Bloch向量的2范数、密度矩阵部分转置和PPT重排后的迹范数与真正纠缠的关系给出了真正多体纠缠的判据,并进一步得到了真正多体纠缠并发度分析的下界.最后文章分析对比了相关结果和一些已有的真正多体纠缠判据，用具体例子的数值实验说明我们得到的判据和真正多体纠缠的并发度下界是有效的，与已有的结论相比，我们得到的结果能识别更多的真正多体纠缠态，对于真正纠缠并发度，我们得到的下界大于已有的相关结果.关键词：真正量子纠缠，可分性，真正纠缠并发度The detection and measurement of genuinequantum entanglementJia Lingxia(Mathematics Engineering)Directed by Associate Prof.Li MingAbstractQuantum information is the physical information taking by quantum system states in quantum mechanics.Quantum entanglement is a most important kind of physical resource in quantum information theory.This paper mainly studies the detection and some measurements of genuine tripartite quantum entanglement.Firstly,the relationship between quantum entanglement and quantum information,the development background and current situation at home and abroad are introduced.The second part of the paper introduces some basic concepts of mathematics and physics, such as matrix decomposition,density matrix,quantum entanglement the definition of genuine tripartite quantum entanglement,some classical quantum entanglement criteria,the construction of correlation tensors with Bloch representation the definition and the lower bound of concurrence,etc.The third part of the thesis studies the genuine tripartite entanglement determination and the estimation of genuine tripartite concurrence by using the Bloch representation,the partial transposition and rearrangement of the density matrix.We analied the2norm of the Bloch vector and density matrix partial transposition and PPT rearrangement.The relationship between the trace norm and the genuine entanglement,we gives a criterion for genuine tripartite entanglement and further obtains the analytical lower bound of the genuine tripartite entanglement.The last part of the thesis analyzes some classical genuine tripartite entanglement ing numerical examples of specific examples,we show that our criterion and the lower bounds of genuine tripartite entanglement concurrence are valid,that is,our criteria can identify more genuine tripartite entanglement and the lower bound of the true degree of concurrency we obtain is larger than the existing related results.Key words:genuine quantum entanglement，separability，GME concurrence目录第一章绪论 (1)1.1量子纠缠简介 (1)1.2国内外研究现状 (2)1.3研究进展及研究结果 (2)第二章基础知识 (4)2.1矩阵分解 (4)2.1.1谱分解 (4)2.1.2奇异值分解 (4)2.2密度矩阵 (4)2.3量子态的Bloch表示 (6)2.4量子纠缠的定义 (7)2.5量子纠缠的判定 (9)2.5.1利用PPT判据检测量子纠缠 (9)2.5.2利用矩阵重排判据检测量子纠缠 (10)2.5.3利用Bloch表示检测量子纠缠 (10)2.6Concurrence (11)2.6.1量子态concurrence的定义 (11)2.6.2量子态concurrence的下界 (13)第三章真正量子纠缠的检测与度量 (16)3.1真正量子纠缠的检测 (16)3.2真正量子纠缠的度量 (19)第四章算例 (24)总结与展望 (28)参考文献 (29)攻读硕士学位期间的科研情况 (33)致谢 (34)中国石油大学（华东）硕士学位论文第一章绪论在量子力学中，量子信息是量子系统“状态”带有的物理信息.通过量子系统的相关性质（如量子的可分性与纠缠性、波粒二象性、不确定性等），将各种数据资料编码成可利用的处理和分析的信息，进行运算及传输信息的崭新的信息方式.其中，量子纠缠是一种区别与经典物理学的量子力学现象，并且它也是是量子信息科学领域中至关重要的物理资源.本章首先介绍了量子纠缠的概念及发展过程，随后阐述量子纠缠研究背景及意义，最后综述该领域的一些研究进展及结果.1.1量子纠缠简介1932年,V on Neumann为非相对论量子理论描述世界奠定了基础.后来Einstein，Podolsky，Rosen以及Schr o dinger第一次认识到量子力学的这种诡异现象.这种量子态的特别之处就在于,在任何条件下,它都不可以被分解成两个子系统状态的张量积形式,这就是量子纠缠的定义.量子纠缠在量子信息[1]与量子计算中都是重要的组成部分.在当今快速发展的信息时代,量子信息有望成为第四次工业革命,因此量子纠缠成为极其重要的资源而且它在量子信息应用方面有着重要的作用.比如利用量子比特共享自身状态,创造出一种超级叠加的量子并行计算、通过传输一个量子比特而传输两个比特的经典信息的量子密集编码[2]、区别与经典信息的量子态携带量子信息进行“超时空传输”的量子隐形传态[3,4]、量子秘钥分发等等,这些应用的实现都需要借助于量子纠缠.然而对于量子纠缠态的研究存在一定的难度,想要完全了解量子纠缠的数学和物理特点还需要很多学者为其做出坚持不懈的努力.随着研究的深入,人们发现量子纠缠并不是使量子计算机超越经典计算机的完全原因.1998年，Knill和Laamme提出了一个量子计算模型[5],在这个模型中有n+1个量子比特,其中n个处在极大混合态,而有一个始终与另外n个是可分的,也就是没有纠缠的,并且这个计算模型能够完成比任何经典计算机指数加速的量子运算.这启发了人们研究量子理论中的各种纠缠度量的存在和作用问题.对于两体纯态系统中的研究得到了相对简单且重要的结果,但是,由于局限性的存在,对于多体混合态而言,我们还没有弄清楚它的本质.最基础的,给定一个多体量子混合态,怎么判断它具有的特性?是可分的还是纠缠的?更深层次的,量子态的纠缠程度怎么来度量?退一步来讲,两体系统中量子concurrence的计算,多体系统中Bloch对量子态的刻画和度量等等,很多问题都需要第一章绪论更加深入的研究.描述多种纠缠态之间的定量可比关系是对其本质的探索.因此，在量子信息理论与应用中,量子纠缠通常被作为量子力学的关键“信息源”.纠缠度量最初的想法与信息交流的可用性[6]与纠缠度量的关系至关紧密.对于较为简单的两体量子系统,学者们已经研究出了一些经典的纠缠度量,例如,子系统约化密度矩阵的冯诺依曼熵、生成纠缠、concurrence等等.文献[7]中Wootters给出了两个量子比特系统的EOF和concurrence的较为简洁且计算量相对较小计算公式，但是对于多体量子系统而言,其子系统维数的增加也会导致EOF和concurrence的计算复杂且困难.到现在为止,只能研究出一些特殊的对称量子态EOF和concurrence的计算公式[8].对于一般的高维数和多体量子态,我们只能研究出concurrence的上下界,来估计两体高维量子态或者多体情形量子态的纠缠度.1.2国内外研究现状量子技术最前沿的两个应用方向就是量子计算和量子通信,不难看出,我国青睐量子通信技术,而美国更倾向于量子计算技术.1993年美国IBM研究人员已经开始研究量子通信技术,欧洲也成立了以英国、法国、意大利等国在内的量子信息物理学研究网.2018年3月,谷歌发布全球首个72量子芯片,如今各大科技公司都将量子计算机视为计算的下一个重大突破,量子计算机可以更快的运行某些算法.在过去的三十多年,我国量子领域的相关技术从最初的起步摸索,到目前为止已经走到了世界的前沿,掌握了世界较为先进的尖端技术.在国内,比较早起步而且系统全面研究量子信息（重点是量子纠缠）理论的有安徽合肥市的中国科技大学、中国科学院的物理所和北京的首都师范大学数学系等.这些学者们已经取得了一些很有价值的应用成果,包括成功研制出量子密钥分配终端一体机、千兆量子安全网关、量子交换机等量子产品；2013年中国量子通信已在潜艇深海实验中取得成功；2016年8月16号，世界首颗量子科学试验卫星“墨子号”在酒泉成功发射升空；2017年9月29日中国科学院举行新闻发布会,宣布世界首条量子保密通信干线“京沪干线”正式开通等,这些都标志着我国对量子信息的科学研究又迈出重要一步.1.3研究进展及研究结果量子纠缠态不仅在量子力学中发挥着至关重要的作用,而且在量子计算和量子通讯中国石油大学（华东）硕士学位论文技术中起着无可替代的作用.刘坤在他的论文[9]中系统全面的介绍了量子纠缠态的特性,从他的定义、一般常见的纠缠态及纠缠度量进行了说明.Peres提出了PPT判据：如果量子态可分,则对于给定的密度矩阵,它的部分转置为半正定的,否则纠缠.Rungta给出了concurrence的定义,concurrence可以度量量子纠缠程度的强弱.Rudolph提出了重排判据：如果量子态可分,则矩阵的迹范数必定小于或等于1,否则量子态是纠缠的.另外,李晓宇在他的博士论文中研究了纠缠态在量子信息处理中的应用,包括存储和提取量子信息,隐藏量子信息以及量子密钥分配等.他对量子信息存储和提取做了一般性的讨论,利用非最大纠缠态的量子信息隐藏方案和正交直积态的量子信息隐藏方案已经被成功提出.真正多体纠缠（GME）是量子纠缠中的一种重要纠缠类型,在比较两体纠缠[10]的量子任务中具有明显的优势,它也是组成测量量子计算的基本成分,而且在多种量子通信协议[11]中也是很有效的,其中包括一些通用的度量任务中的秘密共享[12,13]、极端自旋压缩器[14]和高灵敏度的多体量子网[15,16]等.然而检测和测量量子纠缠是非常困难的.人们推导出了一系列线性和非线性[17,18]、一般的并发度[19,20]和Bell不等式[21]等定理来证明真正多体纠缠,并且开发出了描述半正定的程序,尽管如此，GME问题仍然没有得到圆满的解决,用量子态的标准张量积和Bloch表示,来刻画两体和多体量子可分性的条件在文献[22,23]中有介绍.人们在检测真正的多体纠缠的框架和多体量子系统的非完全可分性中引入了任意维数.在文献[24]中已经证明了相关张量与最大违反的关系,在文献[25]中给出贝尔不等式和并发度的关系.由量子信息任务的不同,能得出不同的纠缠度量可以度量各种不同类型的纠缠.总的来说,两体量子态纠缠度量的研究理论已经基本成形,但是还缺少可操作性的纠缠度量.多体量子态的纠缠度量,相对来讲显得特别棘手.第二章基础知识第二章基础知识线性代数是研究量子纠缠需要用到的最重要的数学知识之一,想要研究量子纠缠问题就要精确掌握这些知识,尤其是矩阵的分解和密度矩阵等基础知识必须要要熟练的掌握.量子的可分性与纠缠性也在本章有了详细的介绍,除此之外还介绍了一些著名的纠缠判据包括PPT 判据、矩阵重排判据和真正纠缠判据及量子度量需要用到的传统的Bloch 和concurrence 的概念特性等知识.2.1矩阵的分解2.1.1谱分解任意一个满足M M MM ++=的线性算子都能写成如下形式:,+Λ=U U M 其中U 为幺正算子，Λ为实对角矩阵.2.1.2奇异值分解线性算子M 都能通过线性变换转化成如下形式:,+∑=W V M ∑是对角矩阵，V 和W 是幺正矩阵，∑的对角元叫做M 的奇异值，也是M M +的特征值的正平方根.此外定义[]∑==+i ii KF MM Tr Mσ并且用它表示矩阵M 的迹范数，用[]+=MM Tr M HS 表示矩阵M 的Frobenius 范数或者Hilbert-Schmidt 范数.2.2密度矩阵定义2.2.1密度矩阵在量子力学中，系统可能处于量子纯态，也可能处于多个量子纯态以某种概率的叠加，在数学上，Hilbert 空间的向量可以描述量子纯态，由于向量的任意叠加还是向量，所以要描述多体量子系统就需要引进密度矩阵的概念.设量子以i p 概率处在一组量子纯态中的某一个,其中i 是一个指标，则{},i i p ψ就叫做一个纯态系综,定义密度矩阵为:中国石油大学（华东）硕士学位论文ρψψ=∑,i i i i p 密度矩阵也被称为密度算子.密度矩阵的特点:矩阵ρ称为一个有系综分解{ψi ip 的密度算子，当且仅当它满足如下条件时:a)迹条件:ρ的迹为一.b)正定性:ρ是一个正算子，即对任意,ψ0.ψρψ≥定义2.2.2约化密度矩阵[26,27]量子信息常用的方法是在两个量子系统A 和B 复合而成的大系统中研究量子态.设Hilbert 空间A H 和B H 表示维数为A d 和B d 的系统A 和系统B 的量子态空间，A H 和B H 的张量积构成维数为A B d d 的复合量子系统AB H ，即此系统中的量子态AB ρ都能用迹为一的正定算子在AB H 来刻画.A和B是复合量子系统，AB ρ是的一个量子态AB H ，则对应子系统()A B H H 的约化密度矩阵为(),,A B AB B A AB Tr Tr ρρρρ⎡⎤⎡⎤==⎣⎦⎣⎦其中()B A Tr Tr 是子系统()B A H H 的偏迹.偏迹运算定义为:()⎡⎤⎡⎤⊗=⎣⎦⎣⎦⎡⎤⎡⎤⊗=⎣⎦⎣⎦1212121212121212,,B A Tr a a b b Tr b b a Tra b b Tr a a b b 且1a 和2a 是子系统A H 中的任意的两个向量，1b 和2b 是子系统B H 中的任意的两个向量.定义2.2.3W 态、Bell 态、GHZ 态在量子信息研究中最广泛应用的几类纠缠态是W 态、Bell 态、GHZ 态等.下面对此做一简单介绍.（1）两粒子体系的量子纠缠中，存在有如下四个量子态：(),100121B A B A AB ±=±ψ第二章基础知识(),001121B A B A AB ±=±ϕ其中AB ψ-叫做单重态，其他三个叫做三重态.这四个量子态构成一组两粒子系统的四维Hilbert 空间的正交完备基，其称作Bell 基(也称为Bell 态).Bell 基是有最大纠缠度的两量子位纯态，常被称为最大纠缠态，即不能再通过任何的方式增大它的纠缠度.（2）纠缠态还可以存在于多体系统中，下面是GHZ 态的形式：(),00011121+=ψGHZ 态具有和Bell 态类似的性质，也就是当检测出其中一个粒子的态是1态，则其他两个粒子一定在1态上，若测得其中一个粒子的态为0态时，另外两个粒子必定处在0态上.（3）三粒子纠缠态中还有一种区别于GHZ 态的纠缠形式：()，10001000131++=ψ称为W 态.W 态与GHZ 态不能通过某种操作相互转换.2.3量子态的Bloch 表示[28]D 维Hilbert 空间的厄米算子都可以用特殊酉群SU(d)的生成元表示.SU(d)可用转换投影算子定义：,k j Pjk =其中,,,1,d i i ⋅⋅⋅=是d H 中的正交本征态.假设），（）（1,1221112++-++++-=l l ll l lP P P P l l ω，kj jk jk P P +=μ，)(kj jk jk P P i -=υ其中.1,11d k j d l ≤≤≤-≤≤可得1-2d 个算子，且这些算子满足[][],2,0ij j i i Tr Tr δλλλ==中国石油大学（华东）硕士学位论文既而生成了SU(d)，其中i λ属于1-2d 个算子.任何d H 中的厄米算子ρ都能用SU(d)生成元表示为∑-=+=112211d j jj r I d λρ的形式，其中I 是单位矩阵，.),,,(112122--∈=d d R r r r r r 叫做Bloch 向量.设d d H H 21⊗∈ρ是一个两体量子态，可以用关联张量表示如下：12122111(),24i i j j ij i j I I t I t I td d dρλλλλ=⊗+⊗+⊗+⊗∑∑∑设d d d H H H 321⊗⊗∈ρ是一个三体量子态，可以用关联张量表示如下：123312132312311()21()41,8i i j j k k ij i j ik i k jk j k ijk i j k I I I t I I t I I t I I d d t I t I t I d t ρλλλλλλλλλλλλ=⊗⊗+⊗⊗+⊗⊗+⊗⊗+⊗⊗+⊗⊗+⊗⊗+⊗⊗∑∑∑∑∑∑∑),(),(21I I tr t I I tr t j j i i ⊗⊗=⊗⊗=λρρλ),(),(123I tr t I I tr t j i ij k k ⊗⊗=⊗⊗=λρλλρ),(),(2313k j jk k i ik I tr t I tr t λλρλρλ⊗⊗=⊗⊗=).(123k j i ijk tr t λλρλ⊗⊗=在下面，我们设)23()13()12()3()2()1(T T T T T T ，，，，，和)123(T 为由231312321,,,,,jk ik ij k j i t t t t t t 和⋅⋅⋅=,2,1,,,123k j i t ijk ，12-d 构成的向量，称为关联向量.2.4量子纠缠的定义量子纠缠是量子力学区别于经典物理的，存在于多量子系统中的奇特现象.量子信息最核心的部分就是量子纠缠.无论某些粒子间相差了多远的距离，某一个粒子的状态都与其他粒子的状态是相互关联的，这种物理现象就叫做量子纠缠.定义2.4.1纯态的可分性对于两体量子纯态B A AB H H ⊗∈ρ是可分的，当且仅当它可以写成BA AB ρρρ⊗=第二章基础知识的形式，其中A ρ和B ρ为约化密度矩阵.这等价于,B B A A AB φφψψρ⊗=其中.,B B A A H H ∈∈φψ相反，如果一个量子态不能写成上式形式就是纠缠态.对于多体量子纯态N N H H H ⊗⋅⋅⋅⊗⊗∈⋅⋅⋅2112ρ是可分的，当且仅当它可以写成,2112N N ρρρρ⊗⋅⋅⋅⊗⊗=⋅⋅⋅的形式，其中1ρ和N ρρ,...,2相应于各个子系统中的约化密度矩阵.这个条件等价于,221112N N N μμφφψψρ⊗⋅⋅⋅⊗⊗=⋅⋅⋅其中N N H H ∈⋅⋅⋅∈μψ,,11为每个子系统中规一化的纯态.相反,如果一个量子态不能写成上式形式就是纠缠态.定义2.4.2混合态的可分性对于两体量子混合态B A AB H H ⊗∈ρ是可分的，当且仅当它可以写成,iB i A i i AB p ρρρ⊗=∑的形式，其中A i ρ和B i ρ相应于子系统A H 和B H 的约化密度矩阵;且需要满足,1,0∑=>ii i p p 这个条件等价于,B i B i A i iA i i AB p φφψψρ⊗=∑其中∑=>iii pp .1,0B i A i φψ,为子系统中规一化的纯态，而且这些纯态不一定是正交的.反过来说，假如一个量子态不能写成上式形式就是纠缠态.对于多体量子态N N H H H ⊗⋅⋅⋅⊗⊗∈⋅⋅⋅2112ρ是完全可分的，当且仅当它可以表示成如下形式:∑⊗⊗⊗=i2112,N i i i i N p ρρρρ 其中N i i ρρ,...,1为相应于子系统N H H ,,1⋅⋅⋅的约化密度矩阵，且需要满足，∑=>ii i p p 1,0中国石油大学（华东）硕士学位论文这个条件等价于∑⊗⊗⊗=i221112,N i N i i i i i i N p μμφφψψρ其中N i i i μφψ,,21⋅⋅⋅为子系统N H H ,,1⋅⋅⋅中的纯态,且需要满足，∑=>ii i p p 1,0相反，如果一个量子态不能写成上式形式就是纠缠态.定义2.4.3真正多体量子纠缠一个纯态N N N H H ⊗⋅⋅⋅⊗∈=⋅⋅⋅⋅⋅⋅1,,1,,1ϕρ两体可分，如果N ,,1⋅⋅⋅ρ可以写成如下形式：,,,1B A N ϕϕϕ⊗=⋅⋅⋅其中,1k j j A H H ⊗⋅⋅⋅⊗∈ϕ,1N k j j B H H ⊗⋅⋅⋅⊗∈+ϕ{}N k k j j j j ,,,,11⋅⋅⋅⋅⋅⋅+为{}N ,,1⋅⋅⋅的任意排列，否则N ，，⋅⋅⋅1ρ为真正纠缠.一个混合态N ,,1⋅⋅⋅ρ两体可分，如果N ,,1⋅⋅⋅ρ可以写成如下形式：,,,1iiii B A ii N q ρρρ⊗=∑⋅⋅⋅其中{}{}{},,,1,,,1,,,1N B A N B N A i i i i ⋅⋅⋅=⋃⋅⋅⋅⊂⋅⋅⋅⊂，1=∑iiq否则N ,,1⋅⋅⋅ρ为真正纠缠.2.5量子纠缠的检测2.5.1利用PPT 判据[29]检测量子纠缠PPT(partial positive transposition)判据是Peres 在1996年给出的一个很强的纠缠判据，也称为Peres 判据.它可以识别很多混合态的纠缠.PPT 判据指出，对任意两体可分的量子态的密度矩阵进行部分转置，得到的仍是一个正定的密度矩阵.如果AB ρ可分，则下面定义的矩阵，μρννρμn m n m AB T AB B≡仍是一个密度算子，即B TAB ρ仍然是一个量子态(同理定义正定矩阵AT AB ρ),部分转置的算子B T 对应于对第二个子系统进行转置.后来，Horodecki 等人[30]证明了此判据是2*2和2*3系统以及某些特殊的高维量子体系的纠缠性的充分必要条件.此外，PPT 判据对于一般的高维量子态而言，纠缠性只第二章基础知识是一个必要条件.2.5.2利用矩阵重排判据[31,32]检测量子纠缠另外一个很强的纠缠判据是基于乘积态的线性条件得到的.这个判据被称之为矩阵重排判据或者computable cross norm(CCN)判据，它独立于PPT 判据.该判据可表示为如下形式:设AB ρ是可分的量子态，则定义(),,,jl ik kl ij R ρρ=矩阵()ρR 的迹范数满足小于或等于1，即().1≤KF R ρ更为重要的是，一些PPT 纠缠态可以用矩阵重排判据检测.它还可以用来构造一些不可分解的量子态.另外，concurrence 的下界也可以由这个判据给出.2.5.3利用Bloch 表示检测量子纠缠[33]李明研究了关于任意维数的多体量子态的纠缠判据.对于粒子量子态，任意的量子态12d d A BH H ρ∈⊗都可以通过Bloch 表示为，∑∑∑∑=-===⊗+⊗+⊗+⊗=1-d 1k 1d 1l 1-d 1l 1-d 1k 21212222211l k kl l l k k t I s I r I I d d λλλλρ这里12d d A B H ,H 为维数分别为21,d d 的向量空间，()(),21,2112l l k k I Tr d s I Tr d r λρρλ⊗=⊗=()，l k kl Tr t λρλ⊗=41T 为元素是kl t 的矩阵，矩阵()()()()⎥⎥⎥⎥⎥⎥⎦⎤⎢⎢⎢⎢⎢⎢⎣⎡=------11111111111121222121212222d 1d d d d d d t t r t t r s s d T如果12d d A BH H ρ∈⊗是可分的，那么对于任意的2212d d ⊗矩阵M 和()()221211d d -⊗-矩阵N 我们可以得到中国石油大学（华东）硕士学位论文()()(),222max21222121M d d d d d dT m klkl kl σ+-+-≤∑()()()，N d d d d t n klkl kl max 2121411σ--≤∑成立，这里的()()N M max max σσ，是矩阵N M ,的最大奇异值.以此类推，可以推广到N 粒子量子系统的量子纠缠检测.2.6Concurrence [34]一个好的纠缠测度E ，往往需要满足下列条件:(1)对可分态.0)(,=ρρE (2)正规化条件:两体d 维系统的量子最大纠缠态的纠缠度满足:.log )(d P E d =+(3)在经典通信(LOCC)操作和局域下纠缠度不增加:.)())((ρρE E LOCC ≤Λ(4)连续性:即当0→-σρ时，.0)()(→-σρE E (5)可加性:n n nE E ⊗⊗=ρρρ),()(表示n 个量子态ρ的张量积.(6)次可加性:).()()(σρσρE E E +≤⊗(7)凸性:)()1()())1((σλρλσλλρE E E -+≤-+其中.10<<λConcurrence 作为可以量化纠缠的度量，满足上述的一些性质，对于两个量子比特系统虽然还缺少可操作性的纠缠度量但是其研究理论已经基本成形.Wootters 已经在文献给出了两个量子比特系统的concurrence 的计算公式[35].多体量子态的纠缠度量，虽然有些棘手但是也形成了一些理论.Sarandy 指出concurrence 对于描述多体量子系统中各种相互作用下[36]的变换中起到至关重要的作用[37].而且，利用concurrence 的上下界还可以对形成纠缠度[38]进行估计[39].因为形成纠缠度量化了制备一个量子态需要的最少物理资源[40]，这是非常关键的.综上，给出精确的concurrence 的值便于更好地理解相应的物理系统.2.6.1量子态concurrence 的定义[41]第二章基础知识设)(B A H H 为)(N M 维的复向量空间，它的一组正交基,1,,(|,1,,)i i M j j N =⋅⋅⋅〉=⋅⋅⋅复合空间A B H ⊗H中的一个纯态能写成M N11||,ij i j a i j ψ===∑∑〉⊗〉且系数ij a C ∈满足11 1.M N i j ij ij a a *==∑∑=关于两个量子比特的复合系统，量子纯态ψ的concurrence 定义为11221221(|)|2,C a a a a ψψψ〉=〈〉=- 其中0y o i i σ-⎛⎫= ⎪⎝⎭是Pauli 矩阵，||,|y y ψσσψψ**〉=⊗〉〉 为ψ的复共轭.设ρ为两个量子比特的混合态，则concurrence 定义为它的系综分解中任意可能的纯态平均值的最小值，即()inf (|)i i iC p C ρψ=∑〉，其中||i i i i p ρψψ=∑〉〈.在文献[42]中可以得到，对两个量子比特的混合态ρ，前面定义的形成纠缠度)(ρE 与concurrence 有很密切的联系，即:其中22()log (1)log (1).h x x x x x =----作者还对于以上定义的concurrence 给出了较为简单的计算公式，即{}1234()max ,0,C ρλλλλ=---其中i λ降序排列的本征值，()().y y y y ρσσρσσ=⊗⊗ 对维数为M 和N (这里设M N <)的两体量子纯态|ψ〉，文献[44]给出了concurrence 的定义:()h ρE =中国石油大学（华东）硕士学位论文(|)C ψ〉=其中[]|.A r ρψψ=T 〈〉设ρ为A B H ⊗H 中的一个量子态.Concurrence 的定义能推广到混合态的普遍情形:{}{},|()min ():||,i i i i i i i p iiC p C p ψρψρψψ〉=∑=∑〉〈Concurrence 的定义还能推广到多体量子态.假设⊗⊗∈21H H ψ，N H ⊗⋅⋅⋅N i d H i i ,,1,dim ⋅⋅⋅==为一个N 体量子纯态，文献[44]给出了ψ的concurrence 定义:,)()22(2)(221∑--=-ααρψψTr C N NN 其中αρ为所有不同的约化密度矩阵.设)(i i N i i C p ψψρ∑=为多体量子系统N H H H ⊗⋅⋅⋅⊗⊗21中的混合量子态，concurrence 同样由系综的Convex roof 给出:{},|()min().i i iNi i p iC p Cψρψψ〉=∑2.6.2量子态concurrence 的下界[44]首先考虑三体量子态.设H 表示一个d 维的Hilbert 空间.量子系统H H H ⊗⊗中的任意一个纯态都能写成下面的形式:,1,,*1,,1,,=∈=∑∑==ijk dk j i ijk ijk dk j i ijk a a C a ijk a ψ它的concurrence 为⨯-=)1(6)(3d dC d ψ,)(∑-+-+-iqm pjk pqm ijk pjm iqk pqm ijk pqk ijm pqm ijka a a a a a a a a a a a或者写成[][][],))(3()1(6)(2322213ρρρψTr Tr Tr d d C d ++--=第二章基础知识其中[][][]ρρρρρρ123132231,,Tr Tr Tr ===为ψψρ=的约化密度矩阵.定义,)(,)(2|133|12pjm iqk pqm ijk pqk ijm pqm ijk a a a a C a a a a C -=-=ψψαβαβ,)(1|23iqm pjk pqm ijk a a a a C -=ψαβ其中,(2|133|12αβαβC C 或者)1|23αβC 中的α和β分别表示相应于子空间1，2和3(1，3和2或者2，3和1)的下指标.设N i i i L ⋅⋅⋅21表示为其对应于子系统N i i i ,,,21⋅⋅⋅的特殊正交群)(21N i i i d d d SO ⋅⋅⋅.从而对于三体量子纯态可以得到[],)))()1(6)(2*1|232*2|132*3|123∑++-=αβαβαβαβψψψψψψψS S S d d C d 且)(),(),(1231|232132|133123|12βααββααββααβL L S L L S L L S ⊗=⊗=⊗=.ρ是任意H H H ⊗⊗中的混合态，concurrence )(ρC 符合[],)())(())(())(()1(6)(22)1(2)1(21|2322|1323|12322ρρρρρταβαβαβαβC C C C d dd d d d ≤++-≡∑∑--其中)(3ρτ为)(2ρC 的下界，{},)4()3()2()1(,0max )(3|123|123|123|123|12αβαβαβαβαβλλλλρ---=C 3|123|123|123|12)4(,)3(,)2(,)1(αβαβαβαβλλλλ按照降序排列，并且是非厄米矩阵3|12~αβρρ的四个非零本征值的平方根，)(.~2|133|12*3|123|12ρρραβαβαβαβC S S =和)(1|23ραβC 的定义相似于)(3|12ραβC .对任何N 体量子态,H H H ⊗⊗∈ρconcurrence )(ρC 符合,)())(()1(2)(22ρρρταβαβC C d m d p PN ≤-≡∑∑其中)(ρτN 是)(2ρC 的下界，∑p表示对于所有可能取到的指标βα,的所有组合求和，{}4,3,2,1,)(,)4()3()2()1(,0max )(=---=i i C p p p p p p αβαβαβαβαβαβλλλλλρ按降序排列，为非厄米矩阵p αβρρ~的4个特征值的非负平方根，p p p S S αβαβαβρρ*~=.对于一个完全可分的多体量子态.0)(,=ρτρN 故0)(>ρτN 说明这个量子态一定包含了某种程度纠缠，可以通过下界中国石油大学（华东）硕士学位论文)(ρτN 识别量子纠缠,因此就有利于我们度量纠缠程度的强弱,得到有效的下界,进一步为量子真正纠缠并发度的研究作出贡献.第三章真正量子纠缠的检测与度量第三章真正量子纠缠的检测与度量目前为止,量子纠缠的研究结果并不理想,对于两体纯态,我们可以简单的判别它是可分态还是纠缠态,对于两体混合态仍有很多问题尚未解决.但是对于多体量子系统我们想判断一个量子态是可分态还是纠缠态就更困难了,因为对于一个量子态来说可能是真正的纠缠态,真正的两体可分态,真正的三体可分态等,本章将从Bloch 表示与concurrence 出发给出真正量子纠缠的检测与度量定理.3.1真正量子纠缠的检测定义3.1.1设k Mi k i k，∑==σ1是n n ⨯矩阵M 的范数，其中，n i i ,,2,1,⋅⋅⋅=σ是M 按降序排列的奇异值.KFnM M =是KF 范数.定义⋅是一个向量或者矩阵的迹范数.设21231,t t 和312t 分别为矩阵中的,)1(,,)1(,ijk k i d j ijk k j d i t t t t ==+-+-和,)1(,ijk j i d k t t =+-这里我们设:，)(31)(312321231kk k k T T T M ++=ρ),(31)(321T T T M ρρρρ++=).)()()((31)(123132231ρρρρR R R N ++=引理3.1.1设},,min{n m d =对一个两体量子态,Bmn A H H ϕ∈⊗有()AT dϕϕ≤和().AT A B R d ϕϕ≤证明由Schmidt分解，设di ii ϕ==∑其中.011≥=∑=i di i u u ，通过Cauchy-Schwarz 不等式计算得()22()((.AT A B iiR d d ϕϕϕϕ==≤=∑引理3.1.2如果一个三体量子纯态是两体可分的，则它满足：）（i如果这个态是完全可分的，则jlmkT ≤中国石油大学（华东）硕士学位论文）（ii 如果这个态在lm j是可分的，则jlmkT ≤）（iii 如果这个态在lm j是纠缠的，则jlmkT ≤证明我们将反复用,23,13,12,12,3,2,1,)1(222)()(=-≤=-≤lm dd T i d d T lm i [45]且对于任意矩阵都有，M k Mk≤则）（i如果这个态是完全可分的，则()()()()()()()()()()()().j j j l m l m l m jlmkT T T T T T T T T T =⊗=⊗=）（ii 如果这个态在lm j是可分的，则()()()()()()j j lm tlm jlmkkT T TTT==≤）（iii 如果这个态在lm j是纠缠的，则()()()()()()jl jl lm tm jlmkkkT TTTT=⊗=≤定理3.1.1设三体量子态.321123d d d H H H H ⊗⊗=∈ρ如果ρ是两体可分的，那么我们可以得到：,321)}(),(max{dN M +≤ρρ其等价于，如果,321)}(),(max{dN M +>ρρ则ρ是GME .。
基于单光子的量子密钥分发方案
基于单光子的量子密钥分发方案量子计算机是未来计算领域的热门话题，而其中最重要的一项技术便是量子密钥分发。
它是基于量子力学原理，利用量子比特（qubit）进行的一种安全通信方式。
在该领域，基于单光子技术的量子密钥分发已成为研究的重点之一。
一、量子密钥分发的原理传统的加密技术是基于数学难题和算法的。
比如说，利用两个大质数的乘积很容易算出，但是将其因数分解却极其困难。
基于此，RSA加密算法和Diffie-Hellman协议等非对称加密技术被提出。
但是，由于计算机技术的发展，这些加密技术已经不能保证完全安全。
量子密钥分发的原理是利用光子的量子特性，确保通信双方可以确定一个共同的秘密密钥，且任何第三方盗窃或窥视此过程都将被立刻发现。
二、基于单光子的量子密钥分发基于单光子的量子密钥分发是一种典型的“BB84”协议。
在该协议中，Alice 和Bob 分别在同一量子态下发送电子通信中所需的比特，但是这个量子态被第三方别偷看或监听，都会引起其崩溃，导致通信双方的比特不匹配，最终通信错误。
具体而言，在BB84协议中，Alice会将原始的信息随机编码成四种不同的光子，分别是水平和垂直方向的偏振光、45度和135度方向的偏振光。
Bob同样会发送不同的偏振光信号。
通过检测这些光子的偏振，Alice和Bob可以比较他们收到的光子，最终得到通信密钥。
但是，如果有第三方窃听该过程，光子的偏振状态就会被测量和改变。
于是Alice和Bob不能快速把他们得到的光子揭示给第三方，而是必须经过多次的比对，以发现是否存在被监听的情况，确保密钥是安全的。
三、基于单光子技术的优势基于单光子技术既安全又灵活，因为它利用的是光子的量子特性，能够避免三次握手、中转服务器和挟持信息等其他传统加密技术的问题。
基于单光子技术具体有以下几个优势：1. 安全性强基于单光子技术的量子密钥分发，由于使用了量子特性，所以被称为绝对安全通信，并且不受信息窃听和窃取的影响，保证了数据的安全性。
Absolutely Maximally Entangled States
Absolutely Maximally Entangled States:Existence and ApplicationsWolfram Helwig and Wei CuiCenter for Quantum Information and Quantum Control (CQIQC),Department of Physics,University of Toronto,Toronto,Ontario,M5S 1A7,CanadaJune 12,2013AbstractWe investigate absolutely maximally entangled (AME)states,which are multipartite quantum states that are maximally entangled with re-spect to any possible bipartition.These strong entanglement properties make them a powerful resource for a variety of quantum information pro-tocols.In this paper,we show the existence of AME states for any number of parties,given that the dimension of the involved systems is chosen ap-propriately.We prove the equivalence of AME states shared between an even number of parties and pure state threshold quantum secret sharing (QSS)schemes,and prove necessary and sufficient entanglement proper-ties for a wider class of ramp QSS schemes.We further show how AME states can be used as a valuable resource for open-destination teleporta-tion protocols and to what extend entanglement swapping generalizes to AME states.1IntroductionEntanglement has been a hot topic since the beginning of quantum mechanics and fueled a lot of discussions,among them most notable the Einstein-Podolsky-Rosen (EPR)paradox [1],which finally led Bell to come up with a method of actually measuring entanglement [2].It was not until the advent of quantum information,however,that entanglement was recognized as a useful resource.Almost all applications in quantum information make either explicit or implicit use of entanglement,which makes it crucial to gain as much insight as possible.[3]While the entanglement of bipartite states is already very well understood[4,5,6],the road to its generalization to more than two parties is paved with many obstacles.Therefore we often have to restrict ourselves to special cases when analyzing multipartite entanglement.A prominent choice are states that extremize the entanglement for a certain measure of entanglement.In this paper we want to do that by focusing on absolutely maximally entangled (AME)states,which are defined as states that are maximally entangled for any possible bipartition.[7,8,9]1a r X i v :1306.2536v 1 [q u a n t -p h ] 11 J u n 2013Definition1.An absolutely maximally entangled state is a pure state,shared among n parties P={1,...,n},each having a system of dimension d.Hence |Φ ∈H1⊗···⊗H n,where H i∼=C d,with the following equivalent properties: (i)|Φ is maximally entangled for any possible bipartition.This means thatfor any bipartition of P into disjoint sets A and B with A∪B=P and, without loss of generality,m=|B|≤|A|=n−m,the state|Φ can be written in the form|Φ =1√d mk∈Z md|k1 B1···|k m B m|φ(k) A,(1)with φ(k)|φ(k ) =δkk .(ii)The reduced density matrix of every subset of parties A⊂P with|A|= n2 is totally mixed,ρA=d− n2 1d n2.(iii)The reduced density matrix of every subset of parties A⊂P with|A|≤n2 is totally mixed.(iv)The von Neumann entropy of every subset of parties A⊂P with|A|= n2 is maximal,S(A)= n2 log d.(v)The von Neumann entropy of every subset of parties A⊂P with|A|≤n2 is maximal,S(A)=|A|log d.These are all necessary and sufficient condition for a state to be absolutely max-imally entangled.We denote such a state as an AME(n,d)state.The simplest examples of AME states occur for low dimensional systems shared among few parties.Starting with qubits,the most obvious one is an EPR pair,which is maximally entangled for its only possible bipartition.For three qubits shared among three parties,we can recognize the GHZ state as an AME state.It is maximally entangled,with1ebit of entanglement with respect to every bipartition.For four qubits,there is no obvious candidate,and in fact it has been shown that for four qubits no AME state exists[9].We can stillfind an absolutely maximally entangled states for four parties,however,by increasing the dimensions of the involved systems.An AME(4,3)state for four qutrits shared among four parties exists,and it is given by[7]|Φ =1√92i,j=0|i |j |i+j |i+2j .(2)This is thefirst indicator that the search for AME states gets more promising as we increase the dimensions of the systems.Completing the characterization of AME states for qubits,it is known that AME states exist for5and6qubits.Explicit forms for them are given in Ref.[7], and it turns out that they are closely related to thefive-qubit error correction code.For7qubits,it is still not known if an AME state exists,whereas for≥8 qubits,it has been shown that no AME states can exist[9,10].In Ref.[7],we showed how AME states can be used for parallel teleportation protocols.In these protocols,the parties are divided into a sets of senders2and receivers,respectively.One of the two sets is given the ability to perform joint quantum operations,while players in the other set can only perform local quantum operations.Under these conditions,a parallel teleportation of multiple quantum states is possible if the set that performs joint quantum operations is larger than the other set.A closer look at these teleportation scenarios then led to the observation that any AME state shared by an even number of parties can be used to construct a threshold quantum secret sharing(QSS)scheme[11, 12,13].The opposite direction was also shown,with one additional condition imposed on the QSS scheme,namely that the shared state that encodes the secret is already an AME state.In this paper,we will give an information-information theoretic proof of this equivalence of AME states and threshold QSS scheme,which shows that the additional condition is not required.We will rather see that it is satisfied for all threshold QSS schemes.We will further give a recipe of how to construct AME states from classical codes that satisfy the Singleton bound[14].This construc-tion can be used to produce AME states for a wide class of parameters,and it even proves that AME states exist for any number of parties for appropriate system dimension.A result that could also be deduced from the equivalence of AME states and QSS schemes and a known construction for threshold QSS schemes[11].We will then show more applications for AME states.Thefirst be-ing the construction of a wider class of QSS schemes,the ramp QSS schemes,of which threshold QSS schemes are a special case.The next one is the utilization of AME states as resources for open-destination teleportation protocols[15]. Finally,we investigate to what extend entanglement can be swapped between two AME states.This paper is structured as follows.In Section2,we show how AME states can be constructed from classical codes,which also also shows the existence of AME states for any number of parties.In Section3,we establish an equiv-alence between even party AME states and threshold QSS schemes,using an information theoretic approach to QSS schemes.Section4shows how to share multiple secrets using AME states.In Section5,we show that AME states can be used for open-destination teleportation.After that,swapping of AME states is investigated in Section6.2Constructing AME States from Classical MDS CodesThere is a subclass of AME(n,d)states that can be constructed from optimal classical error correction codes.A classical code C consists of M codewords of length n over an alphabetΣof size d.For our purposes,the alphabet isgoing to beΣ=Z d and thus C⊂Z nd .The Hamming distance between twocodewords is defined as the number of positions in which they differ,and the minimal distanceδof the code C as the minimal Hamming distance between any two codewords.For a given length n and minimal distanceδ,the number of codewords M in the code is bounded by the Singleton bound[14,16]M≤d n−δ+1.(3) Codes that satisfy the Singleton bound are referred to as maximum-distance separable(MDS)codes.They can be used to construct AME states:3Theorem2(a).From a classical MDS code C⊂Z2md of length2m and minimaldistanceδ=m+1over an alphabet Z d,an AME(2m,d)state can be constructed as|AME =1√d mc∈C|c (4)=1√d mc∈C|c1 1···|c m m|c m+1 m+1···|c2m 2m.(5)Proof.The code C satisfies the Singleton bound,which means the sum contains a total of M=d2m−δ+1=d m terms.Furthermore,any two of these terms differ in at least one of thefirst m kets because the code has minimal distance δ=m+1.Hence the sum contains each possible combination of thefirst m basis kets exactly once.Moreover,for any two different terms,the last m kets must also differ in at least one ket and are thus orthogonal.This means the state has the form of Equation(1)with respect to the bipartition into thefirst m and last m parties.The same argument works for any other bipartition into two sets of size m,hence the state is absolutely maximally entangled.An analogous argument shows that a similar construction for an odd number of parties results in an AME state.Theorem2(b).From a classical MDS code C⊂Z2m+1d of length2m+1andminimal distanceδ=m+2over an alphabet Z d,an AME(2m+1,d)state can be constructed as|AME =1√d mc∈C|c (6)=1√d mc∈C|c1 1···|c m+1 m+1|c m+2 m+2···|c2m 2m+1.(7)Proof.The code contains M=d m terms.Each of the terms differ in at least one of thefirst m+1and last m terms.Thus,with the same argument as above, this is an AME state.Trivial states of that form are d-dimensional EPR states,which are repre-sented by the code with codewords00,11,...,(d−1)(d−1).This code has n=2,δ=2,M=d1.For n=3,we canfind the GHZ states for arbitrary dimensions, which can be constructed from the code000,111,...,(d−1)(d−1)(d−1),which hasδ=3and M=d1.As already mentioned in the introduction,for n=4 no AME state exists for d=2,however for d=3the AME(4,3)state given in Equation(2)can also be constructed from an MDS code,the[4,2,3]3ternary Hamming code.A wide class of MDS codes is given by the Reed-Solomon codes and its generalizations[17,16,18],which give MDS codes for n=d−1,n=d,and n=d+1,for d=p x being a positive power of a prime number p.From the Reed-Solomon codes,MDS codes can also be constructed for n<d−1[14].This shows that AME states exist for any number of parties if the system dimensions are chosen right.At this point we would like to mention that after posting a preliminary ver-sion of our last paper on this subject[7],it has been brought to our attention by4Gerardo Adesso that the results of this section have already been previously dis-covered by Ashish Thapliyal and coworkers,and were presented at a conference in 2003[19],but remained unpublished.3Equivalence of AME states and QSS schemes In Ref.[7],we showed that AME(2m,d )states,i.e.,AME states shared between an even number of parties,are equivalent to pure state threshold quantum secret sharing (QSS)schemes that have AME states as basis states and share and secret dimension equal to d .Here we will give an information-theoretic proof of this equivalence,which shows that the requirement that the basis states of the QSS scheme are AME states is redundant,as it follows from this proof that these states are always absolutely maximally entangled.Before stating the theorem and the proof,we give a short motivation why AME states and QSS schemes are related.Consider an AME(2m,d )state shared among an even number of parties.If we take any bipartition into two sets of parties A and B ,each of size m ,a d m dimensional state can be teleported from one set to the other due to the maximal entanglement between A and B .Moreover,we have shown in Ref.[7],that the teleportation can be performed in such a way that each party in the sending set B performs a local teleportation operation on their qudit,while the parties in the receiving set A perform a joint quantum operation to recover all m teleported qudits.This is depicted in Figure 1for the case of m =4.This also works if only one party in B ,which we call the dealer D ,performs the teleportation operation,while the others do nothing.Then the teleported d -dimensional state can still be recovered by the players in set A .Furthermore,this also works for any other bipartition into sets A and B of size m ,with D ∈B ,without changing the teleportation operation D has to perform,but now the parties in A can recover the teleported state (see Figure 2).This means that any set with m parties can recover the state.Moreover,the no-cloning theorem guarantees that the complement of a set that can recover the state has no information about the state.Hence all sets with less than m parties cannot gain any information about the state.This,however,are exactly the requirements for a threshold QSS scheme,therefore we have constructed a ((m,2m −1))threshold QSS scheme from the AME(2m,d )state.To formally show this,and moreover that it also works in the opposite direction,meaning that a ((m,2m −1))threshold QSS scheme is always related to an AME(2m,d )state,we will use the information theoretic description of QSS schemes as introduced in Ref.[13].Let us quickly review the framework for a pure state ((m,2m −1))threshold QSS scheme [11].A secret S is distributed among the players P ={1,...,2m −1}such that any set A ⊆P with |A |≥m can recover the secret,while any set B ⊂P with |B |<m cannot gain any information about the secret.We further only consider the case where the dimension d of the secret is the same as the dimension of each player’s share.The secret is assumed to lie in the Hilbert space H S ∼=C d ,and the share of party i in H i ∼=C d .The encoding is described by an isometryU S :H S →H 1⊗···⊗H 2m −1.(8)The secret S is chosen randomly and thus is described by ρS =1/di |i i |.We5Figure 1:(Color online)Parties in B (green)perform local teleportation op-erations,parties in A (red)can recover teleported states by performing a joint quantumoperationFigure 2:(Color online)After D (blue)performs her teleportation operation,any set of m parties (red),A ,A ,A etc.,can recover the teleported state.Any set of parties with m −1or less parties (any set consisting only of green parties)cannot gain any information about the teleported state.consider its purification by introducing a reference system R such that |RS =1/√d i |i |i ∈H R ⊗H S .Let ρRA denote the combined state of the reference system and a set of players A ⊆P after U S has been applied to the secret.Then the players A can recover the secret,if there exists a completely positive map T A :H A →H S such that [13,20]1R ⊗T A (ρRA )=|RS .(9)This can be stated in terms of the mutual informationI (X :Y )=S (X )+S (Y )−S (X,Y )(10)as follows:Definition 3.An isometry U S :H S →H 1⊗···⊗H 2m −1creates a ((m,2m −1))threshold QSS scheme if and only if,after applying to the system S of the pu-rification |RS ,the mutual information between R and an authorized (unautho-rized)set of players A (B )satisfiesI (R :A )=I (R :S )=2S (S )if |A |≥m (11)I (R :B )=0if |B |<m.(12)6Here S is the von Neumann entropy,and because of S(i)≥S(S)[13],we haveS(S)=S(R)=S(i)=log d.(13) From Equations(10)to(12)it immediately follows thatS(R,A)=S(A)−S(R)if|A|≥m(14)S(R,B)=S(B)+S(R)if|B|<m.(15)Theorem4.For a state|Φ the following two properties are equivalent:(i)|Φ is an AME(2m,d)state.(ii)|Φ is the purification of a((m,2m−1))threshold QSS scheme,whose share and secret dimensions are d.Proof.(i)→(ii):We need to show that for an AME(2m,d)state Equations(11) and(12)are satisfied,where R can be any of the2m party.This follows directly from the definition of the mutual information,Equation(10),and Defintion1 (v).(ii)→(i):Consider an unauthorized set of players B,with|B|=m−1. Then the set is B∪i is authorized for any additional player i/∈B,and from Equation(14)we haveS(B,i,R)=S(B,i)−S(R)(16)On the other hand,using the Araki-Lieb inequality[21]S(X,Y)≥S(X)−S(Y) and Equation(15)givesS(B,i,R)≥S(B,R)−S(i)=S(B)+S(R)−S(i).(17)Combining the last two equations and using S(S)=S(R)=S(i)showsS(B,i)≥S(B)+S(i),(18)where equallity must hold due to the subadditivity of the entropy S(X,Y)≤S(X)+S(Y).This means that the entropy increases maximally when adding one player’s share to m−1shares.The strong subadditivity of the entropy[21]S(X,Y)−S(Y)≥S(X,Y,Z)−S(Y,Z)(19)states that adding one system X to a system Y increases the entropy at least by as much as adding the system X to a larger system Y∪Z that contains Y. So in our case,adding one share to less than m−1shares increases the entropy by at least S(i),and since this is the maximum,it increases the entropy exactly by S(i).Hence,starting out with a set of no shares,and repeatedly adding one share to the set until the set contains any m shares and is authorized,shows that any set of m shares has entropy mS(i).This shows that the entropy is maximal for any subset of m parties and thus|Φ is an AME(2m,d)state.Corollary5.The encoded state U S|S of a specific secret|S with a((m,2m−1))threshold QSS protocol with share and secret dimension d is an AME(2m−1,d)state.74Sharing multiple secretsIn the previous section,we outlined how an AME state can be used to construct a QSS scheme.The role of the dealer is assigned to one of the parties and he performs a teleportation operation on his qudit,which encodes the teleported qudit onto the qudits of the remaining parties such that the criteria for a QSS scheme are met.While Theorem 4shows the equivalence of AME states and QSS schemes,the actual protocol for the encoding and decoding operations has been presented in Ref.[7].Note that in the described scenario,the role of the dealer can be assigned to any player.Thus one may ask,what happens if more than one of the players assumes the role of the dealer.The answer is that,given an AME(2m,d )state,up to m players are able to independently encode one qudit each onto the qudits of the remaining players in such a way that results in a QSS scheme with a more general access structure.For a secret sharing scheme with a general access structure,each set of players falls into one of three categories [22,23].1.Authorized :A set of players is authorized,if it can recover the secret2.Forbidden :A set of players is called a forbidden set,if the players cannot gain any information about the encoded secret3.Intermediate :A set of players is classified as an intermediate set,if they cannot recover set secret,but may be able to gain part of the information.This means that the reduced density matrix of that set of players depends on the encoded secret,but not enough as to recover the secret.A special kind of access structure is a (m,L,n )ramp secret sharing scheme[24].Here n is the total number of players,m is the number of players needed to recover the secret,and L is the number of shares that have to be removed from a minimal authorized set to destroy all information about the secret.In terms of the above defined set categories that means that any set of m or more players is authorized,any set of m −L or less players is forbidden,and any set consisting of more than m −L ,but less than m players is an intermediate set.This is the access structure we get from an AME(2m,d )state if more than one party assumes the role of the dealer.Theorem 6.Given an AME(2m,d )state,a QSS scheme with secret dimension d L and a (m,L,2m −L )ramp access structure can be constructed for all 1≤L ≤m .Proof.The encoding of the secret is done by assigning the role of dealer to L of the 2m players.For simplicity we choose them to be the first L players.Each of them performs a Bell measurement on their respective qudit of the AME state and one qudit of the secret.The Bell measurement is described by the general d -dim Bell states |Ψkl and the unitaries U kl that transform among them [25]|Ψqp =1√d j e 2πijq/d |j |j +p (20)U qp = j e 2πijq/d |j j +p |,(21)8where the kets are understood to be mod d .For a secret |s and outcomes (q 1,p 1)...(q L ,p L )for the Bell measurement of the dealers,the initial AME(2m,d )state is transformed to|ΦS =1√ k ∈Z m ds qp ,k 1···k L |k L +1 B 1···|k m B m −L |φ(k ) A .(22)Heres qp ,k 1···k L = k 1···k L |U †q 1p 1⊗···⊗U †q L p L |s ,(23)and the partition of the remaining 2m −L parties into two sets A and B of size m and m −L ,respectively,is arbitrary.After obtaining their measurement outcomes,the dealers broadcast their results to all of the remaining players.This concludes the encoding process.To show that any set of m or more players is authorized,it suffices to show that set A in Equation22can recover the secret.They can do so by applying the unitary operationU =(U q 1p 1⊗···⊗U q L p L ⊗1)V(24)withV =k ∈Z m d |k 1 ···|k m φ(k )|,(25)to their system.This changes the state toU |ΦS =1√d m −L (k L +1,...,k m )∈Z m −L d |k L +1 B 1···|k m B m −L |s A |k L +1 A L +1···|k m A m (26)where A ={A 1,...,A L }.Thus the players in set A have the secret in their possession.It immediately follows from the no-cloning theorem that B ,and thus any set of size m −L or less,cannot have any information about the secret since all information is located in the complement set.Alternatively,this also follows from the observation that the reduced density matrix of B is always completely mixed,independent of the secret.The last thing left to show is that all sets with more than m −L but fewer than m players are indeed intermediate sets.To see that,consider the case L =1,where a set C of m −1players is not authorized to recover the secret.If one more player in the complement of C assumes the role of the dealer,the scheme is changes to L =2.This operation does not change the fact that C cannot recover the first secret,and thus it is still not authorized for L =2.This argument can be continued to any other 1<L ≤m by adding more dealers.Hence a set of m −1(or fewer)players is not authorized to recover the secret for all value of 1≤L ≤m .That a set of more than m −L players is not forbidden follows from the fact that information cannot be lost and thus the complement of a forbidden set has to be authorized.However,we just argued that the complement of a set of more than m −L players is not authorized (since it consists of less than m players).Hence any set with more than m −L and fewer than m players is an intermediate set.9A closer look at the proof shows us that it actually is not absolutely necessary for the initial state to be maximally entangled with respect to any bipartition,but only for bipartitions for which all dealers are in the same set.In fact,we can generalize the proof of Theorem 4to the case of ramp QSS to show that this is a necessary and sufficient condition for the construction of (m,L,2m −L )ramp QSS schemes.Theorem 7.For a state |Φ ∈H P ⊗H R ,shared between 2m −L players P ,each holding a qudit,and L reference qudits,the following two properties are equivalent:(i)|Φ is maximally entangled for any bipartition for which the L referencequdits are in the same set.(ii)|Φ is the purification of a (m,L,2m −L )ramp QSS schemes.The encodedsecret of the ramp QSS scheme has dimension d L ,and each share has dimension d .The proof is a straightforward generalization of the proof of Theorem 4and is provided in Appendix A.5Open-destination teleportationGiven a state with such high amount of entanglement as the AME state has,one cannot help thinking about ways of using these resources for teleportation protocols.In Ref.[7]we already showed how AME states can be used for two different teleportation scenarios that require either sending or receiving parties to perform joint quantum operations,while the other end may only use local quantum operations.Another teleportation scenario that uses genuine multipartite entanglement,and has already been demonstrated experimentally [15],is open-destination tele-portation.In this scenario,a genuinely multipartite entangled state is shared between n parties,each in the possession of one qudit.One of the parties,the dealer,performs a teleportation operation on her qudit and an ancillary qudit |Φ .After this teleportation operation,the final destination of |Φ is still un-decided,thus open-destination teleportation.The destination is decided upon in the next step,where a subset A of the remaining parties P performs a joint quantum operation on their qudits such that a player in P \A ends up with the state |Φ –up to local operations that depend on measurement outcomes of the dealer and parties A .Here we want to show that open-destination teleportation can also be performed with AME states.Assume that an AME(n,d )state has been distributed among n parties.One of the n parties is assigned the role of the dealer.She performs a Bell measurement on her qudit and the secret |S = a i |i .This transforms the state to|S |Φ →|ΦS =1√d m (k,i )∈Z m da pq,i |k 1 B 1···|k m −1 B m −1|φ(k,i ) A ,(27)where pq labels the outcome of the Bell measurement and has to be made public.The remaining n −1parties that share the resulting state have been divided10into two sets A and B of size n/2 and m−1= n/2 −1,respectively.Now,after the teleportation operation has been completed,the parties in set Amay choose one party B i∈B as thefinal destination for the state|S .Then, after performing the joint unitary operation of Equation(25)followed by a Bellmeasurement on qudits A i and A m with outcome rs,the party B i ends up withthe state|ΦB i =U†rs U†pq|S ,which can be easily transformed to|S if themeasurement results pq and rs are known.Note that with the parallel teleportation protocol introduced in Ref.[7],also one of the parties in A can be chosen to receive the state|S .Thus,after the dealer’s teleportation operation is completed,any set of size greater or equal n/2 can choose any of the remaining n−1parties as thefinal destination of the teleportation.116Swapping of AME statesEntanglement swapping [26]is a very useful tool for the application of entan-glement in communication.By making a Bell measurement on Bob’s side,two entangled states shared between Alice and Bob,and Bob and Charlie,respec-tively,can be transformed into an entangled state shared by Alice and Charlie.Employing this procedure in quantum repeaters [27]allows entangled states to be used for long distance communications.In this section,we show to what ex-tent a generalization of the entanglement swapping protocol can be constructed to allow swapping of entanglement between absolutely maximally entangled states shared between different parties.Assume that parties {1,2,...,2n }share an AME(2n,d )state,|Φ 1,...,2n = |i 1···i n 1,...,n |φ(i 1,...,i n ) n +1,...,2n (28)= |i 1···i n 1,...,n U |i 1···i n n +1,...,2n ,(29)where U is a unitary transformation with U |i 1···i n =|φ(i 1,...,i n ) .Suppose parties {n +1,...,3n }also share an AME(2n,d )state|Φ n +1,...,3n = |i 1···i n n +1,...,2n U |i 1···i n 2n +1,...,3n .(30)Now each of the parties {n +1,...,2n }performs a Bell measurement on their qudits from both AME states.Without loss of generality,we can assume the measurement result is (q,p )=(0,0)(see Equation (20)for the notation),since other measurement outcomes produce the same state up to local transforma-tions.Then the state shared by the parties {1,...,n,2n +1,...,3n }becomes|Φ 1,...,n,2n +1,...,3n = |i 1···i n 1,...,n U 2|i 1···i n 2n +1,...,3n (31)Consecutive applications of the above procedure gives the following lemma:Lemma 8.Suppose each group of parties {1,...,2n },{n +1,...,3n },···,{mn +1,...,(m +1)n }shares an AME(2n,d )state,|Φ = |i 1···i n U |i 1···i n .(32)Then,if each of the parties {n +1,n +2,...,mn }performs a Bell measurement on their two qudits,the resulting state shared by the parties {1,...,n,mn +1,...,(m +1)n }is locally equivalent to|Φ 1,...,n,mn +1,...,(m +1)n = |i 1···i n 1,...,n U m |i 1···i n mn +1,...,(m +1)n (33)Proof by induction.The case for m =2is demonstrated in the above discussion already.If the lemma holds for m ,for m +1the two remaining states,after the parties {n +1,n +2,...,mn }performed their Bell measurements,are |Φ 1,...,n,mn +1,...,(m +1)n = |i 1···i n 1,...,n U m |i 1···i n mn +1,...,(m +1)n (34)12。
美国伦斯勒理工学院太赫兹研究中心的远距离太赫兹探测技术研究
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图 １ 远 距 离太 赫 兹传 感技 术 的实 验原 理 图
图 ２ 利 用 ＴＨ — Ｅ ｚＲＥ Ｆ技 术 测 量 炸 药 成 分 ４ Ａ— ＤＮＴ 的 吸 收 光 谱
他 们 是 通 过 调 整 基 频 激 光 脉 冲
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１ ｍ处独 自完成 了等离子体声 １ 波 探 测 实 验 ，这 只是 受 到 了实 验室 可用 空 间的限制。随着 与 可疑危 险物 之 间距 离的增加 ， 该 系 统 的另 一 个 优 势 将 体 现 出 来 ，即该 系 统 并 不 需 要 通 过 直 接视 线来采集 信 号，因为麦克 风在各种 情况下均 可获取 音频 信息。 Ｂ ｎ ｍｎＣｏｇ ｅｊ ｉ ｌ ｈ的这项发 ａ ｕ 明 突破 了远 距 离 太 赫 兹光 谱 测 量 的基 本 限 制 ， 它 将使 人 们 可 以从 一 段 安 全 距 离 外 进 行 大 气 环境监测、 囱排放 监测、 烟 可疑 包裹检查 以及地雷探 测等。
马斯克的星链计划英语文章
马斯克的星链计划英语文章English:Elon Musk's Starlink project aims to provide global high-speed broadband coverage by deploying a constellation of satellites in low Earth orbit. The ultimate goal is to bridge the digital divide by offering internet access to remote and underserved areas around the world. Unlike traditional internet providers, Starlink's satellites operate closer to the Earth, reducing signal latency and delivering faster connectivity. This technology has the potential to revolutionize internet access in rural communities, on ships, airplanes, and in disaster-stricken regions where traditional infrastructure is lacking. With thousands of satellites already launched and more planned for deployment, Starlink is steadily expanding its coverage and improving its service quality. Despite facing challenges such as regulatory hurdles and concerns over space debris, Elon Musk remains committed to his vision of creating a global satellite internet network that is fast, reliable, and accessible to all.中文翻译:马斯克的星链计划旨在通过在近地轨道部署一组卫星网络，提供全球高速宽带覆盖。
尼康 TIRF 细胞成像共享资源 (CISR) 742B Light Hall 快速指南说明书
Nikon TIRFCell Imaging Shared Resource (CISR) 742B Light Hall Quick Guide 1. Sign in to the log book.2. Turn on switches 1-4 in numerical order. The computer must be OFF before starting 1-4.∙1 is the power strip on the left wall.∙ 2 is the key on the top laser box. ∙3 is the key on the bottom laser box.∙4 is the green button on the power strip on the left and tothe back of the microscope.3. The computer should start up.4. Log in to the computer usingyour VUNetID and password.5. Start NIS -Elements software.6. Login to NIS -Elements using your first name and no password.1324561 32 45 61. There are preset layouts along the bottom of the screen for Widefield (WF), TIRF, and Bleaching. These will open the appropriate windows for each type of imaging.2. Along the top of the screen arepreset Optical Configurations . You will see all the default configurations. You may dupli-cate a configuration by right clicking. Rename for yourself, and now you are able to change this new duplicated configuration to fit your imaging needs. This new configura-tion will only be visible under your named login.3. To reuse settings from a previous image, open the image and right click within the image. You will have choice for “reuse camera settings”, “reuse acquisition settings”, or “reuse XYZ”.4.To view your image on the screen, use the green arrow “live” button on the top left. Stop with the red circle. Capture a single time point with the camera button. 5. Autoscale can be useful for viewing images when setting exposure time and laser power.6. For fast imaging, you may wish to adjust the frame size. In a live image, choose ROI. You may choose a preset size or define an ROI.7. Images may besaved as ei-ther .nd2 (Nikonformat) or TIF. Movies will save auto-matically ac-cording to your settings in the ND Acquisitionwindow.1 32 13 24 41.Choose objective (10x dry, 20x dry, or 60x Oil immersion)on left side of scope or in software. Use Nikon oil.2.Add oil to objective if using 60x TIRF lens.3.Loosen screws on stage to adjust for sample. Place samplein holder and tighten screws.4.On either side of scope, choose Coarse, Fine, or Extra Fineto move focus with focus knobs. Turn knob toward user tobring objective up.5.XY joystick also has Coarse, Fine, and Extra Fine formovement. Twist joystick to toggle between choices.132132JoystickPerfect Focus (PFS)1. To find your sample by bright light through the oculars, choose either DIC or BF from the top menu bar.2. To find your sample by fluorescence through the oculars, choose “FITC Eyes” configura-tion from top menu bar.3. Click “Spectra” to turn on widefield illumi-nation to eyes. Ensure that E100 is selected under Ti Pad under Light Path.4. You should be able to see fluorescence through the oculars. This filter show both Green and Red excitation.132 4 13 246.To visualize sample on screen, choose WF tab at bottomof software, and choose WF optical configuration at top(DAPI WF, FITC WF, TRITC WF, CY5 WF, CFP WF, YFP WF).7.Start “Live” with green arrow button.8.Focus with focus knobs on scope.9.Perfect Focus (PFS) may be used to find and hold correctfocal plane.10.Choose PFS ON button on front of scope. While thisbutton is blinking, focus with the focus knob. When PFSstops blinking, focal plane is found. Now use the PFS wheelfor fine focusing.1 32456132456JoystickPerfect Focus (PFS)81231. For live cell imaging, turn on orange power button on the incubator above the laser boxes.2. Three heaters will come to their appropriate tem-peratures. Top Heater will reach 43, Stage Heater will reach 39, and Bath Heater will reach 41. The Lens Heater needs to be switched on separately. Dicuss your needs with CISR staff.3. If not already in place, put the heated stage adap-tor in place. Use lab tape to hold in place.4. Ensure there is sufficient water in the heated stage water bath. Use dI H2O.5. Turn on the CO2 tank on the wall by the main CISR door. Turn on with the main silver knob.6. Check the CO2 indicator on the front of the incu-bator box to ensure CO2 is on.7. Although temperatures will be ready within 5-10 minutes, for optimal environmental conditions, al-low temperature and CO2 to equilibrate for 30 minutes.1231 231.Choose WF tab at the bottom of screen, and choose optical configuration at the top of the screen to match your fluorophore of interest. 2. To adjust signal, adjust Spectra % output as well as exposure time in Zyla camera window. 3. Use PFS to focus sample.4. Choose “Live ” green arrow to see image on screen.5. For single time point, click “Capture ”. Repeat for multiple channels, and merge to create multi -channel image. Merge can be found under File.6. For time -lapse acquisition, use ND Acquisition window.7. Set -up multiple channels under Wavelength. Choose each channel under Optical Configuration.8. Set -up time -lapse under Time. Choose Define. Inter-val is time between images, and Duration is total time.1 2 45 713 24 6 7 31256H-TIRFAutomated TIRF alignment in software1.Raise microscope condenser to make visualizing light easier.2.H-TIRF alignment is done in the software.3.For H-TIRF, open Ti-LAPP H-TIRF Pad.4.Adjust Angle until you see the laser spot on the wall.5.Continue adjusting until the light is overhead.6.Focus the spot to the smallest possible spot.7.Set Direction to 180.8.Adjust Angle again until you see TIRF signal on sample. You will see a bright signal andthen nothing. Adjust Angle back until see image again.9.Adjust Direction to fine tune across the best region of your sample.M-TIRFManual TIRF Alignment1.M-TIRF is done manually with adjustments on the microscope.2.Angle3.Focus4.Direction1.After finding an image by widefield, choose TIRF layout at bottom of screen and TIRF optical configuration at topof screen (488 H -TIRF, 561 M -TIRF, or General M -TIRF).In the previous sections you should have found cells and focused and adjusted TIRF angle.2. Optimize signal by adjusting laser power in the LU -NV Nidaq Pad window and exposure time in the Zyla camera window.3. For single time point, capture image with “capture” cam-era button along top of screen.4. For time -lapse imaging, open ND Acquisition window.5. First tab in ND Acquisition is for time -lapse. Interval is de-lay between images. Duration is total time -lapse. For shortest possible interval, choose “no delay” for interval.6. Choose RUN NOW in ND Acquisition window.1324561 32 45 61. Switch configuration to 561M@H -TIRF or 640M@H -TIRF on top of screen.2. Open the following 3 windows — Triggered Acquisition, ND Acquisition, and Ti -LAPP H -TIRF Pad3. For Triggered Acquisition add channel. For each channel add 3 lines —LineFilter WheelLU -NV NIDAQ Switcher .4. Choose appropriate excitation wavelength for Line .5. For 488, FilterWheel =1 and Switcher =1.6. For 561, FilterWheel =2 and Switcher =4.7. For 640, FilterWheel =3 and Switcher =4.8.Set exposure time in Triggered Acquisition window. Exposure time must be the same for both channels. 9. Set time -lapse parameters in ND Acquisition window. Open Define window.10. C heck “Enable Triggering” in Triggered Acquisition window.11. E nsure that the Lower Turret Layer in the Ti -LAPP Pad shows both H -TIRF and TIRF highlighted . 12. C lick “RUN NOW” in ND Acquisition window.1 3 24 5 6132 45661. Choose the “Bleaching” tab at the bottom of the screen.2. In addition to the TIRF set -up on the previous page, open the Bruker Miniscanner window.3. Choose laser for bleaching and set parameters (% and dwell time) in both the Bruker Miniscanner window and the LU_N4 Padwindow. 4. On right side of image window, right click on ROI icon to choose ROI shape. Draw ROI.5. Right click on ROI and choose “Use as Stimulation ROI ”.6. Set exposure time in camera window.7. Set bleaching and time -lapse in ND Sequence Acquisition window. Bleaching can be Sequential or Simultaneous (next page).12 5 34 1 2 53 47. For Sequential bleaching, set up actions in ND SequenceAcquisition window. For example, add #1 ND Acqui-sition, #2 Stimulation, #3 ND Acquisition. Open Define window for each to set interval and delay.8. For Simultaneous bleaching, set up actions in ND Se-quence Acquisition using Simultaneous Stimulation. Open Define window to set interval and delay. Stimula-tion time will be set based on ROI size and dwell time in miniscanner window. 9. For bleaching, ensure that the following buttons are active:A. Under Filters , choose Galvo on Turret 2(blue box, second from left)B. In Ti -LAPP Pad window, choose FRAP on Upper Turret LayerC. Under menu bar at top of screen, turn on AOTF. 10. RUN NOW. 1 2 43 1 24 3Check the CISR scheduling calendar to see if anyone issigned up after you. If another user is coming with 1 hour,please log out of the software, sign out in the log book, andleave the microscope and lasers ON.If no one is coming after you, follow the next steps.1.Close NIS software.2.Shut down the computer.3.Turn off power strip 4 (green)4.Turn off laser box 3 (bottom)5.Turn off laser box 2 (top).6.Turn off power strip 1 (left wall).7.Sign out in log book.e again soon!Updated 05/2016。
三体量子纯态可分与纠缠的刻画
三体量子纯态可分与纠缠的刻画梁文婷;陈峥立【摘要】Characterization of the full separability for tripartite quantum pure states and several corollaries are given out by their Schmidt decomposition.Two theorems to characterize the full separability for tripartite quantum pure states by conditional entropy and mutual information are given and then several necessary and sufficient conditions to characterize the different kinds of separability and entanglement for them are obtained.Several typical examples are given.%利用三体量子纯态的Schmidt分解给出其完全可分的等价刻画以及若干推论.借助相对熵和互信息给出三体量子纯态完全可分的等价刻画,进而得到三体量子纯态是可分或纠缠的充要条件,并给出几个典型的例子.【期刊名称】《陕西师范大学学报（自然科学版）》【年(卷),期】2017(045)003【总页数】5页(P24-28)【关键词】三体量子纯态;完全可分;vonNeumann熵;Schmidt分解【作者】梁文婷;陈峥立【作者单位】陕西师范大学数学与信息科学学院, 陕西西安 710119;陕西师范大学数学与信息科学学院, 陕西西安 710119【正文语种】中文【中图分类】O177.1量子纠缠的概念最先由薛定谔和爱因斯坦等在文献[1-2]中提出。
单片集成美国MIT和圣地亚国家实验室基于天线阵列的思想，研发出单片太赫兹激光器
单片集成美国MIT和圣地亚国家实验室基于天线阵列的思想，研发出单片太赫兹激光器美国麻省理工学院（MIT）和圣地亚国家实验室采取全新激光器设计理念，构造出单片太赫兹激光器，可显著减少功耗和尺寸，以及发出更密集的光束，对推动激光器的实用化至关重要。
性能指标新的激光器是由37个微纳工艺制造激光器单片集成的阵列，所占面积不足1mm2，由于所有激光器发出的激光都是“锁相”的，所以功耗极低。
该激光器阵列可工作在3THz，彼此之间仅以相当于其基本波长(100µm)的间距隔开。
研究人员表示，通过具有明确相位关系的远场辐射彼此锁相，这些激光器阵列就能以最高450mW A-1斜率效率以及近绕射极限的波束散射，以脉冲方式联合输出6.5mW单模激光。
激光器结构示意图思路来源研究思路受微波领域的天线工程技术激发，而非局限于传统的光电领域。
微波领域所用天线阵列一般都可实现窄波束形成。
研究人员将纳米激光器设计为定义明确的网格图案，利用纳米激光器发散的波束图案来获取和阵列中其他激光器的强耦合，形成比个别激光器单独实现的总合更集中和强大的光束。
相位同步在新的阵列中，每一个激光器都有一个由内部电流产生的磁场，以使激光器所发激光的相位同步。
形成光束时，激光器阵列锁相是一种极其高效率的方法，可提高输出功率以及降低发射阈值。
单片集成研究人员在《自然光子学》杂志上发表了其研究成果，文中对比了已有的的四项锁相技术，并表示其用于微米尺寸均有缺陷：一些需要将光电器件紧密放置，增加制造难度；另一些需要片外光电器件，其与激光器的相对位置必须精确。
而新研究出的激光器则是单片集成。
横向捕获在数十年的发展中，对芯片级激光器的研究一直活跃，潜在应用包括计算机、环境生物传感用芯片到芯片通信等。
随着激光器尺寸的减小，所发激光变得分散，变得像一个小天线。
如果芯片级激光器只试图向一个方向发出激光，那它在横向上的激光就被浪费了，并增加了功耗。
而新设计的阵列则重新捕获了横向上的激光，并重新按垂直于阵列的方向再次发射出去，使光束比其他实验性质芯片级激光器的光束要密集得多。
星链技术原理范文
星链技术原理范文星链技术（Stellar）是一种去中心化的开源区块链协议，旨在支持快速、低成本的跨境支付和资金转移。
它由Jed McCaleb创立，目前由非营利组织Stellar Development Foundation（SDF）管理。
该技术的核心原理是通过分布式账本和一系列共识算法来实现网络中的信任与协作。
星链技术采用可扩展的一致性协议，即联邦拜占庭共识算法（Federated Byzantine Agreement，简称FBA），来达成网络上节点之间的一致性。
这种协议的特点是，节点只需要信任少数其他节点，而不需要信任整个网络。
在星链网络中，每个节点都可以选择自己信任的节点（称为「信任节点」），并与之建立连接。
通过联邦拜占庭共识算法，网络中的节点可以就交易的有效性和顺序达成一致意见，从而实现去中心化的可信交易。
星链技术的分布式账本由一系列「帐本」（ledger）组成，每个帐本代表一段时间内的所有交易记录。
帐本之间通过共识算法确保交易的一致性。
在星链中，帐本按照时间顺序形成一个链，并由一组节点（或者称为「验证节点」）维护和验证。
验证节点之间通过交换信息来达成共识，并更新帐本。
新的帐本包括了最新的交易记录和账户余额，它们通过数字签名和哈希值等加密算法来确保安全性。
星链技术的另一个重要组成部分是「合约」（contracts）。
合约是一段运行在星链上的代码，它定义了一系列规则和条件，并在符合条件时执行相应的操作。
星链中的合约可以实现各种功能，比如发行资产、进行众筹、实现去中心化交易等。
合约的执行是由星链上的所有节点来共同完成的，确保了系统的公平性和安全性。
为了实现可扩展性和低成本的跨境支付，星链技术引入了一种称为「路径支付」（path payments）的机制。
路径支付允许用户在转账时选择最佳的支付路径，将其资金自动进行一系列的货币兑换，以达到最终的支付目标。
这种机制主要通过星链上的「资产」（assets）和「市场」（markets）来实现。
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the shares with correlated Gaussian white noise[8]. Choos-
ing the Gaussian noise to have the same correlations as the
quadrature entanglement, the shares can be expressed as
√
aˆ1 = (aˆin +aˆEPR1+δN )/√2
(1)
aˆ2 = (aˆin −aˆEPR1−δN )/ 2
(2)
aˆ3 = aˆEPR2 +δN ∗
(3)
ψin
secret state
dealer
பைடு நூலகம்OPA
pump OPA
1:1
φ
player 1
1:1
φ
player 2
EPR entangled
ond entangled beam aˆEPR2 form the three shares which are distributed to the players. The entangled state ensures that
the secret is protected from each player individually and is
berg picture of quantum mechanics, a quantum state can be represented by the field annihilation operator aˆ = (Xˆ + + iXˆ −)/2, where Xˆ ± = Xˆ ± + δXˆ ± are the amplitude (+) and phase (-) quadratures, with variances of V ± = (Xˆ ±)2 .
(Dated: February 1, 2008)
We demonstrate a multipartite protocol to securely distribute and reconstruct a quantum state. A secret quantum state is encoded into a tripartite entangled state and distributed to three players. By collaborating, any two of the three players can reconstruct the state, whilst individual players obtain nothing. We characterize this (2, 3) threshold quantum state sharing scheme in terms of fidelity, signal transfer and reconstruction noise. We demonstrate a fidelity averaged over all reconstruction permutations of 0.73 ± 0.04, a level achievable only using quantum resources.
Tripartite Quantum State Sharing
Andrew M. Lance,1 Thomas Symul,1 Warwick P. Bowen,1 Barry C. Sanders,2 and Ping Koy Lam1 1Quantum Optics Group, Department of Physics, Faculty of Science, Australian National University, ACT 0200, Australia 2Quantum Information Science Group, Department of Physics and Astronomy, University of Calgary, Alberta T2N 1N4, Canada
complete basis for all quantum states, arbitrary states can be
shared by this scheme.
The quantum states of interest in this paper reside at the fre-
quency sidebands of an electromagnetic field. In the Heisen-
arXiv:quant-ph/0311015v2 19 May 2004
Secret sharing[1] is a powerful technique in computer science, which enables secure and robust communication in information networks, such as the internet, telecommunication systems and distributed computers. The security of these networks can be enhanced using quantum resources to protect the information. Such schemes have been termed quantum secret sharing[2]. Many applications in quantum information science, however, require the distribution of quantum states. One such example are quantum information networks, which are expected to consist of nodes where quantum states are created, processed and stored, connected by quantum channels[3]. It is of paramount importance that the quantum channels in these networks allow the robust and secure distribution of quantum states between nodes. Cleve et al.[4] proposed the secret sharing of quantum states as a protocol that provides these capabilities, overcoming failures or conspiracies by nodes. We term this quantum state sharing to differentiate from the quantum secret sharing of classical information. In (k, n) threshold quantum state sharing[4], the “dealer” node encodes a secret state into an n-party entangled state and distributes it to n “player” nodes. Any k players (the access structure) can collaborate to retrieve the quantum state, whereas the remaining n − k players (the adversary structure), even when conspiring, acquire nothing. This scheme provides quantum information networks with a secure framework for distributed quantum computation and quantum communication.
In our dealer protocol, the dealer interferes the secret state
aˆin with one of a pair of EPR entangled beams aˆEPR1 on a 1:1 beam splitter (Fig. 1). The two output fields and the sec-
beams player3
player1 & 2
φ
ψout
1:1
φ 2:1 G
ψout φ 50:1
AM
LO
player2 & 3
FIG. 1: Schematic of the (2, 3) quantum state sharing scheme. ψin: secret quantum state, OPA: optical parametric amplifier, G: electronic gain, AM: amplitude modulator, LO: optical local oscillator. x:y: beam splitter with reflectivity x/(x + y) and transmitivity y/(x+y).
generated by interfering a pair of amplitude squeezed beams
aˆsqz1 and aˆsqz2[7]. The dealer can further enhance the security of the scheme by displacing the coherent amplitudes of