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Detrimental DecoherenceGil Kalai∗Hebrew University of Jerusalem and Yale UniversityNovember23,2007AbstractWe propose and discuss two conjectures on the nature of informa-tion leaks(decoherence)for quantum computers.These conjectures, if(or when)they hold,are damaging for quantum error-correction as required by fault-tolerant quantum computation.Thefirst conjecture asserts that information leaks for a pair of substantially entangled qubits are themselves substantially positively correlated.The second conjecture asserts that in a noisy quantum computer with highly entangled qubits there will be a strong effect of error synchronization.We present a more general conjecture for arbitrary noisy quantum systems:prescribing(or describing)noisy quantum systems at a state ρis subject to error E which“tends to commute”with every unitary operator that stabilizesρ.1Quantum computers and the threshold the-oremQuantum computers are hypothetical devices based on quantum physics.A formal definition of quantum computers was pioneered by Deutsch[1],who also realized that they can outperform classical computation.The idea of a quantum computer can be traced back to works by Feynman,Manin,and others,and this development is also related to reversible computation and connections between computation and physics that were studied by Bennett in the1970s.Perhaps the most important result in thisfield and certainly a major turning point was Shor’s discovery[2]of a polynomial quantum algorithm for factorization.The notion of a quantum computer along with the associated complexity class BQP has generated a large body of research, in theoretical and experimental physics,computer science,and mathematics. For background on quantum computing,see Nielsen and Chuang’s book[3].Of course,a major question is whether quantum computers are feasible. An early critique of quantum computation(put forward in the mid-90s by Landauer[4,5],Unruh[6],and others)concerned the matter of noise:[N]The postulate of noise:Quantum systems are noisy.The foundations of noisy quantum computational complexity were laid by Bernstein and Vazirani in[7].A major step in showing that noise can be handled was the discovery by Shor[8]and Steane[9]of quantum error-correcting codes.The hypothesis of fault-tolerant quantum computation (FTQC)was supported in the mid-90s by the“threshold theorem”[10,11, 12,13],which asserts that under certain natural assumptions of statistical independence on the noise,if the rate of noise(the amount of noise per step of the computer)is not too large,then FTQC is possible.It was also proved that high-rate noise is an obstruction to FTQC.Several other crucial requirements for fault tolerance were also described in[14,15].2The study of quantum error-correction and its limitations,as well as of various approaches to fault-tolerant quantum computation,is extensive and beautiful;see,e.g.,[16,17,18,19].Concerns about noise models with statistical dependence are mentioned in several places,e.g.,[21,22].Specific models of noise that may be problematic for quantum error-correction are studied in[23].Current FTQC methods apply even to more general models of noise than thosefirst considered,which allow various forms of time-and space-statistical dependence;see[24].The purpose of this paper is to present two conjectures concerning de-coherence for quantum computers which,if(or when)true,are damaging for quantum error-correction and fault-tolerance.We will now state these conjectures informally.Thefirst conjecture concerns entangled pairs of qubits.[A]A noisy quantum computer is subject to error with the property thatinformation leaks for two substantially correlated qubits have a sub-stantial positive correlation.We emphasize that Conjecture[A](and Conjecture[D]below)refer to part of the overall error affecting a noisy quantum computer(or a noisy quantum system),which we call detrimental.Other forms of errors and,in particular,errors consistent with current noise models may also be present. (We conjecture that the effects of detrimental errors described by Conjectures [B]and[C]cannot be remedied by additional errors of a different nature.)Error-synchronization refers to a situation where,while the error rate is small,there is a substantial probability for errors affecting a large fraction of qubits.[B]In any quantum computer at a highly entangled state there will be astrong effect of spontaneous error-synchronization.We will refer informally to a pure state of a quantum computer that up to a small error is induced by its“marginal distribution”on small sets of3qubits as“approximately local.”We pose a related conjecture to the two conjectures above regarding the effect of detrimental decoherence:[C]The states of noisy quantum computers are approximately local.Section2gives more background on noise and fault-tolerance.The main Section3is devoted to mathematical formulations of the above conjectures. In the Appendix,stronger versions of Conjecture[A]are formulated and some connections with Conjectures[B]and[C]are indicated.Section4dis-cusses extensions of these conjectures to a more general framework.Wefirst consider more general quantum systems and pose and discuss the following extension:[D]A description(or prescription)of a noisy quantum system at a stateρis subject to error described by a quantum operation E that tends to commute with every unitary operator that stabilizesρ.(Here,“tends to commute”reflects a small bias towards commutativity that will be described formally in Section4.)We also briefly discuss classical noise.Section5discusses examples that may give the conjectured behavior and actual models of noise that may demonstrate them.Section6discusses related aspects of computational complexity and Section7concludes.2Quantum computers,noise and fault toler-anceThe state of a digital computer having n bits is a string of length n of zeros and ones.As afirst step towards quantum computers we can consider (abstractly)stochastic versions of digital computers where the state is a (classical)probability distribution on all such strings.Quantum computers are similar to these(hypothetical)stochastic classical computers and they4work on qubits(say n of them).The state of a single qubit q is described bya unit vector u=a|0>+b|1>in a two-dimensional complex space U q.(Thesymbols|0>and|1>can be thought of as representing two elements of abasis in U q.)We can think of the qubit q as representing′0′with probability|a|2and′1′with probability|b|2.The state of the entire computer is a unit vector in the2n-dimensional tensor product of these vector spaces U q’s for theindividual qubits.The state of the computer thus represents a probabilitydistribution on the2n strings of length n of zeros and ones.The evolution ofthe quantum computer is via“gates.”Each gate g operates on k qubits,andwe can assume k≤2.Every such gate represents a unitary operator on U g,the(2k-dimensional)tensor product of the spaces that correspond to these kqubits.Moving from a qubit q to the probability distribution on‘0’and‘1’that itrepresents is called a“measurement”and it can be considered as an additional1-qubit gate.We will assume that measurements of qubits that amount to asampling of0-1strings according to the distribution these qubits representis thefinal step of the computation.The postulate of noise is essentially a hypothesis about approximations.The state of a quantum computer can be prescribed only up to a certainerror.For FTQC there is an important additional assumption on the noise,namely,on the nature of this approximation.The assumption is that thenoise is“local.”This condition asserts that the way in which the state ofthe computer changes between computer steps is statistically independent,for different qubits.We will refer to such changes as“storage errors”or as“qubit errors.”In addition,the gates that carry the computation itself areimperfect.We can suppose that every such gate involves a small number ofqubits and that the gate’s imperfection can take an arbitrary form,so thatthe errors(referred to as“gate errors”)created on the few qubits involvedin a gate can be statistically dependent.(Of course,qubit errors and gateerrors propagate along the computation.)5The basic picture we have of a noisy computer is that at any time during the computation we can approximate the state of each qubit only up to some small error termǫ.Nevertheless,under the assumptions concerning the errors mentioned above,computation is possible.The noisy physical qubits allow the introduction of logical“protected”qubits that are essentially noiseless.Our conjectures apply to the same model of quantum computers but they require a more general notion of errors.They require that the storage errors will not be statistically independent(in fact,they should be instead very dependent)or that the gate errors will not be restricted to the qubits involved in the gates and will be of sufficiently general form.(Note that the errors may also reflect the translation from the ideal notion of quantum computers to a physical realization.)3A mathematical formulationIn this section we give a mathematical formulations for Conjectures[A],[B], and[C].Our setting is as follows.We have a quantum computer running on n qubits.The ideal(or“intended”)state of the computer is pure.We want to propose a picture for noisy quantum computation based on this model.The errors can be described by a unitary operator on the computer qubits and the neighborhood qubits or as a quantum operation E on the space of density matrices for these n qubits.We will not give a specific model of detrimental error but rather describe some of its expected properties.3.1Two qubitsWefirst describe a measure of information leak.For a stateρof the computer and a set A of qubits letρ|A be the induced state on A.Consider a quantum operation E.Note that when the stateτof the quantum computer is a tensor product pure state then for every set A of6qubits,S(τ|A)=0.Here,S(∗)is the(von Neumann)entropy function;see, e.g.,[3],Ch.11.The information leak of the noise operator E from the set of qubits A,w.r.t.τ,can be measured by the entropy S((E(τ)|A).For a tensor product stateτand a qubit a define L E(a;τ)=S(E(τ)|a));more generally, for a set A of qubits defineL E(A;τ)=S(E(τ)|A)).We will now state mathematically a version of Conjecture[A].Our settingis as follows.Letρbe the“intended”(“ideal”)pure state of the computer and consider two qubits a and b.We use as the(rather standard)measureof entanglement between qubits at pure statesENT(ρ;a,b)=S(ρ|a)+S(ρ|b)−S(ρ|{a,b}).As a measure of correlation of information leaks we useEL E(a,b;τ)=L E(a;τ)+L E(b;τ)−L E({a,b};τ).Conjecture[A]can be formulated as follows:For every tensor product stateτ,EL E(a,b;τ)≥K(L E(a;τ),L E(b;τ))·ENT(ρ;a,b),(1)where K(x,y)/min(x,y)2>>0when x and y are positive and small.(K(x,y)= 0,when min(x,y)=0so that relation(1)tells us nothing about noiseless entangled qubits.)In the Appendix we will describe and motivate several stronger forms of Conjecture[A],and point out alternative mathematical formulations.A simple extention that we would like to mention at this point is to pairsof qudits rather than pairs of qubits.The term qudit is used to denote a unitof quantum information in a d-level quantum system.Relation(1)extendsto qudits without any change.This applies,in particular,to two disjoint setsof qubits in a quantum computer.7Remark:Consider two qudits a and b,with d and d′possible levels respectively.The ideal pure state of this pair of qudits is represented by a d by d′matrix.Our conjecture(roughly)asserts that when the state is not represented by(or close to)a rank one matrix then neither is the error.(Or at least part of the error.)We expect that in wider contexts it is not reasonable to expect noisy data described by general matrices to be well approximated up to a rank-one error matrix.3.2Error synchronizationA simple way to describe error-synchronization is in terms of the expansion of the quantum operation E in terms of multi-Pauli operators.A quantum operation E can be expressed as a linear combinationE= v I P I,where I is a multi-index i1,i2,...,i n,where i k∈{0,1,2,3}for every k,P I is the quantum operation that corresponds to the tensor product of Pauli operators described by the multi-index I on the individual qubits,and v I are vectors.We can describe the error distribution of E byf(t)=: { v I 22:|I|=t},and regard f(t)t as the error rate.Suppose that the error rate is a.All noise models studied in the original papers of the“threshold theorem,”as well as some extensions that allow time-and space-dependencies(e.g.,[24]),have the property that f(t)decays exponentially(with n)for t=(a+ǫ)n,where a is the error rate andǫ>0 is anyfixed real number.In contrast,we say that E leads to error-synchronization if f(≥t)is substantial for some t>>a.We say that E leads to a strong error-synchronization if f(≥t)is substantial for t=1/2−δwhereδ=o(1)as n8tends to infinity,and to very strong error-synchronization if f(≥t)is substan-tial for t=3/4−δwhereδ=o(1)as n tends to infinity.A random unitary operator on the qubits of the computer with or without additional qubits representing the environment admits a very strong error-synchronization.3.3CensorshipHere is a suggestion for an entropy-based mathematical formulation for Con-jecture[C].We remind the reader that in this section we always assume that the“ideal”state of the quantum computer(before the noise is applied)is a pure state.Some adjustments to our conjectures will be required when the ideal state itself is a mixed state.Letρbe a pure state on a set A={a1,a2,...,a n}of n qubits.DefineENT(ρ;A)=−S(ρ)+max S(ρ∗),whereρ∗is a mixed state with the same marginals on proper sets of qubits asρ,i.e.,ρ∗|B=ρ|B for every proper subset B of A.Next,defineENT(ρ)= {ENT(ρ;B):B⊂A}.In this language a way to formulate the censorship conjecture is: Conjecture[C]:There is a polynomial P(perhaps even a quadratic polyno-mial)such that for any quantum computer on n qubits,which describes a pure stateρ,ENT(ρ)≤P(n).(2)94Extensions4.1General quantum systemsThe purpose of Section3was to describe formally the conjectures on decoher-ence of quantum computers based on the basic model for such a computer.In the context of general quantum systems these conjectures are thus somewhat arbitrary.(In particular,we always talk about Hilbert spaces of dimensions 2m.)The main idea behind the conjectures is that the error-independence assumption(for different qubits)amounts to an extremely strong dependence of the errors on the tensor product structure of the Hilbert space describing the state of the computer.It can be useful to suggest and examine formula-tions of our conjectures which do not depend on the tensor product structure of the Hilbert space in question.We want to consider quantum physical systems described by a complex Hilbert space V.Our conjectures suggest that if E represents the error for stateρand E′represents the error for state U(ρ),for a unitary operator U on V,then E′will be“close”to U−1EU.In particular,this implies that if U(ρ)=ρthen E′is“close”to U−1EU;hence UE is“close”to EU.In other words,E and U“tend”to commute if U(ρ)=ρ.Here is afirst attempt at a formal conjecture.We will restrict our at-tention to the case where the error is described by a quantum operation E which is a convex combination of unitary operators.[D]There is anα>0such that a prescription(or description)of a noisyquantum system at a stateρis subject to error E with the property that for every unitary operator U such that U(ρ)=ρEU−UE 2≤(1−α)√Remark:Greg Kuperberg pointed out that at a thermodynamics equi-librium a certain limiting error E will actually commute with every U that stabilizesρ.One possible way to regard Conjecture[D]is as a statement referring to non-equilibrium thermodynamics.14.2Classical noiseConjectures[A]and[B]were motivated and originally formulated in[26]also for“natural”noisy classical correlated systems.For example,the analog of[A]asserts that in a noisy system the errors for two highly correlated elements tend to be substantially correlated.Because of the heuristic(or subjective)nature of the notion of noise in classical systems(and of the notion of probability itself),such a formulation,while of interest,leads to several difficulties.Understanding noise and the study of de-noising methods span wide areas. (For example,in machine learning we can see the example where text and speech represent respectively the intended(ideal)and noisy signals.)Certain statistical methods of de-noising are based on assumptions that run contrary to[A].However,our conjectures are in agreement with insights asserting that such statistical de-noising methods will leave a substantial amount of noise uncorrected.Moreover,“natural”examples of noisy highly correlated classical systems exhibit a moderate degree of dependence and appear to be in agreement with Conjecture[C].5Examples and models5.1Unprotected quantum circuitsA basic remaining challenge is to present concrete models of noise that sup-port our conjectures.Wefirst point out that error-synchronization is a familiar phenomenon for error propagation of unprotected quantum programs(or circuits).Take the standard model of independent errors and suppose that the error rate is so small that it accumulates at the end of the computation to a small constant-rate error.It is instructive to see in this context that error-synchronization (and also[A]and[C])are often created.(This goes back to Unruh[6].) Understanding the nature of errors described by ordinary models of noise applied to unprotected programs is of further interest.We should offer a precise definition of“unprotected programs.”A random circuit leading to a given stateρor a random perturbation of a specific circuit leading toρmay serve this purpose.In such a model the errors for a certain stateρof the computer do depend systematically on the state itself,and understanding this further may be of interest.Remark:For noise propagation for unprotected programs the error rate is also related to the intended state.In this paper we assume that the error rate(in each computer cycle)is small andfixed.Trying to understand systematic relations between the error rate itself and the intended stateρmay be of interest.A natural informal conjecture would be[E]In any noisy quantum computer the more entangled the intended stateis,the higher the(detrimental)error rate.Conjecture[E]is close to the negation of the FTQC hypothesis and as such it cannot be very useful.If FTQC fails then propagation of errors will make the error rate dependent on the amount of computation leading toρ.A useful form of[E]should relate directly(not through computational notions)12the error rate to an entanglement measure and perhaps be formulated for general quantum systems.One possibility for such a connection is that when the evolution of a quantum system is prescribed,the rate of detrimental errors depends on the overall space of unitary operators which describe the incremental changes along the evolution.235.2ModelsAs much as error propagation for unprotected programs may supply useful insights it is not directly relevant to our conjectures.We emphasize that a model for decoherence that supports conjectures[A]and[B](and[E]) should already exhibit[A]and[B](and[E])for the“new errors”—either storage errors or gate errors4or both—and thus be quite different from current models and current perceptions regarding noise.Models that satisfy our conjectures may be based on the storage-errors(in a single computer cycle)being represented by a rather primitive(but quick)stochastic quantum program(or circuit).Remarks:1)Such noise models can be regarded as a further step in the direction considered recently by Aharonov,Kitaev,and Preskill[24](and a few earlier works).In these works,interactions between nearby qubits thatlead to statistical dependence between the noise acting on them is considered and it is shown that the threshold theorem prevails if the independence as-sumption still applies to faraway qubits.Interactions between nearby qubits expressed by a quick quantum circuit may lead to errors that are not covered by the assumptions of[24].2)Klesse and Frank[28]described a physical system in which qubits (spins)are coupled to a bath of massless bosons and thet reached(after certain simplifications)a noise model with error-synchronization.3)The earlier models suggested by Alicki,Horodecki,Horodecki,and Horodecki[23]appear to be relevant to our conjectures.4)Let me also mention the relevance of cluster states defined by Briegel and Raussendorf(see,[29]).The description of cluster states involves an array of qubits located on the vertices of a rectangular lattice in the plane (or in space).Cluster states are“generated”by local entanglement between pairs of nearby qubits on the lattice grid.They can be regarded as the quantum analogs of the Ising and Potts classical models.Controlled creation and manipulation of cluster states can be important for building quantum computers.On the other hand,cluster states and the local processes leading to them can possibly serve as a basis for concrete models of detrimental decoherence.6Computation complexityScott Aaronson’s interesting“Sure/Shor challenge”[30]ask for restrictions on feasible(physical)states for quantum computers which do not allow for polynomial time factoring of integers and at the same time do not violate what can already be demonstrated empirically.This looks like a difficult challenge.In a similar spirit,while it looks intuitively correct that our con-jectures are damaging for quantum computation,proving it,and especially proving a reduction all the way to the classical model of computation,is not14going to be easy.A realistic task would be to show that our conjectures exclude fault tol-erance based on linear quantum error-correction,e.g.,deriving relations(1) and(2)(or even(3))for any form of“protected qubits”obtained by linear quantum error-correction.5A more ambitious goal than excluding quantum linear error-correction would befinding a reduction of noisy quantum computation(with detrimen-tal errors)to the computational power of log-depth quantum circuits.(This will still fall short of Aaronson’s challenge in view of a result by Cleve and Watrous[31].)Such reductions are known under the standard assumptions on noise,for reversible quantum computation[15],and when the error rate is above45%[20].When we insist on small error rate it may well be the case that log-depth polynomial size quantum circuits represent the true complexity power of quantum computers with detrimental errors.Consider a log-depth circuit, and suppose that the storage(and gate)errors demonstrate perfect error-synchronization.If we run the computation a polynomial number of times, with high probability there will be no errors in one of the runs.If we replace a given log-depth circuit by a larger one capable of correcting local errors we may reach polynomial size(or quasi-polynomial size)circuits that are immune to low-rate errors of the kind considered in this paper.7ConclusionIf(or when)true,our conjectures on the nature of information leaks(deco-herence)for quantum computers are damaging to the possibility of storing and manipulating highly entangled quantum qubits.The conjectures do not contravene quantum mechanics and,to the best of my knowledge,established physics phenomena.Nor do our conjectures contravene with the feasibility of classical 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原子跃迁频率英语Atomic transition frequency refers to the frequency at which an electron transitions between energy levels in an atom. This phenomenon is crucial in various fields of physics, including quantum mechanics, spectroscopy, and atomic clocks. Understanding atomic transition frequency is essential for studying the behavior of atoms and developing advanced technologies.The concept of atomic transition frequency is rooted in the quantum mechanical description of atoms. According to quantum theory, electrons in an atom can only occupy certain discrete energy levels. When an electron absorbs or emits energy, it transitions between these energy levels. The frequency of this transition is directly related to the energy difference between the initial and final states of the electron.In spectroscopy, atomic transition frequency plays a key role in analyzing the interaction of atoms with electromagnetic radiation. By measuring the frequencies at which atoms absorb or emit light, scientists can determine the energy levels of the atoms and gain insights into their structure and behavior. Spectroscopic techniques have been instrumental in studying the properties of atoms and molecules, as well as in identifying unknown substances.Atomic clocks are another application of atomic transition frequency. These highly precise timekeeping devices rely on the stable oscillation of atoms between energy levels to measure time accurately. By counting the number of transitions that occur within a certain period, atomic clocks can maintain time standards with incredible precision. Atomic clocks are used in various scientific and technological applications, including global positioning systems, telecommunications, and space exploration.The study of atomic transition frequency has also led to significant advancements in quantum information processing. Quantum computers, which harness the principles of quantum mechanics to perform complex calculations, rely on the manipulation of atomic states to store and process information. By controlling the transition frequencies of atoms,researchers can encode and manipulate quantum bits (qubits) to perform computations that are beyond the capabilities of classical computers.In conclusion, atomic transition frequency is a fundamental concept in physics with far-reaching implications. From spectroscopy to atomic clocks to quantum computing, the understanding and manipulation of atomic transition frequencies have revolutionized our understanding of the natural world and enabled the development of advanced technologies. By continuing to study and explore the properties of atoms at the quantum level, scientists can unlock new possibilities for scientific discovery and technological innovation.。
光学专业词汇大全Iris – aperture stop虹膜孔俓光珊retina视网膜Color Blind 色盲weak color 色弱Myopia – near-sighted 近视Sensitivity to Light感光灵敏度boost推进lag behind落后于Hyperopic – far-sighted 远视Dynamic Range 动态范围critical fusion frequency 临界融合频率CFF临界闪变频率visual sensation视觉Chromaticity Diagram色度图Color Temperature色温HSV Model色彩模型(hue色度saturation饱和度value纯度CIE Model 相干红外能量模式Complementary Colors补色Bar Pattern条状图形Heat body 热稠化approximate近似violet紫罗兰Body Curve人体曲线Color Gamut色阶adjacent邻近的normal illumination法线照明Primary colors红黄蓝三原色Color saturation色饱和度Color Triangle颜色三角Color Notation颜色数标法Color Difference色差TV Signal Processing电视信号处理Gamma Correction图像灰度校正Conversion Tables换算表out of balance失衡wobble摇晃back and forth前后clear (white) panel 白光板vibrant震动fuzzy失真quantum leap量子越迁SVGA (800x600)derive from起源自culprit犯人render呈递inhibit抑制,约束stride大幅前进blemish污点obstruction障碍物scratch刮伤substance物质实质主旨residue杂质criteria标准parameter参数adjacent邻近的接近的asynchrony异步cluster串群mutually互助得algorithm运算法则Chromatic Aberrations色差Fovea小凹Visual Acuity视觉灵敏度Contrast Sensitivity对比灵敏度Temporal (time) Response反应时间rendition表演,翻译animation活泼又生气ghost重影Parallax视差deficient缺乏的不足的Display panel显示板NG.( Narrow Gauge)窄轨距dichroic mirror二色性的双色性的Brewster Angle布鲁斯特角Polarized Light极化光Internal reflection 内反射Birefringence 双折射Extinction Ratio 消光系数Misalignment 未对准Quarter Waveplates四分之一波片blemish污点瑕疵Geometric几何学的ripple波纹capacitor电容器parallel平行的他tantalum钽(金属元素) exsiccate使干燥exsiccate油管,软膏furnace炉子镕炉electrolytic电解的,由电解产生的module 模数analog类似物out of the way不恰当pincushion针垫拉lateral侧面得rectangle长方形fixture固定设备control kit工具箱DVI connector DVI数局线Vertical垂直的horizontal 水平的interlace隔行扫描mullion竖框直楞sawtooth锯齿toggle套索钉keypad数字按键键盘tangential切线diagnostic tool诊断工具sagittal direction径向的cursor position光标位置3Yw'/#p3`ray aberration光线相差weighting factor权种因子variables变量for now暂时,目前.眼下check box复选框Airy disk艾里斑exit pupil出[射光]瞳optical path difference光称差with respect to关于diffraction limited 衍射极限wavefront aberration波阵面相差spherical aberration球面象差paraxial focus傍轴焦点chromatic aberration象差local coordinate system局部坐标系统coordinate system坐标系orthogonal直角得,正交的conic sections圆锥截面account for解决,得分parabolic reflector拋物面反射镜radius ofcurvature曲率半径spherical mirror球面镜geometrical aberration几何相差incident radiation入射辐射global coordinate总体坐标in terms of根据按照reflected beam反射束FYI=for your information供参考Constructive interference相长干涉phase difference相差achromatic singlet消色差透镜Interferometer 干涉仪boundary constraint边界约束,池壁效应radii半径Zoom lenses变焦透镜Beam splitters分束器discrete不连续的,分离的objective/eye lens物镜/目镜mainframe主机rudimentary根本的,未发展的photographic照相得摄影得taxing繁重的,费力得algebra代数学trigonometry三角学geometry几何学calculus微积分学philosophy哲学lagrange invariant拉格朗日不变量spherical球的field information场信息Standard Lens标准透镜Refracting Surface折射面astigmatism散光HDTV高清晰度电视DLV ( Digital Light Valve)数码光路真空管，简称数字光阀diffraction grating衍射光珊field angle张角paraxial ray trace equations近轴光线轨迹方称back focal length后焦距principal plane主平面vertex顶点,最高点astigmatism散光,因偏差而造成的曲解或错判medial中间的,平均的variance不一致conic圆锥的,二次曲线field of view视野collimator瞄准仪convolution回旋.盘旋,卷积fuzzy失真,模糊aberrated异常的asymmetry不对称得indicative可表示得parabolic拋物线得suffice足够,使满足specification规格,说明书straightforward易懂的,直接了当的,solidify凝固,巩固. Constraints 约束,限制metrology度量衡field coverage视场,视野dictate口述, 口授, 使听写, 指令, 指示, 命令, 规定irradiance发光, 光辉,辐照度aerial空气得,空中得halide卤化物的monochromatic单色的,单频的polychromatic多色的aspherical非球面的spherical球面的alignment列队,结盟power(透镜)放大率equiconvergence 同等收敛EFL(effective focal length)有效焦距workhorse广为应用的设备biconvex两面凸的global optimization整体最优化concave凹得,凹面得cylindrical圆柱得solid model实体模型Modulation Transfer Function调制传递函数in the heat of在最激烈的时候protocol协议,规定triplet三重态sanity心智健全zinc锌,涂锌的selenide 硒化物,硒醚miscellaneous各色各样混在一起, 混杂的, 多才多艺的versus与...相对polynomial多项式的coefficient系数explicit function显函数" wYgi%distinct清楚的,截然不同的emanate散发, 发出, 发源rudimentary根本的,未发展的intersection角差点PRTE=paraxial ray trace equation旁轴光线轨迹方程achromats 消色差透镜cardinal points基本方位separations分色片dashed虚线blow up放大overlay覆盖,覆盖图multiplayer 多层的humidity 湿度float glass浮法玻璃square one 出发点,端点square up to 准备开打,坚决地面对reflecting telescope 反射式望远镜diagnostic tools诊断工具Layout plots规划图Modulation transfer function调制转换功能FFT快速傅里叶变换Point spread function点传播功能wavelength波长angle角度absorption吸收system aperture 系统孔径lens units透镜单位wavelength range波长范围singlet lens单业透镜spectrum光谱diffraction grating衍射光栅asphere半球的LDE=Lens data editor Surface radius of curvature表面曲率半径surfacethickness表面厚度material type材料种类semi-diameter半径focal length焦距aperture type孔径类型aperture value孔径值field of view视场microns微米F, d, and C= blue hydrogen, yellow helium, red hydrogen lines, primary wavelength主波长sequential mode连续模式object surface物表面The front surface of the lens透镜的前表面stop光阑The back surface of the lens透镜的后表面The image surface 像表面symmetric相对称的biconvex两面凸的The curvature is positive if the center of curvature of the surface is to the right of the vertex. It is negative if the center of curvature is to the left of the vertex.如果曲率中心在最高点的右边,曲率值为正,如果曲率中心在最高点的左边,则曲率为负image plane像平面Ray Aberration光线相差tangential direction切线方向sagittal direction径向paraxial focus旁轴的Marginal 边缘的spherical aberration球面像差Optimization Setup最优化调整variable变量mathematical sense 数学角度MFE= Merit Function Editor, Adding constraints增加约束focal length焦矩长度operand操作数the effective focal length有效焦矩primary wavelength主波长initiate开始spot diagram位图表Airy disk 艾里斑axial chromatic aberration轴向色差with respect to关于至于exit pupil出射光瞳OPD=optical path difference光学路径差diffraction limited衍射极限chromatic aberration色差chromatic focal shift色焦距变换paraxial focus傍轴焦点axial spherical aberration轴向球差(longitudinal spherical aberration 纵向球差:沿光轴方向度量的球差)lateral spherical aberration垂轴球差(在过近轴光线像点A‵的垂轴平面内度量的球差)coma、comatic aberration彗差meridional coma子午彗差sagittal coma弧矢彗差astigmatism像散local coordinate system本地坐标系统meridional curvature of field子午场曲sagittal curvature of field弧矢场曲decentered lens偏轴透镜orthogonal直角的垂直的conic section圆锥截面account for说明,占有,得分stigmatic optical system无散光的光学系统Newtonian telescope牛顿望远镜parabolic reflector抛物面镜foci焦距chromatic aberration,色差superpose重迭parabola抛物线spherical mirror球面镜RMS=Root Mean Square均方根wavefront波阵面spot size光点直径Gaussian quadrature高斯积分rectangular array矩阵列grid size磨粒度PSF=Point Spread Function点扩散函数FFT=Fast Fourier Transform Algorithm快速傅里叶变换Cross Section横截面Obscurations昏暗local coordinates局部坐标系统vignette把…印为虚光照Arrow key键盘上的箭头键refractive折射reflective反射in phase同相的协调的Ray tracing光线追迹diffraction principles衍射原理order effect式样提出的顺序效果energy distribution能量分配Constructive interference相长干涉dispersive色散的Binary optics二元光学phase advance相位提前achromatic single消色差单透镜diffractive parameter衍射参数Zoom lenses变焦透镜Athermalized lenses绝热透镜Interferometers干涉计Beam splitter分束器Switchable component systems可开关组件系统common application通用symmetry对称boundary constraint边界约束multi-configuration (MC) MC Editor (MCE) perturbation动乱,动摇index accuracy折射率准确性indexhomogeneity折射率同种性index distribution折射率分配abbe number离差数Residual剩余的Establishing tolerances建立容差figure of merit质量因子tolerance criteria公差标准Modulation Transfer Function (MTF)调制传递函数boresight视轴，瞄准线Monte Carlo蒙特卡洛Tolerance operands误差操作数conic constant ]MC1"{_qT圆锥常数astigmatic aberration像散误差Mechanical tilt机械倾斜，机械倾角Tolerance Data Editor (TDE)公差资料编辑器compensator补偿棱镜estimated system performance预估了的系统性能iteratively反复的，重迭的statistical dependence统计相关性sequential ray trace model连续光线追迹模型imbed埋葬，埋入multiple多样的，多重的，若干的Non-Sequential Components不连续的组件Corner cube角隅棱镜，三面直角透镜Sensitivity Analysis灵敏度分析Faceted reflector有小面的反射镜emit发射，发出nest嵌套overlap交迭outer lens外透镜brute force强力seidel像差系数aspect ratio长宽比MRA边缘光线角MRH边缘光线高度asynchronous不同时的,异步Apodization factor变迹因子hexapolar六角形dithered 高频脉冲衍射调制传递函数（DMTF），衍射实部传递函数（DRTF），衍射虚部传递函数（DITF），衍射相位传递函数（DPTF），方波传递函数（DSWM）logarithmic对数的parity奇偶% Uc,I elongitudinal aberrations 纵向像差赛得系数: 球差（SPHA，SI），彗差（COMA，S2），像散（ASTI，S3），场曲（FCUR，S4），畸变（DIST，S5），轴向色差（CLA，CL）和横向色差（CTR，CT）.横向像差系数:横向球差（TSPH），横向弧矢彗差（TSCO），横向子午彗差（TTCO），横向弧矢场曲（TSFC），横向子午场曲（TTFC），横向畸变（TDIS）横向轴上色差（TLAC）。
Quantum entanglementMaciej LewensteinMaciej Lewenstein has obtained his degree in Physics from Warsaw University. From 1980 he worked at the Center for Theoretical Physics of the Polish Academy of Sciences. He received his doctoral degree in 1983 at the University of Essen and habilitation in 1986 in Warsaw. He became a full Professor in Poland in 1993. In 1995 he joined “Service de Photones, Atomes et Molecules” of CEA in Saclay. In 1998 he became a full professor and a head of the quantum optics theory group at the University of Hannover. In 2005 he started a new theory group at the “Insitut de Ciencias Fotoniques” in Barcelona. His research interests include: quantum optics, quantum information and statistical physics.Chiara MacchiavelloChiara Macchiavello finished her degree in Physics in 1991 and her PhD in 1995 at the University of Pavia. She held a post-doctoral for two years at the University of Oxford. Since 1998 she has been an Assistant Professor at the University of Pavia.Her research interests include quantum information processing and quantum optics.Dagmar BrussSince 2003 Dagmar Bruss is a professor at the Institute of Theoretical Physics at the University of Duesseldorf, Germany. Her research interests include the foundations of quantum information theory, classification of entanglement and quantum optical implementations of quantum computation.AbstractEntanglement is a fundamental resource in quantum information theory. It allows performing new kinds of communication, such as quantum teleportation and quantum dense coding. It is an essential ingredient in some quantum cryptographic protocols and in quantum algorithms. We give a brief overview of the concept of entanglement in quantum mechanics, and discuss the major results and open problems related to the recent scientific progress in this field.IntroductionEntanglement is a property of the states of quantum systems that are composed of many parties, nowadays frequently called Alice, Bob, Charles etc. Entanglement expresses particularly strong correlations between these parties, persistent even in the case of large separations among the parties, and going beyond simple intuition.Historically, the concept of entanglement goes back to the famous Einstein-Podolski-Rosen (EPR) “paradox”. Einstein, who discovered relativity theory and the modern meaning of causality, was never really happy with quantum mechanics. In his opinion every reasonable physical theory should exhibit a so called local realism.Suppose that we consider two particles, one of which is sent to Alice and one to Bob, and we perform independent local measurements of “reasonable” physical observables on these particles. Of course, the results might be correlated, because the particles come from the same source. But Einstein wanted really to restrict the correlations for “reasonable” physical observables to the ones that result from statistical distributions of some hidden (i.e. unknown to us and not controlled by us) variables that characterize the source of the particles. Since quantum mechanics did not seem to produce correlations consistent with a local hidden variable (LHV) model, Einstein concluded that quantum mechanics is not a complete theory. Erwin Schrödinger, in answer to Einstein’s doubts, introduced in 1935 the term “Verschränkung” (in English “entanglement”) in order to describe these particularly strong quantum mechanical correlations.Entanglement was since then a subject of intense discussions among experts in the foundations of quantum mechanics and philosophers of science (and not only science). It took, however, nearly 30 years until John Bell was able to set the framework for experimental investigations on the question of local realism. Bell formulated his famous inequalities, which have to be fulfilled in any multiparty system described by a LHV model. Alain Aspect and coworkers in Paris have demonstrated in their seminal experiment in 1981 that quantum mechanical states violate these inequalities. Recent very precise experiments of Anton Zeilinger’s group in Vienna confirmed fully Aspect’s demonstrations. All these experiments indicate the correctness of quantum mechanics, and despite various loopholes, they exclude the possibility of LHV models describing properly the physics of the considered systems.Entanglement has become again the subject of cover pages news in the 90’s, when quantum information was born. It was very quickly realized that entanglement is one of the most important resources for quantum information processing. Entanglement is a necessary ingredient for quantum cryptography, quantum teleportation, quantum densecoding, and if not necessary, then at least a much desired ingredient for quantum computing.At the same time the theory of entanglement is related to some of the open questions of mathematics, or more precisely linear algebra and functional analysis. A solution of the entanglement problem could help to characterize the so called positive linear maps, i.e. linear transformations of positive definite operators (or physically speaking quantum mechanical density matrices, see below) into positive definite operators.Entanglement of pure statesIn quantum mechanics (QM) a state of a quantum system corresponds to a vector |Psi> in some vector space, called Hilbert space. Such states are called pure states. One of the most important properties of QM is that linear superpositions of state-vectors are also legitimate state-vectors. This superposition principle lies at the heart of the matter-wave dualism and of quantum interference phenomena.Entanglement is also a result of superposition, but in the composite space of the involved parties. Let us for the moment focus on two parties, Alice and Bob. It is then easy to define states which are not entangled. Such states are product states of the form |Φ>= |a>|b>, i.e. Alice has at her disposal |a>, while Bob has |b>. Product states obviously carry no correlations between Alice and Bob. Entangled pure states may be now defined as those which are superpositions of at least two product states, such as|Φ> = α1|a1>|b1> + α2|a2>|b2> + etc.but cannot be written as a single product state in any other basis. All entangled pure states contain strong quantum mechanical correlations, and do not admit LHV models.Entanglement of mixed states and the separability problemVerify whether a given state-vector is a product state or not is a relatively easy task. In practice, however, we often either do not have full information about the system, or are not able to prepare a desired state perfectly. In effect in everyday situations we deal practically always with statistical mixtures of pure states. There exists a very convenient way to represent such mixtures as so called density operators, or matrices. A density matrix rho corresponding to a pure state-vector |Φ> is a projector onto this state. More general density matrices can be represented as sums of projectors onto pure state-vectors weighted by the corresponding probabilities.The definition of entangled mixed states for composite systems has been formulated by Reinhard Werner from Braunschweig in 1989. In fact, this definition determines which states are not entangled. Non-entangled states, called separable states, are mixtures of pure product states, i.e. convex sums of projectors onto product vectors:ρ = Σι pi|ai>|bi><ai|<bi|, (*)where 0 ≤ pi ≤ 1 are probabilities, i.e. Σιpi= 1. The physical interpretation of thisdefinition is simple: a separable state can be prepared by Alice and Bob by using local operations and classical communication. Checking whether a given state is separable or not is a notoriously difficult task, since one has to check whether the decomposition (*) exists or not. This difficult problem is known under the name of “separability or entanglement problem”, and has been a subject of intensive studies in the recent years.Simple entanglement criteriaThe difficulty of the separability problem comes from the fact that rho admits in general an infinite number of decompositions into a mixture of some states, and one has to check whether among them there exists at least one of the form (*). One of the most powerful necessary conditions for separability has been found by one of the fathers of quantum information, the late Asher Peres. Peres (Technion, Haifa) observed that since Alice and Bob may prepare separable states using local operations, Alice may safely reverse the time arrow in her system, which will change the state, but will not produce something unphysical. In general, such a partial time reversal is not a physical operation, and can transform a density operator (which is positive definite) into an operator that is no more positive definite. In fact this is what happens with all pure entangled states. Mathematically speaking partial time reversal corresponds to partial transposition of the density matrix (only on Alice's side). We arrive in this way at the Peres criterion: If a stateρis separable then its partial transposition has to be positive definite.This criterion is usually called positive partial transpose condition, or shortly PPT condition. Amazingly, the PPT condition is not only necessary for separability, but it is also a sufficient condition for low dimensional systems such as two qubits (dimension 2x2)and a system composed of one qubit and one qutrit (dimension 2x3). In higher dimensions, starting from 2x4 and 3x3, this is no longer true: there exist entangled states with positive partial transpose, which are called PPT entangled states.There exist several other necessary or sufficient separability criteria which have been established and frequently discussed in recent years. For example, states that are close to the completely chaotic state (whose density operator is equal to the normalized identity) are necessarily separable. There exist also other criteria that employ entropic inequalities, uncertainty relations, or an appropriate reordering of the density matrix (so called realignment criterion) etc. There exists, however, no general simple operational criterion of separability that would work in systems of arbitrary dimension.Entanglement witnessesThe set of all states P is obviously compact and convex. If ρ1 and ρ2are legitimate states,so is their convex mixture. The set of separable states S is also compact and convex (seeFigure 1). From the theory of convex sets and Hahn-Banach theorem we conclude that for any entangled state there exists a hyperplane in the space of operators separating rhofrom S. Such a hyperplane defines uniquely a Hermitian operator W (observable) which has the following properties: The expectation value of W on all separable states, <W> ≥ 0, whereas its expectation value on ρ is negative, i.e. <W>ρ< 0.Figure 1Such an observable is for obvious reasons called entanglement witness, since it “detects” the entanglement of ρ. Every entangled state has its witnesses; the problem obviously is to find appropriate witnesses for a given state. To find out whether a given state is separable one should check whether its expectation value is non-negative for all witnesses. Obviously this is a necessary and sufficient separability criterion, but unfortunately it is not operational, in the sense that there is no simple procedure to test for all witnesses.Nevertheless, witnesses provide a very useful tool to study entanglement, especially if one has some knowledge about the state in question. They provide a sufficient entanglement condition, and may be obviously optimized (see Figure 2) by shifting the hyperplane in a parallel way towards S.Figure 2Bell inequalitiesAfter introducing the concept of separability and entanglement for mixed states, it is legitimate to ask what is the relation of mixed state entanglement and the existence of a LHV model, which requires that the state cannot violate any of the Bell-like inequalities. Let us discuss an example of such inequalities, the so called Clauser-Horne-Shimony- Holt inequality for two qubits. Let us assume that Alice and Bob measure two binary observables each, namely A 1, A 2, and B 1, B 2. The observables are random variables taking the values +1 or − 1, correlated possibly through some dependence on local hidden variables. It is easy to see that in the classical world, if B 1 + B 2 is zero, then B 1 − B 2 is either +2 or −2, and vice versa. Therefore if we define s = A 1(B 1 + B 2 ) + A 2 (B 1 − B 2 ) , we obtain that 2 ≥ s ≥ −2. This inequality holds also after averaging over various realizations. On the other hand, it can be shown that by taking suitable sets of observables for Alice and Bob we can find pure and even mixed quantum states that violate this inequality.Are Bell-like inequalities similar in this respect to witnesses, i.e. for a given entangled state can one always find a Bell-like inequality that “detects” it? The answer to this question is no, and has been already given by R. Werner in 1989. Even for two qubits there exist entangled states that admit an LHV model, i.e. cannot violate any Bell-like inequality.This observation indicates already that there is more structure in the “eggs” of Figure 1 and Figure 2. Separable states are evidently inside the PPT egg, according to the Peres condition. They admit an LHV model, i.e. they are also inside the LHV egg. But what about PPT entangled states? Do they violate some Bell-like inequality? Peres has formulated a conjecture that this not the case, and there is a lot of evidence that this conjecture is correct, although a rigorous proof is still missing.The distillability problem and bound entanglement Above we have classified quantum states according to the property of being either separable or entangled. An alternative classification approach is based on the possibility of distilling the entanglement of a given state. In a distillation protocol the entanglement of a given state is increased by performing local operations and classical communication on a set of identically prepared copies. In this way one obtains fewer, but “more entangled”, copies. This kind of technique was originally proposed in 1996 by Bennett and coworkers in the context of quantum teleportation, in order to achieve faithful transmission of quantum states over noisy channels. It also has applications in quantum cryptography as a method for quantum privacy amplification in entanglement based protocols in the presence of noise, as pointed out by David Deutsch and coworkers from Oxford.The distillability problem poses the question whether a given quantum state can be distilled or not. A separable state can never be distilled because the average entanglement of a set of states cannot be increased by local operations. Furthermore, the positivity of the partial transpose ensures that no distillation is possible. Thus, a given PPT entangled state is not distillable, and is therefore called bound entangled. There mayeven exist undistillable entangled states which do not have the PPT property. However, this conjecture is not proved at the moment.The first example of a PPT entangled state has been found by Pawel Horodecki from Gdansk in 1997. These states are so called edge states, which means that they cannot be written as a mixture of a separable state and a PPT entangled state. Particularly simple families of states have been suggested by Charles Bennett and coworkers at IBM, New York. They have found the so called unextendible product bases (UPB), i.e. sets of orthogonal product state-vectors, with the property that the space orthogonal to this set does not contain any product vector. It turns out that the projector onto this space is a PPT state, which obviously has to be entangled since it does not contain any product vector in its range (note that all state-vectors in the decomposition of a separable state ρinto a mixture of product states belong automatically to the range of ρ).The existence of bound entanglement is a mysterious invention of Nature. It is an interesting question to ask whether bound entanglement is a useful resource to perform quantum information processing tasks. It was shown so far that this is not the case for communication protocols such as quantum teleportation and quantum dense coding (i.e.a protocol that allows to enhance the transmission of classical information, using entanglement). However, surprisingly, it is possible to distill a secret key in quantum cryptography, starting from certain bound entangled states.Entanglement detectionAs discussed above, entanglement is a precious resource in quantum information processing. Typically in a real world experiment noise is always present and it leads to a decrease of entanglement in general. Thus, it is of fundamental interest for experimental applications to be able to test the entanglement properties of the generated states. A traditional method to this aim is represented by the Bell inequalities, a violation of which indicates the presence of entanglement. However, as mentioned above, not every entangled state violates a Bell inequality. So, not all entangled states can be detected by using this method.Another possibility is to perform complete state tomography, which allows determining all the elements of the density matrix. This is a useful method to get a complete knowledge of the density operator of a quantum system, but to detect entanglement it is an expensive process as it requires an unnecessary large number of measurements. If one has certain knowledge about the state the most appropriate technique is the measurement of the witness observable, which can be achieved by few local measurements. A negative expectation value clearly indicates the presence of entanglement.All these methods have been successfully implemented in various experiments. Recently another method for the detection of entanglement was suggested based on the physical approximation of the partial transpose. It remains a challenge to implement this idea in the laboratory because it requires the implementation of non local measurements.Entanglement measuresWhen classifying a quantum state as being entangled, a natural question is to quantify the amount of entanglement it contains. For pure quantum states there exists a well defined entanglement measure, namely the von Neumann entropy of the density operator of a subsystem of the composite state. For mixed states the situation is more complicated. There are several different possibilities to define an entanglement measure. The so called entanglement cost describes the amount of entanglement one needs in order to generate a given state. An alternative measure is the entanglement of formation, which is a more abstract definition. A further possibility to quantify entanglement is given by the minimum distance to separable states. Finally, motivated by physical applications, one can introduce the distillable entanglement which quantifies the extractable amount of entanglement.Unfortunately all of these quantities are very difficult to compute in general. For example, in order to determine the entanglement of formation one has to find the decomposition of the state that leads to the minimum average von Neumann entropy of a subsystem and this is a very challenging task. So far a complete analytical formula for the entanglement of formation only exists for composite systems of two qubits.Entanglement in multipartite systemsSo far, we have restricted ourselves to the case of composite systems with two subsystems, so called bipartite systems. When considering more than two parties, i.e multipartite systems, the situation becomes much more complex. For example, for the most simple tripartite case of three qubits, a pure state can be either completely separable, or biseparable (i.e. one of the three parties is not entangled with the other two), or genuinely entangled among all three parties. The latter class again consists of inequivalent subclasses, the so called GHZ and W states. This concept can be generalized to mixed states. For more than three parties it is easy to imagine that the number of subclasses grows fast.In recent years there has been much progress in the creation of multipartite entangled states in the laboratory. The existence of genuine multipartite entanglement has also been demonstrated experimentally by using the concept of witness operators.Even if the full classification of multipartite entanglement is a formidable task, certain classes of states, the so called graph states, have been completely characterized and shown to be useful both for quantum computational and quantum error correction protocols. Moreover, a deeper understanding of entanglement has proved to be very fruitful in connection with statistical properties of physical systems. All of these problems are discussed in more details in other sections of this publication.References[1] Einstein, P. Podolsky and N. Rosen, Phys. Rev. 47, 777 (1935)[2] J.S. Bell, Physics 1, 195 (1964)[3] P. Horodecki, Phys. Lett. A 232, 333 (1997)[4] M. Lewenstein et al., J. Mod. Opt. 47, 2481 (2000)[5] A. Peres, Phys. Rev. Lett. 77, 1413 (1996)[6] E. Schrödinger, Naturwissenschaften 23, 807 (1935)[7] R.F. Werner, Phys. Rev. A 40, 4277 (1989) Contact information of the author of this article Maciej LewensteinInstitut de Ciènces Fotòniques (ICFO)C/Jordi Girona 29, Nexus 2908034 BarcelonaSpainEmail: maciej.lewenstein@icfo.esChiara MacchiavelloIstituto Nazionale di Fisicadella Materia, Unita' di Pavia Dipartimento di Fisica "A. Volta"via Bassi 6I-27100 PaviaItalyEmail: chiara@unipv.itProf. Dr. Dagmar BrussInst. fuer Theoretische Physik IIIHeinrich-Heine-Universitaet Duesseldorf Universitaetsstr. 1, Geb. 25.32D-40225 Duesseldorf,GermanyEmail: bruss@thphy.uni-duesseldorf.de。
