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The asymptotic entanglement cost of preparing a quantum state

a r X i v :q u a n t -p h /0008134v 1 31 A u g 2000

The Asymptotic Entanglement Cost of Preparing a Quantum State

Patrick M.Hayden 1,Micha l Horodecki 2and Barbara M.Terhal 3

Centre for Quantum Computation,Clarendon Laboratory,Parks Road,Oxford,OX13PU,UK;

2Institute of Theoretical Physics and Astrophysics,University of Gda′n sk,80-952,Gda′n sk,Poland;

IBM Watson Research Center,P.O.Box 218,Yorktown Heights,NY 10598,US.

Emails:patrick.hayden@https://www.wendangku.net/doc/8512562426.html,,michalh@iftia.univ.gda.pl,terhal@https://www.wendangku.net/doc/8512562426.html,

(February 1,2008)

We give a detailed proof of the conjecture that the asymptotic entanglement cost of preparing a state ρis equal to lim n →∞E f (ρ?n )/n where E f is the entanglement of formation.

I.INTRODUCTION

One of the central issues in quantum entanglement theory is to determine how to optimally convert between di?erent entangled states shared by distant observers Alice and Bob.More precisely,one is interested in the conversion of m pairs of particles,with each pair in a state ρ,into n pairs,each in another state ρ′,by means of local quantum operations and classical communication (LOCC)[1],so that the asymptotic ratio m/n is minimal.Of course,the perfect transformation by LOCC

ρ?m →ρ′?n

is usually impossible.Therefore,one permits imperfections and requires only asymptotically perfect transformations:

the state ρ?m is transformed into some state ρ′n ,which for large n approaches ρ′?n

with a chosen distance measure D :

lim n →∞

D (ρ′n ,ρ

′?n

)=0.If the ?nal state ρis a two-qubit singlet state 12(|01 ?|10 )(which we will denote by |Ψ? ),then the process of

conversion is called distillation [2].If,instead,it is the initial state that is in the form of singlets,then one refers to formation [1].In this paper we are interested in the latter process.We will call the optimal asymptotic yield the entanglement cost,and denote it by E c .In Ref.[3]it was shown that for a pure state ρ=|ψ ψ|,E c is equal to the entropy of either of its reductions,e.g.ρA =Tr B (ρ).Thus,to produce |ψ ?n one needs m ≈nS (ρA )singlets,i.e.the

initial state has to be |Ψ? ?nS (ρA )

.That result suggested the following stochastic method for the production of ρout of singlets [1].One decomposes ρinto an ensemble ρ= i p i |ψi ψi |of pure states.Then one picks a state |ψi according to the probability distribution {p i },makes the state |ψi from initially shared singlets,and ?nally forgets the identity of the state.Therefore,one needs on average i p i S (ρA i )singlets (where ρA

i is the reduction of |ψi ).One can choose the ensemble which minimizes the above average,with the corresponding minimal cost being called the entanglement of formation of ρ[1],which we will denote by E f (ρ).

The above scenario can be improved if we realize that it might be more economical to produce the state ρ?n all at once than it is to produce its n constituents one by one (see e.g.[4]).Thus,the proposed optimal cost should be revised to [5]

E ∞f

(ρ)=lim

n →∞

E f (ρ?n )

To begin,let us sketch our approach.In our proof,we will?rst show that E∞f(ρ)≥E c(ρ),i.e.that there exists a formation protocol that achieves the asymptotic rate m

1?F(ρ,ρ′)with F(ρ,ρ′)=Tr

n≈

E f(ρn)

n≤E f(|Ψ? Ψ?|?m)

.(5)

Now,by defnition,the left-hand-side of the equation(4)tends to E∞f while the last term of estimate(5)tends to E c, hence we obtained the required inequality.

Since we used only two properties of the entanglement of formation,we can rephrase the result in a more general setting.Consider any function f,which can be regularized,i.e.for which the limit f∞(ρ)=lim n→∞f(ρ?n)

n|≤δand D(L(|Ψ? Ψ?|?m),ρ?n)≤?

,(6)

where|Ψ? is the singlet state in C2?C2,L is an LOCC superoperator and D is again the Bures distance.

Our main result is the following:

Theorem1The asymptotic entanglement cost of preparing a stateρis given by

E c(ρ)=lim

n→∞

E f(ρ?n)

where E c(ρ)is de?ned in Eq.(6)and E f(ρ)is the entanglement of formation ofρ,de?ned as

E f(ρ)=min

E={p i,|ψi } i p i E(|ψi ψi|),(8) and E(|ψ ψ|)is the von Neumann entropy of the reduced density matrix of|ψ .

To make sense of the above claim,we note that the sequence E f(ρ?n)

p i|ψi and they are such that every state√

1??1.(11) Consider the density matrixρ

and its decomposition in terms of the states|ψs .Each state|ψs can be obtained

T(n)

δ1

from a set of EPR pairs by entanglement dilution[3].In particular,let|ψs be a state in which every state|ψi occurs

(2)/k times.Starting from a set of maximally entangled EPR pairs,we do entanglement dilution p i n±δ1n log p

for each state |ψi ?p i n ±δ1n log p i (2)/k .For any δ2and ?2greater than zero there exists an n such that starting from (p i n +δ1n log p i (2)/k )[E f (|ψi ψi |)+δ2]EPR pairs we can obtain an approximation to |ψi ?p i n ±δ1n log p i (2)/k which has square-root-?delity larger than 1??2.Since there are k states in the optimal ensemble (and k is ?nite),we can

therefore approximate the state |ψs with square-root-?delity | ψ|ψ′ |>(1??2)k

,starting from n i p i E f (|ψi ψi |)+n (O (δ1)+O (δ2))EPR pairs,with ?2→0,δ1→0,and δ2→0for n →∞.

The approximation of ρT (n )δ1

by ρ′

T (n )δ

= s ∈T (n )δ1

|ψ′s

ψ′s |,where |ψ′s is the approximation of |ψs which we obtain by entanglement dilution starting from the set of EPR pairs,has the property that

F (ρT (n )δ1

,ρ′T

(n )

δ1

)≥

s ∈T (n )δ

| ψs |ψ′

s |≥(1??2)k ≡1??3,(12)

where ?3→0for n →∞,since k is ?nite.Furthermore,since we can make every state |ψ′

s starting from a

given

set of

EPR

pairs,we can make any convex combination,for example ρ′T (n )δ

,of the states |ψ′

s (see [20]and also [21]),

starting from this same set of EPR pairs.

Finally,we can use the triangle inequality for the Bures metric,and Eqs.(11)and (12)to obtain that

D (ρ?n ,ρ′

T (n )δ1

)≤2

1??1+2√n

.Now suppose that the lemma does not hold,so that lim A n >E c (ρ).It follows that there exists an

integer N ,such that for all k >N ,

|A k ?lim A n |

lim A n ?E c (ρ)

|≤?(17)

and D (ρ?k ,ρk )

lim A n ?E f (ρk )k

<2?,(18)

by using Eqs.(15)and (16).

This gives E f (ρk )>k (lim A n ?2?).On the other hand,by Eq.(17),we have

E f (|Ψ? Ψ?|?m )=m

(19)

which ?nally yields

E f (ρk )?E f (|Ψ? Ψ?|?m )>k (lim A n ?E c (ρ)?3?)=k ?,

(20)

a contradiction since the entanglement of formation cannot increase under LOCC operations.2

III.ALTERNATIVE DEFINITIONS OF THE ASYMPTOTIC COST

While E c(ρ)is perhaps the most natural function to associate with the asymptotic cost of preparing a bipartite state,other de?nitions would have been consistent with our discussion in the introduction.An example of a di?erent but perhaps useful de?nition of the asymptotic entanglement cost is the following:

E alt(ρ)=inf{E|??>0,δ>0,?N|(?n>N?m,L,|E?

1?(1??2)k=2.(22)

The protocol,therefore,generally fails to produce a good approximation for large k.This example suggests that the existence of a threshold N beyond which approximations to some given?delity are always possible is a very di?cult condition to satisfy.Nonetheless,by applying the results of the previous section,it is actually easy to see that the de?nitions E c and E alt are equivalent,so that the extra condition can be met without increasing the asymptotic unit cost.First,notice that the argument of Lemma2actually also works for E alt so that E alt(ρ)≤E∞f(ρ).Next,since the de?nition of E alt is more stringent than that of E c,we have that E c(ρ)≤E alt(ρ).Combining these inequalities with the result of Lemma3,we get

E∞f(ρ)≤E c(ρ)≤E alt(ρ)≤E∞f(ρ),(23) so that these two de?nitions of the entanglement cost always agree.

A?nal pair of alternative de?nitions for the asymptotic entanglement cost would use the trace distance

d(ρ,σ)=

D(ρ,σ)2=1?F(ρ,σ)≤d(ρ,σ)≤ D216,(25)

which show that d and D are equivalent metrics and bounds them in terms of each other by functions that are independent of the dimension of the underlying Hilbert space,H.In other words,d and D are equivalent metrics even in the asymptotic regime.To be concrete,applying the right-hand inequality to Eq.(13),for example,is su?cient to show that Lemma2also holds for the trace distance.In order to prove Lemma3,we used the continuity relation, Eq.(3),but e?ectively only required the weaker inequality

|E(ρ)?E(σ)|≤B D(ρ,σ)log dim H+C,(26) where B and C are constants and H is the supporting Hilbert space.Applying Eq.(25),however,gives an inequality of the form

|E(ρ)?E(σ)|≤B

IV.CONCLUSIONS

We have given two rigorous de?nitions of the asymptotic cost of preparing a bipartite mixed stateρand shown them both to be equal to the regularized entanglement of formation,E∞f(ρ),resolving an important conjecture in the theory of quantum entanglement.Furthermore,we have shown that this asymptotic cost is fairly insensitive to the choice of metric on density operators.In particular,the Bures distance and trace distance result in identical asymptotic costs.

An important problem left open by this work is the question of actually evaluating E∞f(ρ).Even the non-regularized function E f(ρ)is notoriously di?cult to calculate;its value is only known for some very special cases[4,23].If it turns out that E f is not additive for tensor products,then,in spite of the results of this paper,determining the asymptotic cost of preparing a state remains quite a formidable problem.

V.ACKNOWLEDGMENTS

We would like to thank Pawe l Horodecki,Michael Nielsen and John Smolin for helpful discussions.PMH is grateful to the Rhodes Trust and the EU project QAIP,contract No.IST-1999-11234,for support.MH acknowledges support of Polish Committee for Scienti?c Research,contract No.2P03B10316and EU project EQUIP,contract No. IST-1999-11053.BMT acknowledges support of the ARO under contract number DAAG-55-98-C-0041.

[22]C.A.Fuchs and J.van de Graaf,Cryptographic distinguishability measures for quantum mechanical states,IEEE T.

Inform.Theory45:(4)1216(1999),quant-ph/9712042.

[23]B.M.Terhal and K.G.H.Vollbrecht,The entanglement of formation for isotropic states,quant-ph/0005062.

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Q ED Queen’s Economics Department Working Paper No.1227 Critical Values for Cointegration Tests James G.MacKinnon Queen’s University Department of Economics Queen’s University 94University Avenue Kingston,Ontario,Canada K7L3N6 1-2010

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Asymptotic Methods Lecturer:Xuesong Wu,Professor Pre-requisites Linear algebra,calculus(including Fourier/Laplace transforms),analysis of complex variable, ordinary and partial di?erential equations. (Knowledge of classical mechanics and?uid dynamics would be helpful but not essential.) Basic competency in English language:familiarity with mathematic terminology in English. Useful references O.M.Bender&S.A.Orszag Advanced Mathematical Methods for Scientists and Engineers. M.H.Holmes Introduction to Perturbation Methods. E.J.Hinch Perturbation Methods. M.Van Dyke Perturbation Methods in Fluid Mechanics. J.Kevorkian&J.D.Cole Perturbation Methods in Applied Mathematics. A.H.Nayfeh Perturbation Methods. Note:This course may be called asymptotic analysis or perturbation methods. Teaching arrangement 30lectures and plus5tutorials.Extensive printed notes; 5Question Sheets plus solutions(no assessed coursework). General description Many mathematical problems arising from sciences and engineering often consist of small or large parameters. In exceptional cases(e.g.certain types of linear ordinary/partial di?erential equations),the problem can be solved exactly,often by means of an integral transformation(https://www.wendangku.net/doc/8512562426.html,place or Fourier transform),and the solution may then be expressed in the form of an integral,referred to as integral representation.Numerical evaluation by using a computer,though straightforward nowadays,may not be most e?cient.Moreover,the interest is often in the analytical property of the solution when the parameter or variable is large,and it is then desirable to derive an ana-lytical formula which approximates the integral for asymptotically large values of the parameter of variable.Such a formula is referred to as asymptotic representation of the integral. In general,the mathematical problems formulated are so complex(e.g.nonlinear)that?nding an exact analytical solution in closed form is impossible.However,because of the presence of a large or small parameter,approximate solutions may be constructed in the form of asymptotic expansions or asymptotic series,and the original system systematically reduced to a simpler one,which can be solved analytically or numerically.The resulting solution represents a?rst-order approximation,which may be further improved by seeking higher-order corrections.A range of powerful techniques and methods have been developed,each pertaining to a speci?c mathematical feature of the problem.They represent some of the most e?ective and important tools,which are being widely used by researches in both natural and engineering sciences. Aims&objectives This advanced course presents a systematical introduction to asymptotic methods,which form one of cornerstones of modern applied mathematics.The main ideas and various techniques

计量术语对照表

英汉对照计量经济学术语 A 校正R2（Adjusted R-Squared）：多元回归分析中拟合优度的量度，在估计误差的方差时对添加的解释变量用一个自由度来调整。对立假设（Alternative Hypothesis）：检验虚拟假设时的相对假设。 AR（1）序列相关（AR(1) Serial Correlation）：时间序列回归模型中的误差遵循AR（1）模型。渐近臵信区间（Asymptotic Confidence Interval）：大样本容量下近似成立的臵信区间。渐近正态性（Asymptotic Normality）：适当正态化后样本分布收敛到标准正态分布的估计量。渐近性质（Asymptotic Properties）：当样本容量无限增长时适用的估计量和检验统计量性质。渐近标准误（Asymptotic Standard Error）：大样本下生效的标准误。渐近t统计量（Asymptotic t Statistic）：大样本下近似服从标准正态分布的t统计量。渐近方差（Asymptotic Variance）：为了获得渐近标准正态分布，我们必须用以除估计量的平方值。渐近有效（Asymptotically Efficient）：对于服从渐近正态分布的一致性估计量，有最小渐近方差的估计量。渐近不相关（Asymptotically Uncorrelated）：时间序列过程中，随着两个时点上的随机变量的时间间隔增加，它们之间的相关趋于零。衰减偏误（Attenuation Bias）：总是朝向零的估计量偏误，因而有衰减偏误的估计量的期望值小于参数的绝对值。自回归条件异方差性（Autoregressive Conditional Heteroskedasticity, ARCH）：动态异方差性模型，即给定过去信息，误差项的方差线性依赖于过去的误差的平方。一阶自回归过程[AR（1）]（Autoregressive Process of Order One [AR(1)]）：一个时间序列模型，其当前值线性依赖于最近的值加上一个无法预测的扰动。辅助回归（Auxiliary Regression）：用于计算检验统计量——例如异方差性和序列相关的检验统计量——或其他任何不估计主要感兴趣的模型的回归。平均值（Average）：n个数之和除以n。 B 基组、基准组（Base Group）：在包含虚拟解释变量的多元回归模型中，由截距代表的组。基期（Base Period）：对于指数数字，例如价格或生产指数，其他所有时期均用来作为衡量标准的时期。基期值（Base Value）：指定的基期的值，用以构造指数数字；通常基本值为1或100。最优线性无偏估计量（Best Linear Unbiased Estimator, BLUE）：在所有线性、无偏估计量中，有最小方差的估计量。在高斯—马尔科夫假定下，OLS是以解释变量样本值为条件的BLUE。贝塔系数（Beta Coefficients）：见标准化系数。偏误（Bias）：估计量的期望参数值与总体参数值之差。偏误估计量（Biased Estimator）：期望或抽样平均与假设要估计的总体值有差异的估计量。向零的偏误（Biased Towards Zero）：描述的是估计量的期望绝对值小于总体参数的绝对值。二值响应模型（Binary Response Model）：二值因变量的模型。二值变量（Binary Variable）：见虚拟变量。两变量回归模型（Bivariate Regression Model）：见简单线性回归模型。 BLUE（BLUE）：见最优线性无偏估计量。 Breusch-Godfrey检验（Breusch-Godfrey Test）：渐近正确的AR（p）序列相关检验，以AR（1）最为流行；该检验考虑到滞后因变量和其他不是严格外生的回归元。 Breusch-Pagan检验（Breusch-Pagan Test）：将OLS残差的平方对模型中的解释变量做回归的异方差性检验。

计量经济学术语.

校正R2（Adjusted R-Squared）：多元回归分析中拟合优度的量度，在估计误差的方差时对添加的解释变量用一个自由度来调整。对立假设（Alternative Hypothesis）：检验虚拟假设时的相对假设。 AR（1）序列相关（AR(1) Serial Correlation）：时间序列回归模型中的误差遵循AR（1）模型。渐近置信区间（Asymptotic Confidence Interval）：大样本容量下近似成立的置信区间。渐近正态性（Asymptotic Normality）：适当正态化后样本分布收敛到标准正态分布的估计量。渐近性质（Asymptotic Properties）：当样本容量无限增长时适用的估计量和检验统计量性质。渐近标准误（Asymptotic Standard Error）：大样本下生效的标准误。渐近t 统计量（Asymptotic t Statistic）：大样本下近似服从标准正态分布的t统计量。渐近方差（Asymptotic Variance）：为了获得渐近标准正态分布，我们必须用以除估计量的平方值。渐近有效（Asymptotically Efficient）：对于服从渐近正态分布的一致性估计量，有最小渐近方差的估计量。渐近不相关（Asymptotically Uncorrelated）：时间序列过程中，随着两个时点上的随机变量的时间间隔增加，它们之间的相关趋于零。衰减偏误（Attenuation Bias）：总是朝向零的估计量偏误，因而有衰减偏误的估计量的期望值小于参数的绝对值。自回归条件异方差性（Autoregressive Conditional Heteroskedasticity, ARCH）：动态异方差性模型，即给定过去信息，误差项的方差线性依赖于过去的误差的平方。一阶自回归过程[AR（1）]（Autoregressive Process of Order One [AR(1)]）：一个时间序列模型，其当前值线性依赖于最近的值加上一个无法预测的扰动。辅助回归（Auxiliary Regression）：用于计算检验统计量——例如异方差性和序列相关的检验统计量——或其他任何不估计主要感兴趣的模型的回归。平均值（Average）：n个数之和除以n。 B 基组、基准组（Base Group）：在包含虚拟解释变量的多元回归模型中，由截距代表的组。基期（Base Period）：对于指数数字，例如价格或生产指数，其他所有时期均用来作为衡量标准的时期。基期值（Base Value）：指定的基期的值，用以构造指数数字；通常基本值为1或100。最优线性无偏估计量（Best Linear Unbiased Estimator, BLUE）：在所有线性、无偏估计量中，有最小方差的估计量。在高斯—马尔科夫假定下，OLS是以解释变量样本值为条件的BLUE 。贝塔系数（Beta Coef?cients）：见标准化系数。偏误（Bias）：估计量的期望参数值与总体参数值之差。偏误估计量（Biased Estimator）：期望或抽样平均与假设要估计的总体值有差异的估计量。向零的偏误（Biased Towards Zero）：描述的是估计量的期望绝对值小于总体参数的绝对值。二值响应模型（Binary Response Model）：二值因变量的模型。二值变量（Binary Variable）：见虚拟变量。两变量回归模型（Bivariate Regression Model）：见简单线性回归模型。 BLUE（BLUE）：见最优线性无偏估计量。 Breusch-Godfrey 检验（Breusch-Godfrey Test）：渐近正确的AR（p）序列相关检验，以AR（1）最为流行；该检验考虑到滞后因变量和其他不是严格外生的回归元。 Breusch-Pagan 检验（Breusch-Pagan Test）：将OLS残差的平方对模型中的解释变量做回归的异方差性检验。

计量经济学术语

A 校正R2（Adjusted R-Squared）：多元回归分析中拟合优度的量度，在估计误差的方差时对添加的解释变量用一个自由度来调整。对立假设（Alternative Hypothesis）：检验虚拟假设时的相对假设。 AR（1）序列相关（AR(1) Serial Correlation）：时间序列回归模型中的误差遵循AR（1）模型。渐近置信区间（Asymptotic Confidence Interval）：大样本容量下近似成立的置信区间。渐近正态性（Asymptotic Normality）：适当正态化后样本分布收敛到标准正态分布的估计量。渐近性质（Asymptotic Properties）：当样本容量无限增长时适用的估计量和检验统计量性质。渐近标准误（Asymptotic Standard Error）：大样本下生效的标准误。渐近t 统计量（Asymptotic t Statistic）：大样本下近似服从标准正态分布的t统计量。渐近方差（Asymptotic Variance）：为了获得渐近标准正态分布，我们必须用以除估计量的平方值。渐近有效（Asymptotically Efficient）：对于服从渐近正态分布的一致性估计量，有最小渐近方差的估计量。渐近不相关（Asymptotically Uncorrelated）：时间序列过

程中，随着两个时点上的随机变量的时间间隔增加，它们之间的相关趋于零。衰减偏误（Attenuation Bias）：总是朝向零的估计量偏误，因而有衰减偏误的估计量的期望值小于参数的绝对值。自回归条件异方差性（Autoregressive Conditional Heteroskedasticity, ARCH）：动态异方差性模型，即给定过去信息，误差项的方差线性依赖于过去的误差的平方。一阶自回归过程[AR（1）]（Autoregressive Process of Order One [AR(1)]）：一个时间序列模型，其当前值线性依赖于最近的值加上一个无法预测的扰动。辅助回归（Auxiliary Regression）：用于计算检验统计量——例如异方差性和序列相关的检验统计量——或其他任何不估计主要感兴趣的模型的回归。平均值（Average）：n个数之和除以n。 B 基组、基准组（Base Group）：在包含虚拟解释变量的多元回归模型中，由截距代表的组。基期（Base Period）：对于指数数字，例如价格或生产指数，其他所有时期均用来作为衡量标准的时期。基期值（Base Value）：指定的基期的值，用以构造指数数字；通常基本值为1或100。最优线性无偏估计量（Best Linear Unbiased Estimator, BLUE）：在所有线性、无偏估计量中，有最小方差的估

计量经济学英汉术语名词对照及解释

计量经济学英汉术语名词对照及解释 A 校正R2（Adjusted R-Squared）：多元回归分析中拟合优度的量度，在估计误差的方差时对添加的解释变量用一个自由度来调整。对立假设（Alternative Hypothesis）：检验虚拟假设时的相对假设。 AR（1）序列相关（AR(1) Serial Correlation）：时间序列回归模型中的误差遵循AR（1）模型。渐近置信区间（Asymptotic Confidence Interval）：大样本容量下近似成立的置信区间。渐近正态性（Asymptotic Normality）：适当正态化后样本分布收敛到标准正态分布的估计量。渐近性质（Asymptotic Properties）：当样本容量无限增长时适用的估计量和检验统计量性质。渐近标准误（Asymptotic Standard Error）：大样本下生效的标准误。渐近t 统计量（Asymptotic t Statistic）：大样本下近似服从标准正态分布的t统计量。渐近方差（Asymptotic Variance）：为了获得渐近标准正态分布，我们必须用以除估计量的平方值。渐近有效（Asymptotically Effcient）：对于服从渐近正态分布的一致性估计量，有最小渐近方差的估计量。渐近不相关（Asymptotically Uncorrelated）：时间序列过程中，随着两个时点上的随机变量的时间间隔增加，它们之间的相关趋于零。衰减偏误（Attenuation Bias）：总是朝向零的估计量偏误，因而有衰减偏误的估计量的期望值小于参数的绝对值。自回归条件异方差性（Autoregressive Conditional Heteroskedasticity, ARCH）：动态异方差性模型，即给定过去信息，误差项的方差线性依赖于过去的误差的平方。一阶自回归过程[AR（1）]（Autoregressive Process of Order One [AR(1)]）：一个时间序列模型，其当前值线性依赖于最近的值加上一个无法预测的扰动。辅助回归（Auxiliary Regression）：用于计算检验统计量——例如异方差性和序列相关的检验统计量——或其他任何不估计主要感兴趣的模型的回归。平均值（Average）：n个数之和除以n。 B 基组、基准组（Base Group）：在包含虚拟解释变量的多元回归模型中，由截距代表的组。基期（Base Period）：对于指数数字，例如价格或生产指数，其他所有时期均用来作为衡量标准的时期。基期值（Base Value）：指定的基期的值，用以构造指数数字；通常基本值为1或100。最优线性无偏估计量（Best Linear Unbiased Estimator, BLUE）：在所有线性、无偏估计量中，有最小方差的估计量。在高斯—马尔科夫假定下，OLS是以解释变量样本值为条件的BLUE 。贝塔系数（Beta Coefficients）：见标准化系数。偏误（Bias）：估计量的期望参数值与总体参数值之差。

Asymptotic locally optimal detector for large-scale sensor networks under the Poisson regim

Asymptotic Locally Optimal Detector for Large-Scale Sensor Networks Under the Poisson Regime Youngchul Sung,Lang Tong,Fellow,IEEE,and Ananthram Swami,Senior Member,IEEE Abstract—We consider distributed detection with a large number of identical binary sensors deployed over a region where the phenomenon of interest(POI)has spatially varying signal strength.Each sensor makes a binary decision based on its own measurement,and the local decision of each sensor is sent to a fusion center using a random access protocol.The fusion center decides whether the event has occurred under a global size constraint in the Neyman–Pearson formulation.Assuming homo-geneous Poisson distributed sensors,we show that the distribution of"alarmed"sensors satis?es the local asymptotic normality (LAN).We then derive an asymptotically locally most powerful (ALMP)detector optimized jointly over the fusion form and the local sensor threshold under the Poisson regime.We establish conditions on the spatial signal shape that ensure the existence of the ALMP detector.We show that the ALMP test statistic is a weighted sum of local decisions,the optimal weights being the shape of the spatial signal;the exact value of the signal strength is not required.We also derive the optimal threshold for each sensor.For the case of independent,identically distributed(iid) sensor observations,we show that the counting-based detector is also ALMP under the Poisson regime.The performance of the proposed detector is evaluated through analytic results and Monte Carlo simulations and compared with that of the counting-based detector.The effect of mismatched signal shapes is also investi-gated. Index Terms—Asymptotically locally most powerful(ALMP), distributed detection,fusion rule,local asymptotic normality (LAN),Neyman–Pearson criterion,spatial Poisson process,spa-tially varying signal. I.I NTRODUCTION A.Detection in Large-Scale Sensor Field W E CONSIDER the detection of phenomenon in a geo-graphical area using a large number of densely deployed microsensors.The sensors measure the phenomenon of interest (POI)and transmit their local data(the binary decision)via wireless channel to a central site for global processing.A spe-ci?c implementation is the Sensor Network with Mobile Access Manuscript received November10,2003;revised July12,2004.This work was supported in part by the Multidisciplinary University Research Initiative (MURI)under the Of?ce of Naval Research under Contract N00014-00-1-0564. Prepared through collaborative participation in the Communications and Net-works Consortium sponsored by the U.S.Army Research Laboratory under the Collaborative Technology Alliance Program,under Cooperative Agreement DAAD19-01-2-0011.The https://www.wendangku.net/doc/8512562426.html,ernment is authorized to reproduce and dis-tribute reprints for Government purposes notwithstanding any copyright nota-tion thereon.The associate editor coordinating the review of this manuscript and approving it for publication was Prof.Yucel Altunbasak. Y.Sung and L.Tong are with the School of Electrical and Computer Engineering,Cornell University,Ithaca,NY14853USA(e-mail:ys87@ https://www.wendangku.net/doc/8512562426.html,;ltong@https://www.wendangku.net/doc/8512562426.html,). A.Swami is with the Army Research Laboratory,Adelphi,MD20783USA (e-mail:a.swami@https://www.wendangku.net/doc/8512562426.html,). Digital Object Identi?er 10.1109/TSP.2005.847827 Fig.1.Sensor network with mobile access points. (SENMA)architecture[30],where a mobile access point or in- terrogator collects local decisions from sensors using random access schemes such as ALOHA;see Fig.1.We assume that the number of sensors in the?eld is large,which makes it necessary that each sensor is inexpensive and has limited computation and communication capability. Detection in a large-scale sensor network faces several chal- lenges not encountered in the classical distributed detection problem.First,inexpensive sensors are not reliable;they have low duty cycles and severe energy constraints.The communi- cation link between a sensor and the central unit is specially weak due to a variety of implementation dif?culties such as synchronization,fading,and interference from other sensors. The probability that the local decision at a particular sensor can be successfully delivered to the central unit can be very low.Second,POI in a wide geographic area generates spatially varying signals,which makes the observation at each sensor lo- cation-dependent and not identically distributed.Furthermore, the strength of POI is unknown in many applications such as the detection of environmental hazards such as nuclear,biological, and chemical(NBC)activities.Third,the scale of the network makes it more practical to deploy sensors randomly without careful network layout.It is thus not possible to predict whether data from a particular sensor can be retrieved by the central processing unit,especially when random access protocols are used.Consequently,the decision rule for each sensor should be optimized before deployment without knowing its exact location and signal strength.In addition,because sensors may expire and the collection process is random,the optimal deci- sion should not critically depend on the number of available sensors or on the collection process. B.Approach and Summary of Results For large-scale sensor networks it is natural to consider asymptotic techniques,and one expects that the central limit theorem will lead to a design under Gaussian statistics.For 1053-587X/$20.00?2005IEEE