in recent years , a smart grid has attracted a tremendous amount of research interest due to its potential benefit to modern civilization .the remarkable development of computer and communication technology has enabled a realization of smart power grid .a key feature of a smart power grid is the change of the power distribution characteristics from a centralized power system to a distributed power system .though centralized power plants still cover the major portion of power demand , the amount of power demand covered by distributed power resources has been increasing steadily .  in distributed power systems , achieving the supply - demand balance , which is one of the fundamental requirements , is a key challenging issue due to its decentralized characteristics . the supply - demand balance problem has been traditionally considered as the economic dispatch which minimizes the total cost of operation of generation systems . however , the traditional dispatch problem is a highly nonlinear optimization problem and thus it has been addressed usually in a centralized manner .distributed control of distributed power systems has been considered in . in the distributed power system ,individual power resources are interconnected with each other through communication and transmission line .thus , for the coordination of power generation and power flow , sharing information is essential .it is more realistic and economic to use only local information .this kind of problem has been solved in consensus .distributed resource allocation method named `` center - free algorithm '' has been considered in . in the algorithm ,since the amount of resource for each node is determined by the sum of weighted difference with its neighbors , we can easily see the equivalence of the algorithm to the consensus .thus , we can consider that the idea of consensus had been already used in the distributed resource allocation problems .  in this paper , power distribution among distributed power resourcesis studied .the problem is classified into three subproblems . in the first problem , which is called _ power coordination _ ,the desired net power is determined . in the second problem and the third problem ,we consider power generation and power flow control of distributed power resources with and without power coordination respectively .note that the power flow control without power coordination is significantly important when the net power and the generation capacities of each power resource are limited .consensus schemes are employed to generate power coordination , power generation and power flow control signals .subsequently , the main contribution of this paper can be summarized as follows .first , the coordination and control of power generation and power flow are precisely defined and formulated .second , two power distribution schemes with and without power coordination are developed , which may be helpful in coordinating actual power generation and power flow among distributed power plants .the proposed approach deals with the desired net power which is not necessarily an average value as in the distributed averaging problem .third , it is shown that the distributed consensus algorithms ensure power distribution among distributed nodes that have net power and generation capacity constraints .the proposed approach can be applied even if the desired net power of some power resources does not satisfy the power generation capacity as opposed to those in .this paper is organized as follows . in section [ sec_prob_form ] , power coordination ,power generation and power flow control problems are formulated .main results of this paper are presented in section [ sec_main_res ] .illustrative examples are provided in section [ sec_example ] and the conclusion is given in section [ sec_conc ] , respectively .1 shows a graph representing interconnected power systems .each node denotes a subsystem , i.e. , an area that consists of generators and loads , and it has its power demand determined by its loads .thus , the desired net power for each generator in the subsystem is required to be determined .the power can be transmitted to its neighboring subsystems since they are interconnected with each other through transmission lines . in this paper, we would like to control the power generation and power flow to satisfy supply - demand balance under net power and generation capacity constraints .let be the desired net power at the -th node ; be the generated power at the -th node ; and be the power flow from the -th node to the -th node , where and .the net power at a node is determined after the net power exchange of generated powers with its neighbor nodes .thus , the net power at the -th node can be defined as , where is the set of neighbor nodes of the -th node .the total generated power and total power demand within the power grid are represented by and , respectively .  throughout this paper ,we need the following assumption to represent a reality of power grid network systems .[ assump ]  1 .each node has the net power capacity constraint such as where indicates the sampling instant of the physical power layer .the net power capacity is associated with the amount of power that each node can handle .thus , the desired net power for each node shall be limited by the net power capacity constraint .2 .  each node has the generation capacity constraint such as the generation capacity is also assumed to be bounded by the net power capacity constraint i.e. , \subseteq \left [ { \underline { { p_i } } , \overline { { p_i } } } \right] ] and  ] which satisfies .thus , this solution is given by in the similar manner , the steady state solution for ^t } \label{eq32}\end{aligned}\ ] ] is given by thus , it follows from and that can be represented by which is equivalent to .since the power coordination law is fully distributed , it can be applied even if some power resources are locally added to or removed from the power grid network under the realizability assumption .the convergence of the algorithm - can be ensured if the sampling of the algorithm is much faster than that of the physical layer .thus , the power coordination scheme of - is feasible in practice . in more detail , let be the time at the -th sample instant in the physical power layer .then , the power coordination - should be completed in .thus , the time interval for iterations - should be chosen to so that the steady state solutions of - can be obtained in .   as mentioned in _ problem [ prob2 ] _ and_ problem [ prob3 ] _ , we attempt to design power generation and power flow in order to make net power be equal to the desired net power at each node with and without power coordination , respectively . from - , the net power of each node can be described as follows : where is the control input at the -th node .it is remarkable that , considering the generation capacity , the power generation control input should satisfy the following constraint where and .define the net power flow at the -th node as follows and define the coordination error at each node as   in the sequel , we provide power generation and power flow control schemes with and without power coordination taking account of two different scenarios mentioned in section [ sec_prob_form ] .   with the power coordination - , the desired net power for each node should satisfy the generation capacity of each node . from ,the coordination error is given by our goal is to design such that the coordination error becomes zero .  with the power coordination - ,if control input is given by then the constraint and can be ensured .  with the power coordination -, can be achieved .thus , we can have in , choosing and by the control law of , we have .therefore , by , the constraint is satisfied. furthermore , by inserting into , we can achieve .[ fig3 ] shows the overall power flow control with power coordination . without loss of generality, it is assumed that node is a leading node that knows the total demand .only the information of generation capacity and the degree of node are exchanged between neighboring nodes . with this information , the desired net power for each nodeis determined by the power coordination .after the power coordination , power generation control input is given by .as a consequence of the control input , is achieved . in this case , because there is no power flow between pairs of nodes .let us assume that the desired net power for each node is individually given without power coordination .but it is supposed that the desired net power satisfies the realizability condition and net power capacity constraints . in this case, the desired net power for some nodes may not satisfy the generation capacity of their node . thus , it is not sufficient to provide only the power generation control for each node .therefore , the power flow control input can be represented as , with enabled in this case .first , we want to design the power generation control to achieve the overall supply - demand balance .now , we provide a lemma to derive the main result of this paper .if the power generation control input is designed by the following law then , the control input will satisfy the constraint and the supply - demand balance is achieved .  for the left - side inequality of ,if is given by , then we can obtain the following inequality :   for the right - side inequality , if is given by , with the substitution of } [ { \underline { { \delta p_{{g_{j } } } } } \left ( k \right ) - \overline { { \delta p_{{g_{j } } } } } \left ( k \right ) } ] + [ { \overline { { \delta p_{{g_{i } } } } } \left ( k \right ) } { - \underline { { \delta p_{{g_{i } } } } } \left ( k \right ) } ] [ { p_d}(k ) ] , we can obtain the following inequality : furthermore , summing up the both sides of over from to yields thus , the proof is completed .  however , as in , requires global information such as , , and . for a decentralized power generation control ,we now provide the following theorem :  the power generation control input can be achieved if the following consensus scheme is used . where are steady state solutions of the following equations :   the consensus scheme and can be represented by ^t } \label{eq61}\end{aligned}\ ] ] as in the proof of _ theorem [ thm1 ] _ , the steady state solution is given by where in element - wise , and with and ^t} ] and $ ] is doubly stochastic , where and .as in the proof of _ theorem [ thm1 ] _ , the steady state solution is given by where in element - wise , and with and which satisfies .furthermore , * 1 * is also the right eigenvector with the associated - eigenvalue because is doubly stochastic .thus , without loss of generality , let , which satisfies .then , the solution of with converges to the average as follows : from , we can obtain because of .thus , we have .hence also , from - , it follows that thus , from and , we have if we choose the interaction control input as , it follows from and that     fig .[ fig4 ] shows the power generation and power flow control scheme without power coordination .as previously mentioned , it is assumed that the desired net power for each node is given under .the power generation control input is given by - .as a consequence of the control input , the supply - demand balance is achieved .next , the power flow between pairs of nodes is determined by - . then , after the power flow control , is achieved .there are iterations for the power generation control input computation and for the power flow control input computation . thus , the time interval for iterations - and - should be chosen to so that the steady state solutions of the iterations can be obtained in .  the constraints of the _ assumption 2.1 _ might be time - varying in renewable power resources such as a wind or a solar system .it is possible for the proposed approach to account for time - varying bounds if the rate of change is not faster than and the realization condition is satisfied .this can be easily verified if the time - varying bounds are substituted into the proposed approach instead of the constant bounds .in this section , two illustrative examples are provided .the distributed power resources are interconnected as depicted in fig .[ fig1 ] . the generation capacity and net power capacity of each nodeare listed in table [ tb_1 ] .  .genearation capacity and net power capacity at each node [ cols="^,^,^",options="header " , ]   this example shows the power distribution with power coordination . in this case , without loss of generality , it is assumed that node is the leading node that knows the total power demand of the distributed power system .first , the total power demand satisfying the realization condition is randomly created and it is depicted in fig .[ fig5 ] ( a ) .then , is determined by power coordination - as shown in fig .[ fig5 ] ( b ) .next , the power generation control input given by is shown in fig .[ fig5 ] ( c ) and the generated power ( i.e. , ) is depicted in fig .[ fig5 ] ( d ) .consequently , the coordination error given by is zero and the supply - demand balance is also achieved as shown in fig .[ fig5 ] ( b ) and fig . [ fig5 ]( d ) .this example shows power distribution without power coordination . in this case , the desired net power for each node is not given by power coordination but they are randomly created under the realization condition as depicted in fig .[ fig8 ] ( a ) .first , the power generation control input are given by - as shown in fig .[ fig8 ] ( b ) and the generated power ( i.e. , ) is shown in fig .[ fig8 ] ( c ) .after the power generation control , a coordination of power flows between pairs of nodes is necessary to make converge to . the power flows between pairs of nodes are determined by - and it is depicted in fig .[ fig8 ] ( d ) .next , we can obtain the net power flow at each node from as shown in fig .[ fig8 ] ( e ) .then , the net power at each node is given by as shown in fig .[ fig8 ] ( f ) .then , the coordination error given by is zero and the supply - demand balance is also achieved as shown in fig .[ fig8 ] ( a ) and fig .[ fig8 ] ( f ) .this paper has addressed power distribution problems in distributed power grid with and without power coordination .first , a power coordination using a consensus scheme under limited net power and power generation capacities was considered .second , power generation and a power flow control laws with and without power coordination were designed using a consensus scheme to achieve supply - demand balance . sincethis paper has provided systematic approaches for power distribution among distributed nodes on the basis of consensus algorithms , the results of this paper can be nicely utilized in power dispatch or power flow scheduling .the authors believe that consensus algorithm - based power distribution schemes of this paper have several advantages over typical power dispatch approaches .the first key advantage is that the power coordination and control can be realized via decentralized control scheme without relying upon nonlinear optimization technique .the second advantage is that the framework proposed in this paper can handle power constraint , generation , and flow in a unified framework .it is noticeable that this paper has considered the power coordination , power generation and power flow control in the higher - level models of grid networks but does not consider lower - level models of grid networks such as current , voltage drop , and line impedance .however , in our future research , it would be meaningful to add links between the higher - level models and the lower - level models .further , it is desirable to investigate various features such as behavior of self - interested customer , power loss in the transmission line , and structures between the physical and cyber layers such as delays , and mismatches between them .though the paper has only focused on power distribution , we believe that the proposed approach can be extended to various attribute distribution problems such as traffic control and supply chain management , because we use a fundamental model describing flow of attribute between pairs of distributed resources as well as the amount of attribute in each distributed resource .for example , in a traffic control , each freeway section named `` cell '' corresponds to each distributed power resource , distribution of vehicle density corresponds to net power of each power resource , desired traffic density corresponds to desired net power , and on - ramp traffic flow corresponds to power generation control input .thus , the goal of traffic control which satisfies desired traffic density corresponds to that of power distribution which satisfies desired net power .it is recommended to see ` byeong - yeon kim , `` coordination and control for energy distribution using consensus algorithms in interconnected grid networks '' , ph.d .dissertation , school of information and mechatronics , gwangju institute of science and technology , 2013 ' for applications to various engineering problems of the algorithms developed in this paper .z. zhang , x. ying , and m.y .chow , `` decentralizing the economic dispatch problem using a two - level incremental cost consensus algorithm in a smart grid environment , '' in _ north american power symposium ( naps ) , 2011_. ieee , 2011 , pp .17 .dominguez - garcia and c.n .hadjicostis , `` coordination and control of distributed energy resources for provision of ancillary services , '' in _ smart grid communications ( smartgridcomm ) , 2010 first ieee international conference on_. ieee , 2010 , pp .537542 .robbins , a.d .domnguez - garca , and c.n .hadjicostis , `` control of distributed energy resources for reactive power support , '' in _ north american power symposium ( naps ) , 2011_. ieee , 2011 , pp . 15 .ahn and k.k .oh , `` command coordination in multi - agent formation : euclidean distance matrix approaches , '' in _ control automation and systems ( iccas ) , 2010 international conference on_. ieee , 2010 , pp . 15921597 .cady , a.d .dominguez - garcia , and c.n .hadjicostis , `` robust implementation of distributed algorithms for control of distributed energy resources , '' in _ north american power symposium ( naps ) , 2011_. ieee , 2011 , pp .

Abstract: this paper presents power coordination , power generation , and power flow control schemes for supply - demand balance in distributed grid networks . consensus schemes using only local information are employed to generate power coordination , power generation and power flow control signals . for the supply - demand balance , it is required to determine the amount of power needed at each distributed power node . also due to the different power generation capacities of each power node , coordination of power flows among distributed power resources is essentially required . thus , this paper proposes a decentralized power coordination scheme , a power generation , and a power flow control method considering these constraints based on distributed consensus algorithms . through numerical simulations , the effectiveness of the proposed approaches is illustrated .


cavity qed experiments offer wide possibilities to vary interaction strengths , resonance conditions and dissipation rates , and thus provide ideal demonstrations of the dynamical evolution of single quantum systems .the combined field and atom degrees of freedom allow , on the one hand , probing of a cavity field by transmission and detection of the state of atoms passing through the cavity , and , on the other hand , probing of the quantum state of atoms located inside a cavity by detection of the transmitted field .recent experiments with superconducting circuit elements and resonators have shown similar measurement capabilities to monitor over time the quantum state of individual quantum systems .we have recently shown how the incoherent jumping of one or two atoms between different hyperfine ground states can be monitored due to the different cavity transmission levels associated with the atomic states .rather than inferring the atomic states directly from the noisy value of the transmission signal , a bayesian approach was used , where the measured signal continuously updates our probabilistic knowledge based on the previous measurements .this was shown to provide more decisive predictions about the state of the system .an extension of the bayesian method to incorporate different unknown physical parameters of the light - atom interacting system has also been demonstrated .  in this paperwe extend the modelling of the system to condition our estimation of the state of the atom at a given time , not only on the optical measurement data acquired up to time but also by the data acquired later .this is a well developed procedure , known as smoothing , in the analysis of hidden markov models ( hmm ) , where methods also exist to combine the state estimation with the estimation of the physical model parameters from the available data .the paper is organized as follows . in sec .ii we give a description of the experimental set - up and the data obtained . in sec iii , we present a summary of the general bayesian method applied in the paper and of the formalism needed to provide the atomic state based on both past and future measurements . in sec .iv , we discuss how system parameters such as atomic transition probabilities and optical signal distributions can be efficiently determined in the hmm framework . in sec .v , we apply our formalism to experimental data , and we show that the atomic state and model parameters are indeed more decisively determined by the full time analysis . in sec .vi , we conclude with a discussion of the results and some further perspectives .in the experiment , we couple single neutral caesium atoms to the mode of an optical high - finesse cavity , see fig .[ fig : setup ] .very few caesium atoms are collected and laser - cooled in a high - gradient magneto - optical trap , and subsequently transferred into a far - detuned standing - wave optical dipole trap . by analyzing a fluorescence imagewe make sure that only a single atom is loaded  . using the dipole trap as an optical conveyor belt , we transport the atom to the center of the mode of an optical cavity with a very high finesse of  .the small mode volume of the cavity ( length 160 m , mode waist radius 23 m ) leads to an effective atom - cavity coupling rate of  , which is larger than the atomic dipole decay rate and the cavity field decay rate .these parameters place the system in the strong - coupling regime , where the single - atom cooperativity is larger than one , and the absorptive and dispersive effects of a single atom lead to a nomal mode splitting of the cavity resonance ( vacuum - rabi - splitting ) and can thus strongly change the cavity transmission  .the cavity resonance is stabilized 30mhz blue detuned to the transition of the d2 line at 852 nm , see fig .[ fig : scheme ] .a weak probe laser beam , resonant with the empty cavity , is coupled into the cavity and populates the cavity mode with 0.3 photons on average . in this blue - detuned weak probing regime ,cavity cooling  counteracts heating of the atom due to photon scattering and thus allows us to observe each atom over several seconds .the transmission of the probe light through the cavity is detected by a single - photon counting module with an overall detection efficiency of 4.4%  .the arrival time of each detected photon is recorded with 50ns time resolution , see fig .[ fig : data ] and combined by software into time bins convenient for presentation of the data .s time intervals .( c ) zoomed time region showing the individual detector clicks , experiencing a pronounced change in click rate as reflected also in ( a ) and ( b).,scaledwidth=90.0% ]  when we place an atom prepared in into the cavity mode , the detected cavity transmission drops from about 30 counts / ms to 5 ... 10 counts / ms , see fig .[ fig : data](a ) , where the data is binned in 1 ms intervals . within a few miliseconds , however ,off - resonant scattering of the probe light fron the level transfers the atom to the ground state with a rate of about 40s .since the optical transitions starting from are detuned by the hyperfine splitting of from the cavity resonance , the atom in this state does not noticeably influence the cavity resonance and the transmission raises to the high level of the empty cavity .the cavity transmission is thus a measure of the atom s hyperfine quantum state , and a quantum jump from to becomes visible as a sudden transition from low to high photon count rate .a very weak laser beam tuned to the transition repumps the atom back to the state with a rate comparable to the pumping rate due to the probe laser . in this waywe obtain a sequence of quantum jumps between and , and in the cavity transmission we observe the corresponding random telegraph signal  .the physical process described in the previous chapter is a good candidate for a hidden markov model analysis , i.e. , as a stochastic process which produces a measurement signal whose statistics is governed by the current state of the ( hidden ) system which is itself governed by a markov evolution process . in our case the signal is the number of photons recorded and the hidden system is the atom , whose state jumps randomly between the hyperfine states .the state and the signal are represented in discrete intervals of time , and for convenience we use integer values to represent these bin time intervals of a finite duration , which is short enough to accurately represent the atomic dynamics .the markov property formally requires that the atomic state is conditionally independent of all variables except and that is conditionally independent of all variables except . in sec .v , we shall comment further on the markovian ansatz in connection with our modelling of the experiment . between measurements ,the atomic system is governed by jumps between its internal states , as described by the _ transition _probabilities , .the signal , on the other hand , is governed by a conditional probability distribution . in this sectionwe show how recursive application of bayes rule , , permits calculation of the probability for the atomic state , conditioned on the measured data . in sec .iv , we show how the calculation can be applied in an iterative manner to re - estimate also the jump rates and the signal conditional probabilities , which are assumed to be known input to the bayesian analysis .  the joint probability for the stochastic variables and to have attained particular values at all instances of time , , can be written as a product of conditional probabilities and a prior atomic state probability ,   we recall that the optical transmission signal is given by the experiment , while the atomic state is not directly observed , and the goal of the present work is to determine the probability distribution conditioned on the experimental results as sharply as possible .let us first present the conventional situation , where one uses the signal data to estimate the current state of the system , by a recursive bayesian update .we shall refer to this as a `` forward estimate '' , because the data acquired until a given instant is used to estimate the state at time .while the current state probabilities depend on the entire signal prior to the current time , they need not be formally computed as a function of these many variables , as the distribution can be updated at every instant of time taking into account only the prior probability and the most recently measured result .we assume an initial prior for the atomic state and we note that given the probability distribution at the subsequent time is given by the transition probabilities , this provides the prior for the bayesian estimate of the state at time , and the probability conditioned on the measurement result becomes where the constant of proportionality can be found by normalization .the combined effect of transition probabilities and bayesian conditional probabilities is repeated at every time step , and such a procedure was succesfully applied in to the case of both one and two atoms inside an optical cavity , and its achievements were compared favorably to the simpler state estimation based on the noisy instantaneous transmission signal in different time bins .any information that we acquire , which contains further information about the atomic state will lead to an improved estimate and a conditional update of . in this sectionwe will focus on the information about available in the measurement signal _ after _ time .this information must be dealt with via appropriate definition and calculation of conditional probabilities and transition probabilities .let us first , however , discuss the meaning of assigning probabilities and making predictions for past events .clearly , we can not go back in time and verify the prediction , and if we already measured the atomic state at time , it would already be known to us and no further refinement of would be relevant .note , however , that in the case of classical probabilities , we assign the probabilities to states that could have been measured and the result could have been copied and stored , so that they could be compared with our later predictions . a meaningful test of `` past predictions '' could then involve the comparison of our prediction with the actual outcome of a measurement which had a result that was not revealed at the time of the measurement .time dependent state estimation based on a full detection record constitutes , indeed , an already well established component , called smoothing , in the theory of hidden markov models . for completenesswe provide a brief summary of this method , following the presentation in .first , it is convenient to introduce two time dependent quantities , a joint and a conditional probability distribution ,   by normalization of , we recognize the forward state estimate , while and together serve to provide the state probabilities , conditioned on the full measurement record here , we have used that and that the probability for observing _ and _ is the probability for observing and multiplied by the probability for observing _ given _ .it follows from the markov property that   to obtain a similar update rule for , we consider from which we obtain where the first equality follows from the general formula , with , and _ and _ the fact that in our case due to the markov property .we have thus obtained the desired recursive equation for : and even though represents a seemingly complicated conditonal probability for the entire signal sequence , the equation is readily solved recursively backwards from the final instant of measurements where .finally , combining the solution for and , we have the state estimate based on the full detection record ( [ eq : alpha_beta_estimate ] ) .it will often be the case that some physical parameters are not precisely known , and that they will have to be determined by the same or a similar experiment as the one revealing the quantum state itself . in the bayesian formalism, it is possible to treat not only the quantum state but also the value of a small set of variables probabilistically . such parameter estimation , which is possible if the parameter space is not too large , was demonstrated in in a simultaneous determination of the atomic state and the transition rates between the atomic states from the photon transmission data set . for a larger number of unknown parameters ,the multidimensional bayesian filter becomes numerically intractable .one may then have recourse to the more direct approaches discussed in this section .consider first the case where we wish to determine the atomic transition probabilities for the transition from state i to state j. we assume that this probability is time - independent , and we therefore write it as in the following .the transitions are revealed through the time - dependent probability that we occupy state and at subsequent time steps , which are in turn , inferred from the measured data , now , since we may also write   this enables the re - estimation of the transition probabilities  where the second equality follows from we recognize that eq.([reest_trans ] ) in a probabilistic manner counts the number of times a transition occurs between and divided by the accumulated occupation of the -state .note , that for the calculation of this quantity , the forward - backward estimated state provides a more accurate estimate of the parameters : since the estimated state is more accurate , the counting of the number of transitions is more accurate , which leads to the more accurate estimate of the transition probabilities .as shown in , the actual photon counting distribution for a given known atomic state may be broader than a poisson distribution , possibly associated with the physical motion of the atom between locations that experience different coupling strengths to the cavity field mode .assuming , for simplicity , that the counting distribution is governed by the markov property and depends conditionally only on the atomic state , a simple procedure determines by counting how often the signal outcome occurs when the atom is in state .since the occupation of state is given probabilistically , this translates into the expression ,   the above procedure is the so - called baum - welch estimation which can be shown to be equivalent to the so - called estimation - maximization ( em ) algorithm , and will always converge to a ( local ) optimal estimate .there is , however , no guarantee that the procedure will find the globally optimal model . as in the determination of the atomic transition rates ,the better the knowledge of the state occupation in state , the better is the estimation of the signal probability distribution . as we shall show with our numerical example , the state estimation based on the full data record is far superior over the only forward bayesian method , and therefore the parameter estimation is also expected to be much more precise .note that the equations in this and the previous section have to be iterated , since the conditional signal probabilities determine the state estimate , which in turn leads to an updated estimate of the signal probabilities .further details of this iterative scheme and on the identification of a suitable hidden markov model will be described in the next section .in this section we apply the procedures reviewed in the previous sections to the data obtained in the experiments described in sec .ii . while the atom has two ground hyperfine ground states , andwould thus seem amenable to an hmm description with two ( hidden ) states , it was observed in that the atom in the state showed super - poissonian counting statistics , var . in this was ascribed to the influence of other degrees of freedom of the atom , e.g. , its spatial position , which would affect its coupling to the cavity field mode . if such degrees of freedom in the system are evolving on a significantly longer time scale , it would violate the basic markov assumption of hmm to treat the purported occupation by a single hidden state .such an effect can be clearly seen in fig .3(a ) when comparing the transmission level around 0.3 to 0.4 s and around 0.8 to 0.9 s. the hmm theory outlined in secs .iii and iv is , however , ideally suited to problems with an unknown number of hidden states with unknown dynamics .it is the basic idea of the method that it allows `` black box '' modelling of an observed signal by states , transition rates and signal properties .we will thus allow more than two states in our modelling of the system , and use the state and parameter estimation to assess their physical meaning afterwards .[ fig : data ] is , indeed , suggestive of a dynamical evolution of the system between different levels of low transmission , and we shall therefore attempt a model of the system with three states .we do not restrict the properties of these states , but we expect that the algorithm will use the available parameter space to represent one non - interacting state and two different states belonging to with slightly different transmission characteristics .  , inferred from the forward bayesian analysis of the measured data . the two states with slightly different transmission characteristics hidden in are indicated with dark and light blue color .( b ) the atomic state populations conditioned on the forward - backward bayesian state estimation . in both figures ,the state is conditioned on the data shown in fig .[ fig : data],scaledwidth=90.0% ]  as described above , we first use the bayesian state update formulas with a more or less random initial ansatz for the rate and signal parameters , and we then subsequently iterate the re - estimation and state estimation process until they have converged .we have tested that this procedure converges to the same values with different initial parameter choices , and the results we show in this section are the ones obtained after consistently identifying the model parameters by this iterative procedure .the forward bayesian estimation of the atomic state applies the recursive update formula ( [ eq : recursive_bayes ] ) for every 50 time bin of the data acquisition , and this determines the time dependent state of the system , shown in the upper panel in fig .[ fig : hmmstateestimate ] . in reality , the hmm method identifies the time dependent population of three different states , but it does not identify which are these hidden atomic state .our estimation of the signal rates associated with the states , however , clearly yield the expected outcome : two hidden states comply with two slightly different low transmission levels and are thus identified with the atom in the hyperfine state , and one state complies with the high transmission of the atomic state . in fig .[ fig : hmmstateestimate ] , we plot the two populations by the light and dark blue shadings , and we observe that most of the time they add to values near zero and unity while , their individual populations fluctuate more .this confirms our intuition that the transmission signal clearly distinguishes high from low transmission , while statistical fluctuations prevent a clear distinction between the two states which are assigned low but quite similar transmission .the measured signal leads to a number of narrow spikes both in the individual and in the total populations .most of the narrow spikes clearly correspond to statistically improbable short intervals spent by the atom in one and the other hyperfine state , and they result from the inability of the forward bayesian update formula to distinguish a short time statistical fluctuation in the transmission signal from an actual change due to the transfer of the atom into a different state .the forward - backward estimate , however , conditions the atomic state on the full measurement record according to eqs.([eq : alpha_beta_estimate],[eq : alpha_update],[eq : beta_update ] ) , and the results are shown in fig .here , indeed , the atomic state populations in the and the states are significantly closer to zero and unity at all times , and most of the spikes in the upper panel of the figure have disappeared .the reason for this is the use of the full detection record , in which , e.g. , a brief dip in the signal is recognized as a statistical fluctuation and thus distinguished from an actual change from high to low transmission .the figure is a clear proof of what we set out to demonstrate : the forward - backward bayesian estimate provides much more decisive predictions for the atomic state than the usual conditioned dynamics .the forward - backward analysis also distinguishes sharper between the two candidate states : the data relevant to the state assigment at any time is approximately doubled when both the past and the future transmission is considered .  naturally , fig .[ fig : hmmstateestimate ] only compares two sets of predictions with each other , and to test theory and confirm its predictions , measurements should be carried out .one simple test of the theory could consist in hiding the data acquired in a single time bin and compare it later with the theoretical predictions , which are governed by .the state probabilities are assigned different values by the forward and the forward - backward recursive expressions ( [ eq : recursive_bayes],[eq : alpha_beta_estimate ] ) , and our improved predictions about the past state of the atom implies an improved ability to guess the photon count in the past time bin .as stated in the beginning of this section , we have iteratively estimated both the atomic state and the transition and transmission probabilities .for the photon transmission , we have applied this recursive scheme to the data from the experiments , assuming that the number of detector clicks in every 50 time bin has an unknown probability distribution , that we update according to eq .( [ eq : emissionreestimation ] ) until the conditioned state dynamics and the parameter re - estimation converged .after about 200 iterations , all rates and signal probabilities were converged to within a tolerance of , and we arrived at the distributions shown in fig . [fig : hmmparameters ] .s obtained after 200 iterations of the re - estimation algorithm described in the text .the photon count distribution with high mean value ( middle , down hatched columns ) is associated with the state , while the hmm algorithm yields two states with low transmission that we ascribe to the hyperfine state ( left and right , shaded and up hatched columns).,scaledwidth=90.0% ]  the histograms shown in fig .[ fig : hmmparameters ] , represent the actual counts in 50 intervals , reported in fig .[ fig : data](b ) , sorted according to the state assignment in fig .[ fig : hmmstateestimate](b ) .the assignment of two different signal probabilities to the state is compatible with the super - poissonian character in the low count signals , and we observe that the two distributions are sufficiently different to be distinguished if the atom resides in one or the other states for long enough .this indeed happens several times during the measurement sequence where the forward - backward state assignment favors one or the other of the states with near unity probability .our analysis also determines the rates of transitions between the three different states in the model .since the baum - welsch expression for the transition rates makes explicit use of the time dependent conditional probability assignments , we see in fig . [fig : hmmstateestimate ] that the forward bayes analysis with its many sharp jumps will strongly overestimate the atomic transition rates , while the forward - backward analysis yields much fewer transitions .indeed , the rates for transitions from the to the states ( and back ) are estimated to 60 ( 80 ) transitions per second , in better agreement with estimates based on the laser - atom parameters made in sec .we have presented a hidden markov model for cavity transmission experiments with an atom jumping between different states .we showed that the standard hmm formalism , taking only the experimental data as input was able to identify states with the expected characteristics and a dynamical jump process between states in agreement with our physical understanding of the dynamics .this lends support to the validity of the state assignment , shown in fig.4 , and it brings promises for use of the same method to analyze a wider range of conditional dynamics experiments .we wish , in particular to highlight the use of the forward - backward estimate which , by incorporating the whole data set rather than only data taken prior to the time at which the state is determined , provided a considerable improvement of the state analysis . due to its significantly sharper predictions , the forward - backward analysis yields also better estimates of the physical process parameters , rates and signal probabilities .it is already a well established procedure in classical hmm theory , and it should become a standard component in the analysis of quantum dynamics .in this paper , the atomic system underwent only incoherent processes , and the conditional dynamics was described by classical probability theory .we have recently shown that a full forward - backward theory can also be established for coherent quantum dynamics of a system subject to repeated or continous measurement . in that theory ,predictions for the outcome of quantum measurements are given by two matrices , which become diagonal with elements equal to the and in eq.(7 ) in the case of incoherent processes .let us , finally , comment on the choice of three states for the hmm theory .this choice was motivated by the observation that only two states would not be compatible with the assumption of markovian dynamics ( the atom in the appeared to have a `` memory '' leading to a slowly varying transmission at the low transmission level ) . with three states ,the model identifies two underlying states , possibly associated with the atomic motion inside the cavity .we have tested the ansatz by further augmenting the model and allowing also four hidden states .that calculation led to a further splitting of the state with transmission characteristics .the pair of states obtained this way , however , show very similar transmission probabilities , and the 4-state model does not appreciably change the assignment of ( total ) time - dependent probabilities of occupation of the and the hyperfine states compared with the 3-state model .we thus conclude that the hyperfine state assignment shown in fig.4 is robust to refinements and changes in the modelling .the authors wish to acknowledge financial support jointly from the ccqed training network of the european commission .aarhus also wishes to thank the villum foundation and the aarhus university research foundation .bonn acknowledges support by the bundesministerium fr forschung und technologie ( bmft , verbund quorep ) , and the integrated project siqs funded by the european commission .m. khudaverdyan , w. alt , i. dotsenko , t. kampschulte , k. lenhard , a. rauschenbeutel , s. reick , k. schrner , a. widera and d. meschede : controlled insertion and retrieval of atoms coupled to a high - finesse optical resonator , new j. phys .10 , 073023 ( 2008 )  m. khudaverdyan , w. alt , t. kampschulte , s. reick , a. thobe , a. widera and d. meschede : quantum jumps and spin dynamics of interacting atoms in a strongly coupled atom - cavity system , phys .103 , 123006 ( 2009 )

Abstract: we analyze the quantum jumps of an atom interacting with a cavity field . the strong atom - field interaction makes the cavity transmission depend on the time dependent atomic state , and we present a hidden markov model description of the atomic state dynamics which is conditioned in a bayesian manner on the detected signal . we suggest that small variations in the observed signal may be due to spatial motion of the atom within the cavity , and we represent the atomic system by a number of hidden states to account for both the small variations and the internal state jump dynamics . in our theory , the atomic state is determined in a bayesian manner from the measurement data , and we present an iterative protocol , which determines both the atomic state and the model parameters . as a new element in the treatment of observed quantum systems , we employ a bayesian approach that conditions the atomic state at time on the data acquired both before and after and we show that the state assignment by this approach is more decisive than the usual conditional quantum states , based on only earlier measurement data .


quantum entanglement ( qe ) is the central feature of quantum mechanics that distinguishes a quantum system from its classical counterpart .it is also the corner stone on which many of the novel applications of quantum mechanics -to quantum computation , quantum information theory , quantum cryptography and quantum teleportation - are based .indeed , it is this promise that has led to a renewal of interest in qe in recent years .the simplest example of qe is afforded by a bipartite system of two spin half ( or pseudo spin half ) particles , where the hilbert space has the minimal dimension four .this system is also called a two qubit system ( 2qs ) in the context of quantum computations . for a 2qs ,there is essentially only one measure of entanglement ; it may be given _e.g. _ , by the von - neumann entropy , or expressed in terms of the sum of bilinears in the eigen values as given by the quantity , or in terms of det . is the reduced density matrix obtained by taking a marginal trace over one of the spin degrees of freedom .states with a vanishing entanglement are separable ( in the strong sense ) : they admit a factorisation . on the other hand , fully entangled states have the maximum correlation , and are collectively designated as bell states .consider now a n - qubit system ( nqs ) , realised as a multipartite system of spin halfs .nqs upto have been prepared experimentally with photons , for example .it is easy to check that for an nqs , , and are not equivalent .further , none of them is exhaustive .a natural question that arises is how one may identify a complete set of entanglement measures for any nqs .there have been several proposals that attempt at answering this question .one note worthy proposal is from sixia yu _et al _ who posit a hierachy of classes of entanglement .the total number grows exponentially with increasing .an alternative approach is to look for physically interesting measures , apart from and , with ( potential ) applicability to quantum computation .this simpler approach would also give direction to the experimentalists to prepare nqs in specified states .in this context , a useful question to ask is what the entanglement property of a 2qs is , when looked upon as a subsystem of an nqs , when ( there would be no new information when ) .there would be such subentanglements , as analogues of two particle correlations . in defining the new measure, we seek to determine the proximity of a given 2qs to the classic bell states mentioned above .wootters has developed a closely related concept called concurrence ( hereafter denoted ) for 2qs , which is required to characterise the so called entanglement of formation ( eof ) .indeed , eof is defined to be the minimum entropy carried by a mixed state , under all possible resolutions of a reduced density matrix of a 2qs ; the set \{ } does not _ necessarily _ form a basis - much less an orthonormal set . , which measures this eof is then defined as follows :  the concurrence for a mixed 2qs is given by where the are the eigenvalues , arranged in decreasing order of magnitude , of the operator +   the so called eof is itself given by  the entanglement of formation is given by +   the above definitions are attractive , and have been used to estimate bipartite entanglement in spin systems .there are also reports that states with nonvanishing have been prepared experimentally .attempts have been made to generalise the definition to higher spin sectors .there is , thus , quite an interest in .the definition of , rather involved in its form , crucially hinges on the concept of eof .however , it is well known that there is no way of inferring how a system is prepared , no matter which property of the density matrix we look at , unless it corresponds to a pure state .thus the physical interpretation of eof is rather dubious , and does not seem to have any operational significance . indeed, suppose the expansion of in eq.1 is replaced by a more general decomposition p_{\{\alpha_i\ } } \vert \psi(\{\alpha_i \})><\psi(\{\alpha_i\ } ) \vert ] being an appropriate integration measure .the parameters could be taken to be the coordinates of the associated compact phase space , for instance .the probabilities have as much physical significance as the discrete variables in eq.1 .but the new expansion would change the criterion for eof , leading to a different expression for concurrence .  with this critique of in mind, we propose an alternative definition of concurrence ( hereafter called ) , with a two fold aim : to attain conceptual and computational simplicity . is required merely to capture the entanglement of a 2qs which is a subsystem of a larger nqs , in a manner that suggests a generalisation to higher spin sectors .we take this up in the next section .as a warm up , and for motivation , consider a 2qs first . let a 2qs be in the state the key step in defining concurrence is in recognising that is a measure of entanglement , equivalent to the standard measures listed above . is much simpler to evaluate , though .our notion of concurrence is inspired by , and is very close in its definition to , .although does _ not _ carry any new information , it does exhibit a symplectic structure which is not apparent in the other two measures .further , as observed by hill and wootters , it can also be written in the form , the inner product of with its conjugate ( equivalently , time reversed ) state which we define thus : the main thrust of the paper is that this form of entanglement for a 2qs needs _ little _ modification in defining concurrence in nqs , .the definition will be given in steps , so as not to have a cluttered notation .    which is obtained by a partial conjugation on the sector ( 12 ) .the definition of concurrences and follow similarly . by definition take values in the range $ ] . also has an in built symplectic structure ; to see this , consider the quadruplet of vectors the expansion is in the ordered basis .the concurrence is now simply written as which naturally generalises the definition of entanglement in a 2qs in a manner appropriate to our purpose . is the transpose of .one can straight away construct quadruplets for by inspection .it is also straight forward to generalise the definition when the parent state is an nqs in a pure state .we write the expression for the sector :   the computational simplicity of is evident , from its very form , if only we demonstrate its viability .a convenient method is to use as a benchmark to test the new definition against .for , the definition of is also motivated by the entanglement measure .figure 1 . gives a comparison of the two definitions , where the states are generated randomly .one sees an overall agreement so long as remains positve , without having to impose the minimality condition by hand . in those regions where has to be declared as vanishing ( when ) in _ definition 1 _ , differs from : indeed, these states are distinguishable , in the sense that each of them has a preferential proximity to one of the bell states which is more clear in fig.2 .the computational advantage is of course evident from the form of . for ,if we start with a nqs , there are such concurrences . even in regions where has no interpretational problem , _ definition 1 _ requires an evaluation of three reduced density matrices , a determination of their partner states , and an evaluation of the eigenvalues of .the necessity of these computations gets obviated in the determination of .we proceed to enlarge the definition of concurrence to higher spin sectors .note that there is no analogue of here , since the criterion for eof which was evolved in is specific to a 2qs .the generalisation in our case is naturally suggested by the form for written above .indeed , let be the state in which an nqs is .we define the conjugate state    the above definition is unfortunately restrictive ; it works only when the subsystem of the nqs is bosonic ; the fermionic states ( odd ) are always orthogonal to their conjugates .apart from this , there is generality with respect to the choice of the qubits since any qubits may be chosen to be in the order given , by a permutation .the number of independent concurrence measures is given by .figure 3 displays the 4q concurrences for states generated from a 6qs . as an illustration , and as an example of the discerning capability of the higher order concurrences which we have defined here ,the shor code will be compared with the steane code , both of which are used in quantum error correction .the shor code is a nine qubit state , written as a direct product of three 3q states each of which have the form , in writing which we have employed the qubit notation : . this entangled state has vanishing concurrences in _ all _ orders .in contrast , the steane code , constructed with seven qubits has a more involved structure ; it is not seperable the way the shor state is . , and is given by . as a manifestation , although all the 2q concurrences vanish , the 4q concurrences survive in the sectors , and all of them attain the maximum allowed value 1 .the 6q concurrences vanish .these features highlight quantitatively the manner in which the two codes differ from each other . at this point, it is good to remember that a complete separability in the strong sense , where the multiqubit state is a direct product of single qubit states , necessarily implies a vanishing concurrence of all orders .the converse is not necessarily true , as exemplified by the state generating the shor code .finally , the question of handling mixed states still remain .not getting into the general case , only the 2q concurrence will be discussed .the general case only involves rewriting the argument with more indices .indeed , given a 2qs in a mixed state , the method is to look upon as having descended from a pure parent state of a higher dimension .significantly , .the determination of parent state is done by simple inspection .it also follows that if has only real components , then it is not difficult to check that although the parent state is not unique , the concurrence determined will be so - reflecting the fact that it is a property of the system , and not of the parent - introduced purely as an intermediate step .this final remark completes the programme undertaken .  in conclusion ,we have redefined in this note the concept of concurrence which has a much simpler form , and is computationally trivial - compared to and its generalisations thereof .it is not based on the notion of eof .while concurrences do not completely characterise a system - which was not our aim - it appears that they do constitute an important subset of entangled states , especially in the two qubit sector .the higher order concurrences also have a significance , as illustrated by the concurrence properties of shor and the steane codes .the true import of entanglements of this kind would manifest with only a more geometric approach , which will be taken up in a subsequent publication .jian - wei pan _et al _ phys .lett . * 86 * , 4435 ( 2001 ) .e. g. _ , n. meyer and n. r. wallach j. math .phys * 43 * , 4273(2002 ) and references therein ; j. eisert and hans j.breigel phys ., * a64 * , 022306 ( 2001 ) .et al _ phys .lett . * 90 * , 080401(2003 ) .w. k. wootters phys .rev . lett.*80 * , 2245(1998 ) .s. hill and w. k. wootters phys .lett . * 78 * , 5022(1997 ) .v. subrahmanyam and a. laxminarayan phys . rev .* a67 * , 052304(2003 ) .nicholas a. peters _ et al _ quant - ph/0308003 ( 2003 ) .d. terhaar , rep .phys . , * xxiv * , 304 ( 1961 ) ; l. d. landau and e. m. lifshitz , quantum mechanics , ( pergammon , 1958 ) , pp 35 - 36 .see also j. von neumann , mathemetical foundations of quantum mechanics ( princeton univesity press , 1955 ) and p. a. m. dirac , proc .soc . , * 25 * , 62 ( 1929 ) clearly, elements of in the ( 12 ) sector are given by which , incidentally gives a three parameter family of three qubit states all of which reduce to the same density matrix in the ( 12 ) sector .the structure is obvious .alexander wong _ et al _ phys* a63 * , 044301(2001 ) .an min wang quant - ph/0208036 ( 2002 ) ; p. rungta , v. buzek , c. m. caves , m. hillery , and g. j. milburn phys . rev . *a64 * , 042315(2001 ) ; heng fan , keiji matsumoto , hiroshi imai j. phys .* 36 * no 14 ( 11 april 2003 ) 4151 - 4158 .michael a. nielsen and isaac i. chuang , quantum computation and quantum information ( cambridge university press , 2000 ) chapter 10 .

Abstract: we refine the notion of concurrence in this paper by a redefinition of the concept . the new definition is simpler , computationally straight forrward , and allows the concurrence to be directly read off from the state . it has all the positive features of the definition given by wootters over and above which it can discriminate between different systems to which the wootters prescription would assign the same value . finally , the definition leads to a natural extension of the notion to multiqubit concurrence , which we illustrate with examples from quantum error correction codes .  0.5 cm  0.5 cm  pacs:03.65.ud , 03.67.-a , 03.67.lx


performance evaluation for digital wireless communication systems in terms of bit error probability ( bep ) and symbol error probability ( sep ) requires a careful characterization of disturbances , such as noise and interference , as well as of the wireless channel impairments due to small - scale and large - scale fading ( see , e.g. , ) .this can result in cumbersome expressions for the error probability ( ep ) which require numerical evaluation .  at a first thought, this fact does not appear a relevant issue from the performance study point of view due to the increasing trend of computational power of computers .on the other hand , these cumbersome solutions do not provide a clear understanding of the performance sensitivity to system parameters , which is of great importance for system design , as well as they are often too complicated for further evaluation or inversion , which is as example needed in order to obtain thresholds in adaptive communication systems ( see , e.g. , ) .moreover , it has to be emphasized that simple parametric approximations and bounds on the performance at lower layers , such as physical layer , can avoid long bit - level simulations in upper level protocols network simulators , provided that they are able to capture the main aspects affecting the performance at lower levels .  mainly , but not only for these reasons , the derivation of approximations and bounds on the exact ep is still of interest in the communication theory community .an example is given by -ary quadrature amplitude modulation ( -qam ) , that is adopted in several standards for wireless communication systems , due to its bandwidth efficiency , and is largely studied in conjunction with adaptive techniques which change modulation parameters to maximize transmission rate for a given target bep in wireless channels . in fact , although early work on -qam dates back to the early sixties , the evaluation of bep for arbitrary is still of current interest .-qam , see . ] to briefly summarize some relevant results for additive white gaussian noise ( awgn ) channel we recall that : parameterized exponential approximations fitting simulative bep are adopted in ; approximations based on signal - space concepts were given in ; an exact method to derive the sep was proposed in ; a recursive algorithm exploiting the relationship among different constellation sizes was developed in ; exact expression of the bep for general was derived in .comparisons among approximations , bounds and the exact solution in fading channels ( with small - scale fading and large - scale fading , i.e. , shadowing ) are given in , where it is shown that , for low and medium values of the signal - to - noise - ratio ( snr ) , approximations depart from exact solutions as the constellation size , , increases .moreover , small differences between exact solution and approximation in awgn channel can become relevant when the instantaneous bep is averaged over small - scale fading .similarly , in systems with multichannel reception , known approximations depart from the exact ep as the diversity order increases .it is well known that bounds carry more information than approximations and also enable system design based on the worst or best case. quite often bounds are tight to the exact ep only for high snrs ( namely asymptotic bounds ) . herewe are interested in deriving simple invertible bounds tight in a given region of interest ( roi ) for the ep ( e.g. , for the bep of uncoded systems typical roi s are  ] ) .  in this paper, we define the concept of locally - valid bounds ( in the following called _ local bounds _ ) , that are tight upper and lower bounds on the ep valid only within a given region of the ep and not for all snrs .this concept is motivated by the fact that there is often a roi for the performance of the system under consideration and it is preferable to have tight bounds in this region instead of bounds valid for all snrs which are far from the exact solution within the roi .the behavior of the ep is important for the definition of local bounds .in fact , the proposed framework is based on its _ log - concavity property_. we recall that a function is log - concave if is concave. stands for natural logarithm . ] in most cases the ep is reported on log - scale and investigated as a function of the signal - to - total disturbance ratio in decibel ( db ) .it is commonly recognized that on this scale the function is concave in several cases of interest . even though it is generally assumed ,as the authors often acknowledge , there is not known formal proof of the long - concavity of the ep ( examples of related issues are : convexity properties in binary detection problems which were analyzed in , and some results for the asymptotic behavior of bounds that were investigated in ) .  in this paperwe introduce the problem of log - concavity for general multidimensional decision regions and we prove this property for a class of signals with constellation on a multidimensional regular grid in the presence of gaussian distributed disturbances , such as thermal noise and interference .in fact , there are several wireless systems and situations in which the interference can be modeled as gaussian distributed ( see , e.g. , ) .after having proved the log - concavity in both awgn and fading plus awgn channels for single and multiple channels reception schemes , as examples of application , it will be shown how to take advantage of this property in order to simplify the derivation of bounds valid for all snrs , as well as to define local upper bound ( lub ) and local lower bound ( llb ) valid in a given roi . moreover , the form of the local bounds and the fact that they are easily invertible also enables the derivation of local bounds for others relevant performance figures such as the ep - based outage probability , which is the probability that the ep averaged over small - scale fading exceed a given tolerable target value , also exploited for the evaluation of the mean spectral efficiency for adaptive modulation techniques .it is finally emphasized that the log - concavity property for the ep can have many others applications , thus its relevance is beyond the applications illustrated in the paper .the rest of the paper is organized as follows : in sec .[ sec : lc ] the log - concavity property of the ep is proved in awgn and in awgn plus fading for systems employing single and multiple channels reception , and in sec .[ sec : lb ] it is applied to define a new class of bounds and local bounds , with a discussion on possible applications .finally , our conclusions are reported in sec .[ sec : conclusions ] .^{k } \bigr ) \nonumber \\ & = & 1-\sum_{k=0}^{d}p_k \ \mu\bigl([-(a/2)e^t,\infty)\bigr)^{d - k }  \mu\bigl([-(a/2)e^t,(a/2)e^t]\bigr)^k . \label{ato2}\end{aligned}\ ] ]in this section the log - concavity property for the ep is discussed first for transmission in awgn channel by highlighting the mathematical structure of the problem in the different applications of digital communications . since we are interested in obtaining general results we base our framework on the origin of detection errors in the presence of gaussian disturbances . within this frameworkwe will then prove the log - concavity property for the class of signals with constellation on a multidimensional regular grid ( e.g. , in the two - dimensional case this class includes the well known -qam constellation ) .finally , we will address the log - concavity property in systems with awgn plus fading channels .consider a set of constellation points on a -dimensional signal space , i.e. let us consider an arbitrary probability distribution on the set and let denote the probability of a point for ( we arbitrarily order points in without loss in generality ) .for each let us define its neighborhood as the set of points closest to in .suppose that we transmit a point with probability in an awgn channel , hence we receive where , and has a standard gaussian distribution on with mean zero and identity covariance matrix .is inversely proportional to the snr . ]we classify each point according to a region that it belongs to , which means that we make an error if or through the change of variable , the total probability of making an error results in stands for probability of event . ] if we denote by a standard gaussian measure on then the distribution of vector is a product measure and , therefore , where we denoted by the region translated by , and by the region .the function is the error probability in the detection of digital signals , either coded or uncoded , as a function of signal - to - noise ratio ( in logarithmic scale ) .proving the log - concavity property of this function is a challenging task .in fact , the log - concavity of single terms in the linear combination of eq .( [ pe ] ) depends on the specific structure of regions and and in any case a possible linear combination of log - concave functions is not necessarily log - concave .  only in few special cases , as example when all the regions have the same measure and a special symmetry around the axis intersecting points and , these issues may be overcome with the help of the prekopa - leindler theorems which states that _ the function , where , is log - concave in if is log - concave in and is a convex subset of .  two examples , one for uncoded system and the other for coded system , will illustrate these simple cases below .in all the other cases an inspection of log - concavity property should be based on the specific properties of the signal set . in the next sec .[ sec : lc - grid - awgn ] we will provide the proof of log - concavity property for the specific case of signals with constellation on a multidimensional regular grid , which covers all the relevant applications based on -qam signaling .  _ example 1 ( -psk ) : _ let us consider a 2-dimensional signal set where the points are regularly placed on a circle .the angular separation between closest points is ( see fig . [fig : region]-left ) .this is the signal set used by -psk signaling .the regions are circular sectors , have the same form and the same measure , and are convex .the same holds for regions , which are concave instead .if we split each region in two parts , and , using the line connecting the points ( 0,0 ) and , all these sub - regions have the same gaussian measure and are convex .since , the log - concavity of a single term has to be checked .the 2-dimensional gaussian measure can be evaluated by using polar coordinates and is the angle with respect to the line orthogonal to boundary . ] in as where describes the boundary of region . since the function is log - concave for  ]] assures that is log - concave with respect to ._ example 2 ( parity check linear block codes and bpsk ) : _ let us consider the -dimensional signal set representing signals obtained by combining a simple parity - check binary block code and binary antipodal modulation .all the points are placed on ( half of ) the vertexes of a -dimensional cube and are equidistant from the origin .each point has closest points or neighbors and each region is bounded by faces in the -dimensional space ( see fig .[ fig : region]-right ) .all regions have the same form and the same measure , and are convex .the same holds for regions , which are concave instead .let us now simplify the example to for better understanding .we have 4 equidistant points placed on 4 vertices of a cube .regions and are bounded by planes intersecting in . if we split each region into three parts , , and , using three half - planes generated by the line connecting the points ( 0,0,0 ) and , all these sub - regions have the same gaussian measure and are convex . since , the log - concavity of a single term has to be checked . by using cylindrical coordinates , is the coordinate along the line orthogonal to boundary , is the angle on the plane orthogonal to -axis . ] in , the 3-dimensional gaussian measure can be evaluated as where with describes the boundary of region , , and is the gaussian q - function . since the function is log - concave is log- concave , whereas is convex . ] for  ] and with given by the following inequality holds   [ lem : non - negativity ] if , for any  ] .] , , making the analysis of not at all straightforward .0.5 truecm   in the above proof of log - concavity of the function the size of the grid was fixed .when we transmit a symbol related to the constellation point with fixed in awgn plus fading channel , the receiver observes where is a random variable ( rv ) representing the channel gain due to fading .this is equivalent to the observation of when the constellation has a random scaling parameter with the same statistics of the fading gain .thus , we show now that when the constellation is scaled with a real random parameter , the average of over is still log - concave in .let us now make the dependence of on explicit , through the change of variable , thus .we obtain the function , which is log - concave as a function of both variables if is log - concave with respect to .hence , what follows is valid for all signal sets with log - concave instantaneous ep function of . to obtain the ep averaged over fading we have to evaluate the expected value of with respect to the rv .  0.5 truecm  [ thm : fading ] if has a log - concave pdf then the average of over , that is , is also log - concave .0.5 truecm  ( _ of theorem [ thm : fading ] _ ) suppose that has a distribution with log - concave density , that is density of the form for some convex function then .] is the average of over since is log - concave in both variables prekopa - leindler inequality implies that is log - concave .theorem [ thm : fading ] shows that if the distribution of has log - concave density then the average over is log - concave .this apply to several cases of interest ( e.g. , single and multiple channel reception in rayleigh , nakagami- , and log - normal fading ) thus leading to log - concave average ep .this can be verified by considering that if the pdf of is given then the pdf of results for nakagami- fading ( having ) the pdf of is given by when . ] from which by we obtain that is log - concave in since is concave and . for log - normal fading ( i.e. , in db is a zero - mean gaussian rv with variance ) the pdf of given by ( ) from which by we obtain that is log - concave in since is positive . for maximal ratio combining of -branches i.i.d .rayleigh fading , the pdf of at the combiner output is given by from which by we obtain that is log - concave in since is positive and is concave .it is also important to remark that the log - concavity property for the ep proved above can have several applications , thus its relevance is beyond what illustrated in the following sections where an application example for bounds and local bounds is provided .in this section it is shown how to take advantage of the log - concavity property.for the derivation of bounds and local bounds which are analytically simple and invertible for further analysis .an example of application will be briefly discussed , addressing local bounds of relevant performance metrics for adaptive -qam systems .however , the application of the log - concavity is not limited to these cases ( e.g. , bounds for multidimensional modulations as well as for -psk can also be derived ) .the main idea to be exploited is that , due to the log - concave behavior proved in sect .[ sec : lc ] , the ep plotted in logarithmic scale versus the signal - to - total disturbance - ratio , , in decibel ( db ) is a concave function ( see , e.g. , fig.[fig : lb ] ) . after having identified the roi ,where the system typically operates , we aim to easily obtain tighter analytically tractable and invertible upper and lower bounds valid in the roi .the roi is defined as the range  ] , with and .let us first consider bounds valid for all snrs , that is for a roi corresponding to snr in the range  ] .let us consider the usual ep , , and the asymptote , both in logarithmic scale as a function of ( in db ) .note that on this scale the ep is concave whereas is linear .it is clear that , since the ep is less than or equal to .furthermore , since and are both decreasing , and the two curves approach at , then .therefore , an upper bound to can be easily defined as the ub on the inverse ep , that is on the value of required to reach a target ep , , is thus given by to define local bounds let us now consider a generic roi $ ] with . by shifting the asymptotic ub to touch the exact solution at extremes of the roi , we define local bounds tighter than previously known bounds , easily invertible , and thus enabling further analysis . at the lower end of the roi , that is for a target ep , we can define (db ) as the difference , in db , between the required for the asymptotic upper bound and the exact solution : then , in linear scale : we now define the lub in the roi as : which is an invertible upper bound within the roi .in fact , for a target ep in the roi , the lub on the required becomes : thus , to define the lub one needs to know the exact required snr at one point , namely , at the lower end of the roi .similarly , one can define the invertible llb , which is a lower bound within the roi , by shifting the ub of referred to .this needs only the knowledge of the required snr for the ep at the upper end of the roi , .the llb is given by : the llb on the required results in   a qualitative example of bounds and local bounds within the roi is reported in fig.[fig : lb ] .at this point it is important to emphasize that , while the lub is , within the roi , a tighter bound than the asymptotic ub and still invertible , the llb obtained by translation of the ub can be worse compared to known lb , but in the other hand , the llb is easily invertible enabling further analysis . can be found using systematic " approaches when exact ep expressions are not available or asymptotic expressions can not be easily deduced from well - know ( but often complicated ) expressions ( see , e.g. , ) . ]  _ remark : _ the log - concavity property opens the way for the definition of other classes of bounds , such as based on tangent in the extremes of the roi or based on saddle - point ( steepest descent method ) .local bounds here proposed have the advantage of being simple and analytically tractable for further analysis .we now discuss a possible application of local bounds on direct and inverse bep to the evaluation of relevant performance metrics for adaptive -qam systems .let us consider as an example , -qam with coherent detection and -branches mrc , whose exact bep averaged over i.i.d .rayleigh fading is given in and its asymptotic upper bound is in the form where depends only on the constellation - size and the diversity order . from one can obtain the upper bound on the inverse bep , which is the bound on the snr required to achieve a target bep equal to , and from this invertible lub and llb through and , respectively .this enables the derivation of llb and lub on the error outage ( eo ) , outage probability based on the ep , which is an appropriate qos measure for digital mobile radio when small - scale fading is superimposed on shadowing ( typically modeled with log - normal distribution ) . in systems with slow adaptive modulation , for a given target bep, , the spectral efficiency ( se ) is a discrete rv with distribution that depends on the snr thresholds and on how they are computed ( i.e. , on the bep expression of the given system configuration ) .let and be the element from the set of possible constellation sizes and corresponding snr threshold ( in db ) , respectively , to achieve a target bep .then , the mean se results in \nonumber \\ & + & \tilde{m}_{j } \left[1 - f_{{\overline{\gamma}}_{\text{db}}}({{\overline{\gamma}}^{\star}}_{\text{db,}j } ) \right ] \,,\end{aligned}\ ] ] where and is the cumulative distribution function ( cdf ) of . by substituting in the required snrs , with lub , , we obtain a llb on the mean se allowing a conservative design of the communication system with different constellation - sizes and diversity orders .in this work we proved an important property of the error probability as a function of signal - to - noise - ratio in db for awgn channel as well as awgn plus fading channels with single and multiple channels reception . in particular , we proved that the error probability is log - concave for a wide class of multidimensional modulation formats which include -qam for two dimensions .this property can have several applications .as example , we exploited log - concavity to derive upper and lower bounds and to define local bounds that are tight in a given region of interest for the error probability .we also discussed an application of local bounds highlighting the possibility of easy computation for the inverse of ep formulas without loosing significant accuracy in the evaluation of figures of merit interesting in wireless communications .however , we believe that the relevance of log - concavity property goes beyond the example provided in the paper and may be exploited for other different purposes .the authors would like to thank the editor and the anonymous reviewers for their suggestions that helped the authors to improve the content of the paper .authors would like also to thank m. chiani , m. win , and o. andrisano for helpful comments and discussions .inequality ( [ eq : inequality_h(h ) ] ) holds for , hence , from now on we will assume that .first , let us prove this inequality in the one - dimensional case . in this case therefore , we need to prove that or which results in the exact equality .let us now consider the case .we write the factor in the second term in ( [ eq : inequality_h(h ) ] ) as and let us think of the left hand side of ( [ eq : inequality_h(h ) ] ) as a homogeneous quadratic form in of the type where is given by \left [ h_m'(h)-\left(1-h\right)h_m''(h)\right ] \nonumber \ , .\label{eq : h_km}\end{aligned}\ ] ]lemma [ lem : inequality_h(h ) ] then follows from lemma [ lem : non - negativity ] .let us start by recalling the following well known identities involving binomial coefficients : as for notation , if is a linear combination of we denote with the coefficient of in . by definition of to finish the proof , it is enough to show that for any  since we have and by plugging , and into we obtain the derivative of the right - hand side with respect to is equal to which is negative if . therefore , the right - hand side attains its maximum for or .the difference for and is and , thus , the maximum is attained at and we need to prove that this is equivalent to using the fact that and , the left hand side of can be rewritten as similarly , the right hand side of is equal to using that by and after some mathematical manipulations we obtain .  finally , comparing expansions for the left and right hand side , that is and , respectively , becomes combining all the coefficients for each power of the left hand side can be written as where notice that the sign of is determined by so that if and if . since , this gives it remains to show that .we observe that results in .then , using , , and we obtain \\  & = & 3d^2 - 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Abstract: a clear understanding the behavior of the error probability ( ep ) as a function of signal - to - noise ratio ( snr ) and other system parameters is fundamental for assessing the design of digital wireless communication systems . we propose an analytical framework based on the log - concavity property of the ep which we prove for a wide family of multidimensional modulation formats in the presence of gaussian disturbances and fading . based on this property , we construct a class of local bounds for the ep that improve known generic bounds in a given region of the snr and are invertible , as well as easily tractable for further analysis . this concept is motivated by the fact that communication systems often operate with performance in a certain region of interest ( roi ) and , thus , it may be advantageous to have tighter bounds within this region instead of generic bounds valid for all snrs . we present a possible application of these local bounds , but their relevance is beyond the example made in this paper .  -2 cm  error statistics , fading channels , local bounds , log - concavity , performance evaluation , probability .