entanglement swapping is a phenomenon where 2 or more qubits which did nt interact in the past are brought into an entangled state .for example , two parties alice and bob share the bell states and such that alice is in possession of qubits 1 and 3 and bob of qubits 2 and 4 . if alice performs a measurement in the bell basis on qubits 1 and 3 she will end up with one of the four bell states .simultaneously bob s qubits 2 and 4 are also brought into a bell state .this measurement can be described by the equation this equation shows that alice s result is completely random , i.e. she will obtain any of the four bell states with equal probability .but the state of the qubits 2 and 4 is completely determined by the result of alice s measurement .this correlation is used in a number of key distribution protocols to share a secret key between alice and bob .since the practical realization of entanglement swapping is rather complex to achieve with today s technology none of these qkd protocols has been implemented yet .also li et al . presented a qkd protocol based on entanglement swapping which not only produces random key bits but also certain key bits .this is achieved by introducing a pauli operation in eq .( [ eq_entswap ] ) as it will be described in detail in section [ sec_origprot ] . as it is shown in this protocolis secure against an intercept / resend attack as well as a collective attack .but we will present an attack strategy in section [ sec_attstrat ] which is based on a 6-qubit state .this state is more complex than the system of the sender and the receiver , but it provides an adversary with full information about the key shared between the two parties .the main idea of this attack is that the adversary eve entangles herself with alice and bob using a state which preserves the correlations of alice s bell state measurement .further this state contains additional information which allows eve to eavesdrop bob s secret result .an attack where eve is entangled with one party was presented by zhang , li and guo on the protocol of cabello but it would nt work for the protocol in since eve would nt stay undetected .we will also make a proposal how to secure the protocol against this new attack strategy .this can be achieved using a hadamard operation and we will describe the new version in detail in section [ sec_modprot ] . in section [ sec_secanal ] we will discuss the security of our new version .alice creates epr pairs , each in a bell state , e.g. . in the followingwe will focus on the simple case , where alice and bob share the states and , which are publicly known . in li et al . start the protocol with preshared entangled states which is a very strong assumption because the qubits have to be shared somehow between the two parties .but in the security analysis in an adversary eve is granted the possibility to interact with the bell states .thus we can conclude that there have to be some qubits in transit between alice and bob .there are two possible ways to share the bell states between the legitimate parties : either one party , e.g. alice , prepares both bell states and sends two qubits to the other party or both parties prepare one bell state and they exchange two qubits .the first scenario is insecure against a simple intercept - resend attack : if alice prepares the bell states and and sends qubits 2 and 4 to bob eve can intercept these qubits and perform a bell state measurement on them .this will bring qubits 1 and 3 also into a bell state which is known to eve .then she forwards qubits 2 and 4 to bob .thus eve has full information about alice s and bob s secret measurement result and is able to eavesdrop the key perfectly .therefore we will assume that alice prepares and bob prepares and that they exchange the qubits 2 and 3 ( cf .picture ( 2 ) in figure [ pic_origprot ] ) .alice randomly chooses one of the four pauli operations , , , or and applies it to qubit 1 ( cf .picture ( 3 ) in figure [ pic_origprot ] ) .alice s operation will generally be described as , where the superscript denotes that it is applied on qubit . as a next step alice performs a bell state measurement on qubits 1 and 3 in her possession ( cf . picture ( 4 ) in figure [ pic_origprot ] ) .  with this measurementthe effect of entanglement swapping arises and entangles bob s qubits 2 and 4 . followingthe equation alice can determine in which state bob s qubits are after her bell state measurement if she applied the operation on qubit 1 .additionally , she can compute also what the state of qubits 2 and 4 would be , if she did nt apply any operation on qubit 1 using eq .( [ eq_entswap ] ) from above .now alice announces to bob that she made a bell state measurement but keeps her result secret .to infere alice s result bob performs a bell state measurement on qubit 2 and 4 in his possession .since he does not know yet which operation alice applied and in which state qubits 1 and 2 have been before the measurement he does not know the exact state of qubits 1 and 3 . with the help of eq .( [ eq_entswap ] ) he can at least deduce in which state qubits 1 and 3 should be if alice did nt apply anything to qubit 1 .this is called alice s imaginary result .  to get the correct state of qubits 1 and 3 bob asks alice about her result but keeps his own result secret .using the information about alice s and his own result bob can deduce which pauli operation alice applied .alice and bob now share information about alice s randomly chosen operation and the result bob obtained , which is also random .they use these two pieces of information to extract a classical raw key .therefore they agree beforehand on a mapping of bell states onto classical 2-bit strings , e.g. as well as of pauli operations onto classical 2-bit strings ,i.e.   if we take the general case with epr pairs alice and bob repeat these steps for all the pairs . in the endthey publicly compare a certain number of bits of the generated raw key to detect an eavesdropper .in this section we discuss an attack strategy where eve is able to perfectly eavesdrop the secret key of alice and bob without being detected .therefore she uses the six qubit state ( cf .picture ( 1 ) in figure [ pic_attack ] ) this state is rather complex and not easy to generate ( cf .appendix ) but it has the special property that it can be written as that means , a measurement on qubits and as well as and yields in the same result as it would be expected from entanglement swapping in eq .( [ eq_entswap ] ) .further the remaining two qubits and are in the same state as the qubits and .eve can use this fact to obtain enough information to deduce the secret measurement results of alice and bob if she somehow manages that the two parties share the state instead of two bell states .that can be achieved easily using entanglement swapping on qubits 2 and 3 in transit together with her six qubits .  when alice sends out qubit 2 and bob qubit 3 eve preforms bell state measurements on both of them with respective qubits from ( cf . picture ( 2 ) in figure [ pic_attack ] ) .a problem is that eve , at first , does nt know which initial state alice prepared .but , as we will show in the next paragraphs , she can overcome this problem fairly easy .eve s first bell state measurement on qubits 2 and is described in eq .( [ eq_cesp_espo_as_es_es1 ] ) .( as already pointed out above , we will use to describe alice s secret pauli operation . ) at this point eve can not clearly say which state the qubits 1 , and are in .all four results of her bell state measurement are equally likely and she has absolutely no information about the operation .nevertheless , eve is able to transform the 6-qubit state in a way that it is always in the state .therefore she performs a operation on qubits and if she gets or does nothing if her result is . to correct the negative signs in case eve obtains a she performs a on qubits and . in case of she first has to apply the operator on qubits and and then the operator on qubits and .the second bell state measurement is performed on qubits 3 and .again , the resulting 6-qubit state is not entirely known but it can be one out of four possible states .we wo nt describe this bell state measurements explicitly but we want to stress that eve is able to bring the 6 qubits in the state .therefore she uses a similar method as described above applying the pauli operations and on qubits , and .now eve sends qubit to alice and qubit to bob and keeps qubits and by herself ( cf .picture ( 4 ) in figure [ pic_attack ] ) .alice and bob , who are not aware of eve s intervention follow the protocol and perform bell state measurements on their respective particles ( cf . pictures ( 5 ) and ( 6 ) in figure [ pic_attack ] ) .if we look at the 6-qubit state , alice s bell state measurement on qubits 1 and changes the state in a way that it leaves the qubit pairs , 4 and , in an entangled state .this can be seen from the alternative description of the state in eq .( [ eq_cesp_espo_as_es_esalice ] ) and picture ( 7 ) in figure [ pic_attack ] ) . alice gets each possible result with equal probability of as she would expect it .further , bob s bell state measurement will end up with certainty in a specific bell state , which is fully dependent on alice s result .due to the special choice of the state the correlation between his and alice s result is preserved as they would expect it if no eavesdropper was present .thus the presence of eve wo nt be detected when alice and bob compare some bits of the raw key .additionally , it leaves the qubits and , which are in eve s possession , in the very same state bob obtains as result ( cf . also picture ( 7 ) in figure [ pic_attack ] ) .this is easy to see if we look at the alternative description in the bell basis of the remaining four qubits , 4 , and after alice s measurement   at this time , eve has full information about bob s secret measurement .that means , eve has as much information as bob and thus can obtain the classical secret raw key in the same way bob does : from her result of the bell state measurement eve can compute alice s imaginary result .moreover , when alice publicly announces her result , eve is able to infere which pauli operation alice has chosen .thus eve knows both parts of the shared secret information .therefore she easily can infere the classical bit string which is used as a raw key by alice and bob and so she is able to obtain the secret key , too .  instead of performing the bell state measurements on the qubit pairs 2 , and 3 , as soon as she receives them , eve has the opportunity to delay her measurement and immediately send qubit to alice ( cf . picture ( 2 ) in figure [ pic_modprotattack2 ] ) . with this strategy eveis able to overcome operations alice applies on qubit 1 but she introduces a much higher error rate as we will describe in section [ sec_secanal ] .another , but rather strong , assumption is that eve is in control of alice s and bob s epr source .that means she is able to distribute the state between her and the two parties without further bell state measurements . in this casealice s measurement on qubits and leaves bob s qubits in a correlated state and eve with the full information about the results , as described above .since alice and bob can prepare their epr source themself this assumption is not very practical .the protocol can be improved by performing a hadamard operation before alice applies her secret pauli operation on qubit 1 ( cf .picture ( 3 ) in figure [ pic_modprot ] ) .the hadamard operation alters the initial state in a way that eve ca nt eavesdrop the secret key without introducing a certain error rate .thus alice and bob can detect her easily as it will be shown in section [ sec_secanal ] .  due to the use of the additional hadamard operationit is not important any more wether alice prepares both bell states or each party prepares a bell state by its own . to stay consistent with the above descriptions of the protocol and the attack we will discuss the scenario where alice prepares the state and bob prepares ( for the simple case ) .alice sends out qubit 2 to bob and he sends qubit 3 to alice .when she receives bob s qubit she randomly applies either the identity operator or the hadamard operator on qubit 1 .this alters the initial state to where is one of the four possible superpositions of bell states , i.e. then alice applies her secret pauli operation on qubit 1 and performs a bell state measurement on qubits 1 and 3 ( cf .picture ( 5 ) in figure [ pic_modprot ] ) .this bell state measurement can be described as   further she publicly tells bob wether she has applied the operator or not .if she did so , bob applies the operator on qubit 2 and otherwise he does nothing ( cf .picture ( 6 ) in figure [ pic_modprot ] ) . as we can see from eq .( [ eq_supbell ] ) a repeated application of the hadamard operator eliminates the superposition and the correlation from the original bell state measurement is preserved . in the end bob performs a bell measurement on qubits 2 and 4 and computes alice s actual and imaginary result as it is described in the original protocol .in the following we will provide a security analysis of the modified protocol .since the application of the hadamard operation is just a minor change of the original protocol , the modified version is also secure against an intercept / resend attack as well as a collective attack , as described in .therefore we will first inspect eve s probability to stay undetected if she follows the same attack strategy just described in section [ sec_attstrat ] .if alice does not apply the hadamard operator eve s attack will be successfull as we already pointed out above .thus we will discuss the case where alice performs the hadamard operation on qubit 1 ( cf .figure [ pic_modprotattack ] ) . due to the fact that the hadamard operation alters the initial state as we have seen in eq .( [ eq_altinitstate ] ) eve s first bell state measurement on qubits 2 and will also change as described in eq .( [ eq_modprot_evebsm ] ) . as it has been shown in section [ sec_attstrat ] eve is able to change the resulting states of her measurements such that she will always end up with a slight variation of her initial state .a similar argument holds for eve s second measurement on qubits 3 and . thus the state after these two measurements is ( cf .picture ( 6 ) in figure [ pic_modprotattack ] ) then eve sends out qubit to alice and qubit to bob .when alice performs her bell state measurement the state can be described as in eq .( [ eq_modprot_alicebsm ] ) ( cf .also picture ( 7 ) in figure [ pic_modprotattack ] ) .  for reasons of simplicity we will assume that alice s result is as it is illustrated in picture ( 8) in figure [ pic_modprotattack ] ( the computations for any other result are analog ) .bob also applies the hadamard operator on qubit in his possession and thus alters the state of qubits and to which can be alternatively written as from this it is easy to see that bob will get the expected result , only with 50% probability . in the other half of the cases he will obtain .alice and bob will detect this error when they check the correlation between their results .that means in of the cases , when alice does nt apply the hadamard operation , eve will stay undetected with certainty . in the other half of the cases when alice performs the hadamard transformation on qubit 1 evewill be detected with probability .these probabilities are valid for the simple case ( ) where alice and bob share only 2 bell states . in general ,alice and bob share entangled qubit pairs and then they will detect eve with the probability which can be brought close to 1 for large .  a possibility for eve to overcome the effect of the hadamard operation is to apply a operation herself on the qubits and before the protocol starts .this changes the initial state to and in the course of the protocol alice and bob s application of the hadamard operator will invert eve s operations . in detail , eq .[ eq_evealtstate ] changes to and further , after bob s application of the hadamard operation which is equal to the state from eq .[ eq_cesp_espo_as_es_esalice ] .thus , the attack reduces to the version described in section [ sec_attstrat ] and eve will obtain full information about the key whenever alice decides to perform the operation . in all the other cases evewill introduce an error equal to eq .[ eq_modprot_error ] due to her hadamard operations , which is easy to see following the arguments in the paragraphs above . at this pointwe want to stress that alice and bob will become suspicious if they find errors only in those cases when alice does ( or does not , respectively ) perform the hadamard operation .thus , a better strategy for eve is to randomly apply the two hadamard operations on qubits and .she will introduce the same error rate but the errors will be equally distributed .another option for eve to invert alice s hadamard operation is to delay her measurement on the qubit pairs 2 , and 3 , until alice publicly announces whether she applied the hadamard operation or not .that means , eve intercepts qubits 2 and 3 in transit between alice and bob and immediately forwards qubit to alice ( cf . picture ( 2 ) in figure [ pic_modprotattack2 ] ) .alice , convinced to have received bob s qubit , applies the hadamard operation and performs the bell state measurement on qubits 1 and ( cf . pictures ( 3)-(5 ) in figure [ pic_modprotattack2 ] ) . due to entanglement swapping qubit 2is now entangled with qubits , , , and , as described in eq .[ eq_secanal_alicebsm ] .  here , eve does not know in which state her qubits 2 , , , , and are because she does not know alice s result .but for our further considerations we assume that alice s result is , i.e. eve is in possession of some state similar to her initial state .next , alice publicly announces that she applied the hadamard operation . having that knowledge eve applies a operation on qubits 2 and in her possession inverting the effect of alice operation and preparing for bob s operation ( cf . picture ( 7 ) in figure [ pic_modprotattack2 ] ) .then she sends qubit to bob and performs a bell state measurement on qubits 3 and which changes the state to .finally , bob inverts the hadamard operation due to alice s public announcement ( cf .picture ( 8) in figure [ pic_modprotattack2 ] ) .similar to eq .[ eq_cesp_espo_as_es_esalice ] bob s bell state measurement on qubits and 4 can be described as it is easy to see that bob s obtains every bell state with equal probability of 25% , i.e. his and alice s results are completely uncorrelated . in case alice does not apply the hadamard operation eve will introduce the same error rate .therefore , the probability eve will be detected is which converges much faster to 1 compared to the first strategy described above ( cf .. [ eq_modprot_error ] ) .we showed that the protocol presented by li et al . is open to an attack strategy where eve entangles herself with both legitimate communication parties . using a special 6-qubit state which preserves the correlated results of entanglement swapping an adversary is able to obtain full information about the key alice and bob share in the end .we also discussed an improvement where alice uses a hadamard transformation on one qubit to secure the protocol .we showed that due to the change of the initial state caused by the hadamard operation an adversary will be detected with a high probability .we would like to thank christian kollmitzer and oliver maurhart for fruitful discussions and comments .further we want to thank our reviewer for showing us an interesting new aspect of this topic .this work was supported by the integrated project secoqc ( contract no .ist-2003 - 506813 ) in the sixth framework program of the european union .eve s initial state from eq .( [ eq_eveinitstate ] ) is rather complex and it might not be producible due to today s physical limitations . nevertheless , we can show that it is , in principle , possible for eve to generate this state using a sequence of hadamard operations and controlled pauli operations .first we assume that eve is in possession of 3 sources emitting bell states .the state of these 6 qubits is eve performs a hadamard operation on qubits and ( indicated by the superscripts and ) , which alters the initial state first to and further to then eve applies a operation on qubit if qubits and are in the bell state .otherwise the identity operator is applied .this action can be described as and it changes the state to next eve performs a operation on qubit and a on qubit if qubits and are in the state .alternatively , she performs a operation on qubit and a operation on qubit if qubits qubits and are in the state .if qubits and are in the state eve applies a operation on both qubits and .these three controlled operation can be stated as using this operator eve is able to bring the 6 qubits into the desired state from eq .( [ eq_eveinitstate ] ) .

Abstract: li et al . presented a protocol [ int . journal of quantum information , vol . 4 , no . 6 ( 2006 ) 899 - 906 ] for quantum key distribution based on entanglement swapping . in this protocol they use random and certain bits to construct a classical key and they claim that this key is secure . in our article we show that the protocol by li et al . is insecure presenting a new type of attack strategy which gives an adversary full information about the key without being detected . this strategy is based on entanglement swapping , too , and manages to preserve the correlation between the measurement results of the legitimate parties . further we present a modified version of the protocol and show that it is secure against this new attack strategy .


the phase of an electromagnetic field can not be measured directly using linear optics and photodetectors . rather than measuring phase directly ,phase measurement schemes rely on measuring quadratures of the field and inferring the phase from these measurements . in a typical experimental implementation, the mode to be measured is passed through a 50/50 beam splitter , in order to combine it with a much stronger local oscillator field .the difference photocurrent from the two output ports of the beam splitter yields a measurement of a particular quadrature .the standard technique for measuring a completely unknown phase is heterodyne detection , where all quadratures are sampled with equal probability .this is achieved by using a local oscillator field with a frequency slightly different from the signal s frequency , so its phase changes linearly with respect to the phase of the signal .more accurate phase measurements can be made using the homodyne technique , where the local oscillator phase is , with the phase of the signal .the problem with this is that it requires initial knowledge of the phase of the signal , and so is not a phase measurement in the strict sense .  to maintain the unbiased nature of heterodyne phase measurements but obtain the increased sensitivity of homodyne measurements ,an adaptive dyne technique can be used . here`` dyne '' detection is used to mean photodetection using a strong local oscillator at a beam splitter .the idea behind adaptive phase measurement schemes is to use the information gained so far during the measurement to estimate the system phase .this is then used to adjust the local oscillator phase to approximate a homodyne measurement as above .  the apparatus for performing these measurements is shown schematically in fig .[ apparatus ] . the signal and a local oscillator with amplitude combined at the beam splitter and the outputs are measured with photodetectors .the outputs from the photodetectors , and , are subtracted and then fed into a digital signal processor that uses these measurements to estimate the phase of the system , and adjusts the phase of the local oscillator via an electro - optic phase modulator .the signal is shown here as from a cavity with a half - silvered mirror , as this is what is considered in the theory in sec .[ method ] .the signal of interest is the difference between the photocurrents at the two ports .we therefore define the signal as here we have used units of time such that the decay constant of the cavity is unity .we divide by a factor of because this is the square root of the mode function for the signal. we can take account of signals with more general mode functions in a similar way .when we take the limit of very large local oscillator amplitude and small time intervals , we find that where is time scaled to the unit interval and is the scaled mean amplitude of the _ system _ ( for which the superscript stands ) .the systematic variation in the coherent amplitude with time due to the mode shape is scaled out .this scaling is explained in more detail in sec .[ method ] .the final term is an infinitesimal wiener increment such that  .it can be shown that just two complex numbers are necessary to encapsulate all of the relevant information in the photocurrent record up to a given time .these are for convenience , we often replace by a third complex number defined in terms of and , generally the best estimate of the phase at time is  .the subscripts are omitted for the final values ( ) .  in adaptive measurement schemes the phase of the local oscillatoris generally taken to be where is the estimated phase of the system at time using the measurement results and .there are a number of possible phase estimates , giving different adaptive schemes .for the mark i scheme , both the running phase estimate and the final phase estimate are taken to be .this is better than heterodyne measurements only if the field is very weak . for the markii adaptive phase measurements the best phase estimate is used at the end of the measurement , but for the intermediate phase estimate is used . this is better than heterodyne measurements for all field strengths .if is generally the best phase estimate , it is apparent from eq .( [ defc ] ) that will only be the best phase estimate if is negligible ( as it is in the case of heterodyne measurements ) . for adaptive phase measurements does not vanish and is generally a much worse phase estimate than .this raises the question of why this relatively poor intermediate phase estimate is used .there are two main reasons for this : ( i ) it is possible to obtain direct analytic results for this case , whereas using a better intermediate phase estimate requires numerical evaluation ; ( ii ) the apparatus required to implement this method is much simpler than that required for a better intermediate phase estimate .even with the relatively poor intermediate phase estimate , the mark ii adaptive scheme introduces a phase variance of just , a good improvement over the heterodyne result of . here is the mean photon number of the field being measured , and the actual measured phase variance is the introduced phase variance plus the intrinsic phase variance .the intrinsic phase variance for a state of mean photon number can be as small as of order  .this is far smaller than the introduced phase variance , so the latter is what limits the accuracy of phase measurements . although the mark ii results are far superior to the standard result of heterodyne detection , it is still possible to improve on the mark ii result , and it is shown in ref . that a theoretical lower limit to the phase variance that is introduced by an arbitrary phase measurement scheme ( based on linear optics and photodetection ) is .  in improving on the mark ii result, the obvious thing to do is to use a better intermediate phase estimate .it turns out that using the best phase estimate actually gives a worse result than the mark ii case , for reasons that we will explain later .the phase estimates that we consider in this paper are therefore intermediate between and the best phase estimate : it is possible to obtain a marked improvement over the mark ii case by using constant values of .we show in sec .[ results ] that a scaling of roughly is possible .one drawback is that the value of required depends on the photon number .we can obtain an even better result if we allow to have a variation in time , and we show in sec .[ results ] that we can obtain phase estimates very close to the theoretical limit if we use this expression does not explicitly depend on the photon number .this method works best if the phase estimates are updated in discrete time steps , and the magnitude of the steps depends weakly on the photon number .a more serious problem with this method is that it tends to produce values of that are too close to 1 .this means that final phase estimates with an error close to occur sufficiently frequently to make a significant contribution to the phase uncertainty .we will show how this problem can be corrected .the paper is structured as follows . in sec .[ theory ] we rederive the ultimate theoretical limit to phase measurements of ref .  .this is necessary to understand how the improved feedback algorithm of eq .( [ ifa ] ) can approach the theoretical limit , as explained in sec .[ impfeed ] . in sec .[ method ] we derive the results necessary for a numerical simulation of this algorithm , and in sec .[ results ] present the results of those simulations .the problem of infrequent results with large errors is identified in sec .[ evaluation ] and a solution proposed and evaluated in sec .[ improved ] .we conclude with a summary and discussion in sec .[ conclude ] .in order to understand how to attain the theoretical limit , we must first understand the reason for the theoretical limit .it can be shown that the probability of obtaining the results , from an arbitrary ( adaptive or nonadaptive ) measurement is ^{2}a\,d^{2}b,\ ] ] where is the state of the mode being measured . here is the pom ( probability operator measure ) for the measurement , and is given by where is what the probability distribution would be if were the vacuum state , and is an unnormalized ket defined by { |{0}\rangle}.\ ] ] this is proportional to a squeezed state : \!\!{|{0}\rangle } \!=\!\left ( 1\!-\!|b|^2 \right)^{-1/4}\!\exp(a{{\alpha}}^*\!/2){|{{{\alpha}},\xi}\rangle},\ ] ] where {|{0}\rangle},\ ] ] and the squeezing parameters are where atanh is the inverse hyperbolic tan function . in terms of thesethe pom is given by where .\ ] ] if the system state is pure , and the probability distribution is given by   for an unbiased measurement scheme the probability distribution for the phase resulting from this equation depends entirely on the inner product between the two states , and not on . to see this , note first that if the measurement is unbiased the vacuum probability distribution will be independent of the phase .second , for the squeezed state , is independent of the phase .this in turn means that is independent of the phase . since  and therefore are independent of the phase .since the probability distribution for the phase depends on the inner product between the two states , the variance in the measured phase will approximately be the sum of the intrinsic phase variance and the phase variance of the squeezed state .the maximum overlap between the states will be when the squeezed state has about the same photon number as the input state .this means that the theoretical limit to the phase variance that is introduced by the measurement is the phase variance of the squeezed state that has the same photon number as the input state and has been optimized for minimum intrinsic phase variance .since the phase variance of a squeezed state optimized for minimum intrinsic phase variance is in the limit of large  , this is also the limit to the introduced phase variance .the photon number of the squeezed state at maximum overlap will be mainly determined by the photon number of the input , but the degree and direction of squeezing ( parametrized by ) will be determined by the multiplying factor .the multiplying factor can be expressed as a function of and , for which we will use the same symbol , even though it is a new function . here is the mean photon number for the state ( and will be close to the photon number of the input state ) , and is with the phase of scaled out .the multiplying factor will tend to be concentrated along a particular line , effectively giving as a function of . in order to obtain the theoretical limit, the measurement scheme must give a multiplying factor that tends to give values of for each that are the same as for optimized squeezed states .we can determine the approximate variation of with in the multiplying factor if we can estimate how it varies for measurements on a coherent state .if we consider measurements on a coherent state with real amplitude , then the maximum overlap with the state will be for .we use without a subscript to indicate the initial coherent amplitude before the measurement .if we are using an adaptive scheme with intermediate phase estimates that are unbiased , it is easy to see that the maximum probability will be for real and therefore also real .these results imply that in turn this gives as since the value of is governed by the multiplying factor , this result for should hold for more general input states .  from ref . the phase variance of a squeezed state is where for real .this is minimized asymptotically as where , for if we use the result obtained for in eq .( [ defzetaan ] ) we find that this result means that in order for the measurement to be optimal , should scale with as for the case of mark ii measurements we have the result that  , which is why these measurements are not optimal .note that if we substitute into the expression ( [ n0result ] ) to find , and substitute that into eq .( [ pvss ] ) , we obtain the correct result for the mark ii introduced phase variance , we have the result that for optimal feedback should decrease with photon number . therefore in order to improve the phase measurement scheme we want one that gives . to see in general how this can be achieved , consider a coherent state with amplitude and determine the ito sde ( stochastic differential equation ) for : dv,\end{aligned}\ ] ] where .in terms of the phase estimate this becomes dv.\ ] ] if we take the expectation value of and simplify we get where .if we use this result the expectation value for the increment in is dv.\ ] ] the first term on its own will give , and in order to get the two sines must have the same sign. this will be the case if the phase estimate is between the actual phase and the phase of .it is for this reason that we consider phase estimates that are intermediate between the best phase estimate and the phase of , i.e. , of the form .\ ] ] in general , smaller values of can be obtained by using smaller values of .this is because tends to be a worse phase estimate , thus making it possible for the sines in eq .( [ sines ] ) to be larger .note that it is far too simplistic to use the best phase estimate ( i.e. , with ) , as we need to adjust in order to make closer to optimal .the easiest input states to use for numerical simulations are coherent states , as they remain coherent with a deterministically decaying amplitude. however , in order to estimate the phase variance that is introduced by the measurement this would be very inefficient , as the phase variance would be dominated by the intrinsic phase variance .it is almost as easy ( and much more efficient ) to perform calculations on squeezed states , as squeezed states remain squeezed states under the stochastic evolution , and only the two squeezing parameters need be kept track of .the best squeezed states to use are those optimized for minimum intrinsic phase variance . for these statesthe total phase variance will be approximately twice the intrinsic phase variance when the measurements are close to optimal .  to determine the sde s for the squeezing parameters, we must first consider the sde for the state .for dyne detection the stochastic evolution of the conditioned state vector is  { |{\psi(t)}\rangle } , { \nonumber}\\ \label{thiseq}\end{aligned}\ ] ] where is the annihilation operator for the mode , is the amplitude of the local oscillator , and is its phase . herethe mode being measured is assumed to come from a cavity with an intensity decay rate equal to unity .the point process has a mean , where the equation given in differs from eq .( [ thiseq ] ) by a trivial phase factor .the form above is given because it is not possible to directly take the limit of large local oscillator amplitude using the form given in .to take the limit of large local oscillator amplitude we approximate the poisson process by a gaussian process where is a gaussian random variable of zero mean and variance .then we find that in the limit of large we have { |{\psi(t)}\rangle},\end{aligned}\ ] ] where we will describe the results for constant . for each mean photon number , was varied to find the value that gave the minimum phase variance .this method does not give results close to the theoretical limit for photon numbers above about 5000 , but the phase variances continue to get smaller as compared to the phase variances for mark ii measurements .this indicates that the results are following a different scaling law , and fitting techniques give the power for the introduced phase variance as . the data and the fitted line along with the heterodyne and mark ii cases and the theoretical limit are shown in fig [ constant ] .these results are a significant improvement over the mark ii case , but are still significantly above the theoretical limit .  in order to improve on this result we must vary during the measurement .the value of that we found to give the best result was the reason for the multiplying factor of is that it is an estimator for .this means that the value of tends to be smaller for larger photon numbers , resulting in smaller values of .the reason for the factor of is that it makes the value of close to zero initially , and very large near the end of the measurement.   plotted as a function of the photon number of the input state .the crosses are the values obtained by stochastic integration and the continuous line is the fitted line . for comparisonwe have also plotted , in order from top to bottom , the variance for heterodyne measurements ( dashed line ) , for mark ii measurements ( dash - dotted line ) , and the theoretical limit ( dotted line).,scaledwidth=45.0% ]  this second factor was found essentially by trial and error , and is thought to be related to the fact that the phase of varies stochastically during the measurement .recall that during the measurement we want the phase estimate to be between the phase of and the phase of .we only have an estimate of the phase of ( the initial phase ) , so if we use a phase estimate that is too close to the actual phase when the phase variance of is large , the phase estimate is likely to be outside the interval between the phase of and the phase of .since the phase variance of increases with time , the value of is increased as well , to prevent this happening .plotted as a function of the photon number of the input state .the phase variance is plotted as a ratio to the theoretical minimum phase variance ( i.e. , twice the intrinsic phase variance).,scaledwidth=45.0% ]  the results for this method are shown in fig .[ ratio ] as a ratio to the theoretical limit .as this shows , the results are very close to the theoretical limit , and even for the largest photon number for which calculations have been performed the phase uncertainty is only about 4% above the theoretical limit . for these calculations the time steps used were approximately where is the theoretical limit to the phase uncertainty . with these time steps the uncertainty due tothe finite step size is approximately 1% .if the integration time step is reduced , while keeping the time interval at which the phase estimates are updated constant , the phase variance converges . if , however , the phase estimates are updated at smaller and smaller time intervals then the phase variance does not converge .for example , the phase uncertainty for measurements on an optimized squeezed state with a photon number of 1577 is if we use the time steps given above . if , however , we use time steps that are 100 times smaller , then the phase variance is , and if the time steps are 1000 times smaller the phase variance is .these results indicate that the phase estimates must be incremented in finite time intervals for this method to give good results , and the size of the time steps that should be used depends on the photon number .the phase variance is not strongly dependent on these time steps , however , and only an order of magnitude estimate of the photon number is required .a problem with determing the phase variance by the method above is that , for highly squeezed states ( that are close to optimized for minimum phase variance ) , a significant contribution to the phase variance is from low probability results around . in obtaining numerical results the actual phase variance for the measurement will tend to be underestimated because the results from around are obtained too rarely for good statistics .it would require an extremely large number of samples to estimate this contribution .however , we can estimate it nonstatistically as follows.  recall that in order to have a measurement that is close to optimum the multiplying factor should give values of for each that are close to optimized for minimum phase uncertainty . to test this for the phase measurement scheme described above, the and were determined from the values of and from the samples .the resulting data along with the line for optimized are plotted in fig .the imaginary part of should be zero for optimum measurements , and is small for these results .therefore in fig .[ zeta ] we have plotted the real part . as can be seen , the vast majority of the data points are below the line , indicating greater squeezing than optimum .this means that if the low probability results around are taken into account the phase variance for these measurements will be above the theoretical limit .   and ( calculated from and ) resulting from measurements on squeezed states of various mean photon numbers .the variation of with for optimum squeezed states is also plotted ( continuous line).,scaledwidth=45.0% ]  first we consider the effect of variations in the modulus of , leaving consideration of error in the phase till later . in order to estimate how far above the theoretical limit the actual phase variance is , we make a quadratic approximation to the expression for the phase variance . from expression for the phase variance of a squeezed state is , for real , taking the derivative with respect to gives taking the second derivative and using the fact that the expression above is zero for minimum phase variance gives this means that for values of close to optimum the increase in the phase variance over the optimum value is the main contribution to the phase uncertainty is , so the increase in the phase uncertainty as a ratio to the minimum phase uncertainty is this estimate indicates that the actual phase variance for the measurement scheme described above can be significantly larger than the intrinsic phase variance .for example , for a mean photon number of about 332 the rms  deviation of from the optimum value is only about 0.16 , but a squeezed state with differing this much from optimum will have a phase variance more than twice the optimum value .this indicates that if the low probability results around are taken into account the introduced phase variance is actually more than twice the theoretical limit .next , we estimate the contribution from error in the phase ( rather than the modulus ) of . for a squeezed state with real intrinsic uncertainty in the zero quadrature is where .since , the intrinsic uncertainty in the phase is if the phase of is small , we can make the approximation clearly the first term in the numerator is just the original phase variance , and the second term is the excess phase variance due to the error in the phase of .therefore the extra phase variance due to error in the phase of is given by   using this estimate on the previous example it can be seen that this is not so much of a problem , with the introduced phase uncertainty being increased by less than 3% by this factor .the problem of the large contribution of the low probability results around can be effectively eliminated in the following way . at each timestep the photon number is estimated from the values of and , and the optimum value of is estimated using the asymptotic formula in .then if ( the real part of ) is too far below the optimum value , rather than using the feedback phase above , we use .\ ] ] using this feedback phase takes directly towards the optimum value . to see this , note that the optimum value of is taking the exponential of the feedback phase given by eq .( [ altfeed ] ) gives , so .the details of exactly when is considered too far below optimum can be varied endlessly , but for the results that will be presented here we use this alternate phase estimate after time and when is the estimated optimum value of and is . using the exponential multiplying factormeans that the alternative feedback is used only toward the end of the measurement . only considering the alternative feedback in the last 10% of the measurement is necessary for the smaller photon numbers , where eq .( [ toohi ] ) is too weak a restriction .another variation from the previous scheme is that , for the larger photon numbers , the values of given by the original expression were reduced .the above correction corrects only for values of that are below optimum , and for the larger photon numbers many of the uncorrected values of tend to be above optimum ( see fig .[ zeta ] ) .the corrections will still work well , however , if we use a dividing factor to bring the uncorrected values below the line .for the second largest photon number tested of around , the best results were obtained when the values of as given by eq .( [ epval ] ) were divided by 1.1 . for the largest mean photon number tested , , the best results were obtained for a dividing factor of 1.2 .the value of that gave the best results with these dividing factors was .for all other mean photon numbers the value of used was .( continuous line ) and the phase of ( dash - dotted line ) . dividing factors of and are used for photon numbers of and , respectively . for these photon numbers , otherwise .the contributions are plotted as a ratio to the theoretical minimum introduced phase uncertainty.,scaledwidth=45.0% ]  the estimated contributions to the phase variance due to error in the magnitude and phase of are plotted in fig .[ contrib ] .as can be seen , the contribution due to error in the magnitude of is very small , around 1.5% for the larger photon numbers tested .the contribution due to the error in the phase of is a bit larger , but it still does not rise above 3% .thus we can see that the introduced phase variance can be made very close to the theoretical limit , within 5% for the largest photon number tested .  with this modified techniquethe phase variance again does not converge as the feedback phase is updated in smaller and smaller time intervals .the phase variance is less dependent on the time step with this technique , however .for example , for a mean photon number of 1577 the total phase variance for measurements on an optimized squeezed state increases by only about 9% as the time steps are reduced by a factor of 1000 .in contrast , the phase variance increases by a factor of 38% for the previous technique .any estimate of an initially unknown optical phase made using standard devices ( linear optical and opto - electronic devices , a local oscillator , and photodetectors ) must have an uncertainty above the intrinsic quantum uncertainty in the phase of the input state .the minimum magnitude of the added phase variance was determined in ref . to scale asymptotically as where is the mean photon number of the input state .previous phase measurement schemes do not approach this theoretical limit . in this paperwe have shown that an adaptive phase measurement scheme not previously considered can attain this theoretical limit .in other words , we have determined what is essentially the best possible phase measurement technique .  in practice ,phase measurements are currently limited by detector inefficiency . for detector efficiency introduced phase variance can not be reduced below  when the mark ii phase variance is less than this there is not likely to be any significant advantage to using a more advanced feedback scheme . for the best photodetectors available today , with around 98% efficiency , the markii phase variance falls below this limit for photon numbers above 1000 . below this photon number the markii phase variance is never more than about 27% above the limits determined using eqs .( [ fund ] ) and ( [ effic ] ) , so only relatively small improvements can be obtained by using a more advanced feedback scheme .nevertheless , the technology is always improving , and there is no fundamental reason why photodetectors can not be built with efficiencies extremely close to 1 .when very efficient photodetectors are developed , the feedback techniques described here have the potential to give great improvements in the accuracy of phase measurements for applications where there is a limitation on the photon number that can be used .the other detrimental factors are relatively minor , although the time delay in the feedback loop will become significant for very short pulses .the primary significance of the result obtained in this paper is theoretical , however , as it represents the culmination of the search for the best optical phase measurement schemes using standard devices .to do any better would require using nonlinear optical devices .for example , it is conceivable that down - converting some portion of the signal field , and then measuring the phase of the down - converted light , could enable the above theoretical limit to be surpassed .this is a question for future work .99 h. m. wiseman , phys .lett . * 75 * , 4587 ( 1995 ) . h. m. wiseman and r. b. killip , phys .a * 56 * , 944 ( 1997 ) .h. m. wiseman and r. b. killip , phys .a * 57 * , 2169 ( 1998 ) .d. berry , h. m. wiseman , and zhong - xi zhang , phys .a * 60 * , 2458 ( 1999 ) . c. w. gardiner , _ handbook of stochastic methods _ ( springer , berlin , 1985 ) .h. m. wiseman , quantum semiclassic . opt . * 8 * , 205 ( 1996 ) .g. s. summy and d. t. pegg , opt* 77 * , 75 ( 1990 ) .d. f. walls and g. j. milburn , _ quantum optics _( springer , berlin , 1994 ) .m. j. collett , phys .scr . * t48 * , 124 ( 1993 ) .m. rigo , f. mota - furtado , and p. f. omahony , j. phys .a * 30 * , 7557 ( 1997 ) .e. polzik , j. carri , and h. j. kimble , phys .. lett . * 68 * , 3020 ( 1992 ) .h. mabuchi ( private communication ) .

Abstract: it is well known that the result of any phase measurement on an optical mode made using linear optics has an introduced uncertainty in addition to the intrinsic quantum phase uncertainty of the state of the mode . the best previously published technique [ h. m. wiseman and r. b. killip , phys . rev .  a * 57 * , 2169 ( 1998 ) ] is an adaptive technique that introduces a phase variance that scales as , where is the mean photon number of the state . this is far above the minimum intrinsic quantum phase variance of the state , which scales as . it has been shown that a lower limit to the phase variance that is introduced scales as . here we introduce an adaptive technique that attains this theoretical lower limit .


luminosity conditions at the lhc pose many unique challenges for potential silicon based track trigger systems .this is true for both level-1 and level-2 trigger applications . among those data formattingis one of major challenges , where hits and clusters from many thousands of silicon modules must first be shared and organized into overlapping eta - phi trigger towers due to finite size of the beam s luminous region along the beam axis and the finite curvature of charged particles in the magnetic field .communication between nodes requires high bandwidth , low latency , and flexible real time data sharing .the first silicon based track trigger at the lhc will be the atlas fast tracker ( ftk ) at level-2  .although atlas ftk is designed for level-1 accept rates up to 100 khz , the data volume per event is quite large since all silicon modules ( more than 86 million channels ) are involved at high luminosity , therefore this is where challenging data formatting issues will be encountered for the first time .we have been developing data formatting solutions for high luminosity lhc conditions and the atlas ftk data formatter system  is the first targeted application .the atlas ftk is organized as a set of parallel processor units within an array of 64 eta - phi trigger towers .the matrix in figure [ fig : input_output ] shows the required data sharing in the data formatting stage first among the atlas ftk eta - phi trigger towers . because the existing silicon tracker and front end electronics were not designed for triggering , the data sharing among trigger towers is quite complex , as shown in the matrix .the data formatter hardware design is dominated by the input and output requirements , and we analyzed the data sharing in early design stage using real beam data with the actual readout cable mapping .the four red boxes in the matrix represent crate boundaries .boards within each crate communicate over the backplane .fiber links are used when boards must communicate across crate boundaries .our analysis shows that the data sharing between trigger towers is highly dependent upon upstream cabling and detector geometry .the ideal data formatter hardware platform should be flexible enough to accommodate future expansion and allow for changes in input cabling and module assignments .one example of such a system is shown in figure [ fig : tower_connection ] , where each trigger tower is represented by a green ball and lines represent data paths .matrix shows the required data sharing in atlas ftk data formatter among 64 eta - phi trigger towers .the four red boxes indicate the assignment of trigger towers to four crates to minimize inter - crate data sharing .the color scale indicates the number of clusters shared between trigger towers per event evaluated with lhc - atlas data taken in 2012 .,scaledwidth=45.0% ]our hardware design process followed a bottom up approach whereby we studied various track trigger architectures .implementations involving full custom backplanes and discrete cables were considered .eventually the full mesh advanced telecommunication computing architecture ( atca )  backplane was found to be a natural fit for the data formatter requirements .the fabric interface of the full mesh backplane enables high speed point - to - point communication between every slot ( figure [ fig : fullmesh ] ) , with no switching or blocking .each line in this diagram represents a channel which consists of up to four bidirectional lanes , which runs at the maximum speed of 40 gb / s .field programmable gate array ( fpga ) devices , which are abundant in local cells , memory , and high speed serial transceivers , were selected for the core processing element on each data formatter board .high speed serializer components in the fpga are directly connected to the full mesh backplane fabric interface channels and also to pluggable fiber transceivers located on a rear transition module ( rtm ) .our first prototype atca board incorporated a pair of fpgas , and thus required a high speed local bus to implement the three types of interconnects described in figure [ fig : internal_connectity ] .the full mesh backplane , when coupled with large fpgas rich in high speed serial transceivers , offers a level flexibility previously unattainable in a conventional shared bus architecture . in effect, the system blurs the distinction between individual fpgas and dramatically increases the processing power and i / o capability .we designed the first prototype , the pulsar iia , to gain experience using the latest fpgas with their high speed transceivers in an atca form factor .the pulsar iia consists of a front board and rear transition module ( rtm ) as shown in figure [ fig : pulsar2a_system_photo ] .in addition we developed an fmc  mezzanine card which includes a smaller fpga , four fiber transceivers , and a socket used for asic testing .a mini - backplane was also developed to facilitate standalone board testing on the bench top .this section describes the design details of each component .further design details also can be accessed on our open web site  .the pulsar iia front board is designed around a pair of xilinx kintex xc7k325 t fpgas .the block diagram of the board is shown in figure [ fig : proto_block ] .each fpga has 16 high speed serial transceivers ( gtx ) which support data rates up to 10 gb / s .of these 16 gtx transceivers 9 connect to the fabric interface , 6 connect to the rtm , and the last gtx transceiver is used for the local bus .each board supports up to four high pin count fmc mezzanine cards , which are connected to the main fpgas using general purpose lvds signal pairs .the fmc form factor has in recent years become a popular choice with xilinx development boards and many third party developers .when the pulsar iia board is used as the ftk data formatter four cluster finder input mezzanine cards will be used .a cortex - m3 microcontroller is used as an intelligent platform management controller ( ipmc ) , which is required on all atca boards .this microcontroller is responsible for the following :  * implementing the ipmi protocol and communicating with the atca shelf manager board(s ) , coordinating hot swap operations , etc . * running telnet and ftp servers which are accessed via the 100base - t base interface ethernet port . * managing firmware images on a micro sdhc flash card . *programming the fpgas via jtag and monitoring over an spi bus . * reading various board temperature and voltage sensors . * communicating with the rtm over an i bus .the pulsar iia rtm conforms to the picmg3.8 standard  and is considered an intelligent - fru device . a small arm microcontroller on the rtm continuously monitors the status of the 8 qsfp+ and 6 sfp+ pluggable transceivers .the microcontroller also communicates with the front board ipmc microcontroller and coordinates hot swap sequencing .each of the pulsar iia fpgas connects to one qsfp+ transceiver and two sfp+ transceivers on the rtm .the pulsar iia supports up to four fmc mezzanine cards with the high pin count ( hpc ) lvds interface .mezzanine cards may contain fpgas , pattern recognition asics , fiber optic transceivers , or any other custom hardware .we developed our fmc test mezzanine card , shown in figure [ fig : test_mezzanine_photo ] , in order to become familiar with the fmc form factor and to study high speed lvds communication between fpgas .the test mezzanine card features a xilinx kintex xc7k160 t fpga , 4 sfp+ transceivers , 128 mb ddr3 , and a 144 pin socket used for testing custom asic chips , primarily aimed at testing pattern recognition associative memory devices  .the fmc connector supports 3.3v and 12v power .an i bus and jtag interface are also provided for slow controls and in - system programming .     the mini backplane ( figure [ fig : mini_backplane ] ) was developed to support stand alone testing of a single pulsar iia board and rtm on the bench top .power is supplied from a 48vdc bench supply .the base interface ethernet port is brought out to an rj45 connector which may then be connected to a pc and used for communicating with the ipmc microcontroller .the mini backplane also loops back each fabric interface port .this loopback feature has proved to be very useful in testing and characterizing the performance of the gtx serial transceivers outside of the crate .figure [ fig : board_level_testing_photo ] shows our bench top test setup , which is used to check basic functionality such power supply operation , ipmc microcontroller communication , fpga programming and gtx loopback tests .the fpga gtx transceivers are driving data in the test - stand system in the fabric interface channels , the rtm channels and the local bus .it turns out that the fpgas achieved a bit error rates ( ber ) less than level in long - running loop back tests .the upper limit of ber is defined to be if there is no single error detected during the test .table [ tab : long_hour_test ] summarized the measured upper limit ..ber upper limit measurement results .the upper limit depends on the length of the long hour test for each channel . [ cols="<,^,^",options="header " , ]   in addition to the ber test , we performed a receiver margin analysis , or eye scan , using the xilinx ibert tool .the kintex-7 gtx transceivers have built - in diagnostic features which provide a mechanism to measure and visualize the receiver eye margin after rx equalizer  .sweeping the receiver sampling point and vertical offset voltage enables the generation of a ber map or statistical eye diagram , where the color represents .figure [ fig : eye_diagram ] shows an example of the measured statistical eye diagram and the open blue region indicates that we can have error free operating points .the size of eye corresponds to the quality of the high speed serial communication after the rx equalization .all gtx transceiver channels have been tested and characterized using the ibert tool .furthermore , the eye scan has been done with a xilinx kintex-7 evaluation kit ( kc705 )  , which provides a `` golden '' reference for comparison studies .comparing the pulsar iia eye diagrams against the xilinx reference design helps us learn more about high speed layout techniques , which will be used in the next iteration of the board ( section [ sec : pulsa2b ] ) .communication over the lvds lines between the fmc mezzanine and the main fpgas has been tested successfully at 400 mhz single data rate ( sdr ) and 200 mhz double data rate ( ddr ) .thirty - four lvds pairs running at this speed yield a bandwidth of 13.6 gb / s , which exceeds the atlas ftk requirements for the data formatter board .a crate - level testing followed the single board - level testing .we set up an atca shelf with a network switch blade in the hub slot and seven pulsar iia front boards as well as rtms in the node slots , as shown in figure [ fig : crate_level_testing_photo ] .    in our system testthe ipmc microcontroller communicates with the switch blade over the base interface network .we first log into the switch and then telnet into each pulsar iia board , where commands are issued to program the fpgas and monitor various board sensors .the default firmware image is setup to drive data over all gtx transceivers to the fabric , rtm and local bus channels .the xilinx ibert tool has also been used in the the crate to test the backplane performance .while our 10 g atca backplane is rated for only 3 gb / s per lane it has proven to work admirably at up to 6.25 gb / s across all slots , as shown in figure [ fig : eye_scan_atca ] .leveraging the experience we gained through designing , building and testing the pulsar iia system we are in the final stages of laying out the next generation board , the pulsar iib ( figure [ fig : pulsariib_block ] and figure [ fig : pulsariib_photo ] ) .the new board design replaces the two kintex k325 t devices with a single large virtex-7 fpga .the gtx transceiver count has increased up to 80 channels , providing a significant bandwidth increase to the rtm , fabric and mezzanine cards .the power regulator sections of the board have been redesigned to handle the increased power required by the virtex-7 fpga .the pulsar iib boards will be used for the atlas ftk data formatter system .we anticipate that the boards will also be used for cms l1 tracking trigger early technical demonstrations  .the pulsar iia is our first atca prototype board and works as designed , as demonstrated by our successful stand - alone and crate - level tests . through this prototype development processwe have gained experience using the latest xilinx fpgas and high speed serializers to communicate over the atca full mesh backplane .furthermore , the pulsar iia boards have successfully interfaced with other atca system components such as commercial switch blades and shelf manager cards .the pulsar iib boards will be used in the atlas ftk data formatter system starting in 2015 .the pulsar iib design forms the basic building block of a high performance scalable architecture , which may find applications beyond tracking triggers , and may serve as a starting point for future level-1 silicon - based tracking trigger r&d for cms , where the full mesh backplane is used most effectively for sophisticated time multiplexing data transfer schemes .our baseline design also works well as a general purpose fpga - based processor board .the design may prove useful in scalable systems where highly flexible , non - blocking , high bandwidth board to board communication is required .ted liu et al ., _ a new concept of vertically integrated pattern recognition associative memory _ , fermilab - conf-11 - 709-e , 2011. published in proceedings of the 2nd international conference on technology and instrumentation in particle physics ( tipp 2011 ) , physics procedia , volume 37 , 2012 , pages 1973 - 1982    j. olsen , t. liu , y. okumura , _ a full mesh atca - based general purpose data processing board : pulsar ii _ , fermilab - conf-13 - 526-cms - ppd , to be published in proceedings of topical workshop on electronics for particle physics 2013 ( twepp-13 ) in the journal of instrumentation ( jinst ) proceedings section

Abstract: high luminosity conditions at the lhc pose many unique challenges for potential silicon based track trigger systems . one of the major challenges is data formatting , where hits from thousands of silicon modules must first be shared and organized into overlapping eta - phi trigger towers . communication between nodes requires high bandwidth , low latency , and flexible real time data sharing , for which a full mesh backplane is a natural solution . a custom advanced telecommunications computing architecture data processing board is designed with the goal of creating a scalable architecture abundant in flexible , non - blocking , high bandwidth board to board communication channels while keeping the design as simple as possible . we have performed the first prototype board testing and our first attempt at designing the prototype system has proven to be successful . leveraging the experience we gained through designing , building and testing the prototype board system we are in the final stages of laying out the next generation board , which will be used in the atlas level-2 fast tracker as data formatter , as well as in the cms level-1 tracking trigger r&d for early technical demonstrations .


recent observational data of the deep surveys ( canada - france redshift survey , lyman - break galaxy surveys , the hubble deep field ) have provided information on the astrophysical properties of galactic objects at different stages of evolution of the universe , making possible to carry out a suitable confrontation of different scenarios of structure formation and galaxy chemical evolution .analytical models of chemical evolution are a very powerful tool to study galaxy formation ( e.g. , chiappini et al . 1997 and references therein ) .however , they are restricted by several hypothesis ( no inflows or outflows , instantaneous recycling , etc . ) , and can not include dynamical and kinematical evolution of the matter according to its nature ( dark matter , baryons ) in consistency with a cosmological model .steinmetz & muller ( 1994 ) have implemented chemical enrichment in a code based on sph for the first time ( see also raiteri , villata & navarro 1996 ) .these works run prepared - cosmological initial conditions where the formation and evolution of one single object is studied .fully - consistent cosmological simulations have the advantage of providing a coherent well - described environment for all objects and a complete record of their formation and evolution .we report here results of a chemical model implemented in a fully cosmological sph code in hierarchical clustering scenarios .we have developed a model to implement metal enrichment in a cosmological context based on the ap3msph code described by tissera et al .( 1997 ) .  a star formation ( sf ) algorithm has been included based on the smidth law .cold and dense gas particles that satisfy the jean s instability criterium are eligible to form stars . each star cluster formed in a given baryonic particle at a sf episodeis given by , where is the integration time - step , the gas density of the particle and the sf efficiency parameter .baryonic particles carry out the information of the different stellar populations formed and the remanent gas mass ( hybrid particles ) .when the gas reservoir of a particle is depleted , it is transformed in a star particle .particles are initially formed by hydrogen and helium in primordial abundances ( , ) .metals are produced and ejected to the interstellar medium at the end of the life of stars .most of the elements are produced by type ii supernovae ( sniie ) , except for the iron that is mainly produced by type i supernovae ( snie ) .each formed can be followed up in time and the number of stars of a given mass estimated by assuming an initial mass function ( imf ) .we adopted a salpeter imf with a lower and upper mass cut - off of and  , respectively .we resort to woosley & weaver ( 1995 ) metal ejecta tables for sneii .the adopted nucleosynthesis prescriptions for type ia sne are taken from thielemann , nomoto & hashimoto ( 1993 ) .type ib sne are assumed to be half the total number of snei and to produce only iron ( per explosion ) .we assume that the life time of binary stars that finish their lives as sni ( ) is gyr .ejected metals are distributed within the neighboring sphere of the particle where a is formed according to the sph technique .hydrogen and helium are proportionally decreased according to the metal mass received by the particle ( see for details mosconi et al . 2000 ) .  given an imf ,the free parameters of our chemical model are , the relative rate of different types of supernovae ( =snrii / snri ) and .the effects of thermal or kinetic energy injection into the ism due to sn explosions are not included in this work .we performed sph simulations consistent with a cold dark matter ( cdm ) spectrum with , , , and .we used particles ( ) in a comoving box of length mpc ( , ) .the simulations performed ( s1 , s2 , s3 , s4 ) share the same initial conditions but have different sf and sn parameters ( see figure 1 for values of these parameters ) .   in hierarchical clustering scenarios , the sf process in glos is affected by mergers and interactions ( tissera 2000 , see also observational evidence in barton et al .1999 ) . in our simulations, we found that sf rate histories can be described as series of starbursts supperposed to a continuous component .for each glo we can also follow the evolution of the metallicity of the stellar component and calculate the age - metallicity relation ( amr ; e.g. rocha - pinto et al .the mean amrs estimated for our glos show the expected trend with high metallicity stars forming at more recent times in agreement with observations .the dispersion found in the simulations indicates the existence of coeval sf sites of different metallicities at a given time .the values and trend of these relations depend on the particular evolutionary history of each glo , and the sf and sn model parameters ( see tissera et al. 2000 for details ) .the chemical abundances in the ism in gas - rich galaxies allows to trace the evolution of individual galaxies , being hii regions and early b - type main sequence objects the most accesible probes of interstellar abundances .fairly steep negative gradients are found : dex kpc within kpc for oxygen ( smartt & rolleston 1997 ) .figure 2 shows abundance gradients of the oxygen for glos 596 and 325 as examples .the curves represent the mass - weighted averages of the metal abundances at each particle position : they clearly show negative abundance gradients .the slopes of the calculated gradients range from to dex kpc , in agreement with observations .the differences in the abundance gradient slopes for a given glo are due to the sf mechanism and sn parameters , and the fact that we are only including gas in hybrid particles .the slopes of glo 325 are less pronounced than those of glo 596 .the different behaviour may be due to the fact that the gas in glo 596 forms a well - defined disk , while glo 325 is a clear spheroid , indicating that their histories of formation and evolution have been very different .  to sum up , in our simulations, glos have different evolutionary history in consistency with a hierarchical clustering scenario that affect their sf rates and chemical evolution .this chemical model can take all these physical processes into account , resulting in a powerful tool to study galaxy formation .barton , e. j. , geller , m. j. , & kenyon , s. j. 1999 , , 195 , 2104 chiappini , c. , matteucci f. , & gratton , r. 1997 , , 477 , 765 gratton , r. , cannetta , e. , matteucci , f. , & sneden , c. 1996 , in formation of the galactic halo , inside out , ed .h. morrison & a. sanajedini , 92 , 307 mosconi , m. b. , tissera , p. b. , lambas , d. g. , & cora , s. a. 2000 , , submitted raiteri , c. m. , villata , m. , & navarro , j. f. 1996 , , 315 , 105 rocha - pinto , h. j. , maciel , w. j. , & flynn , c. 2000 , , submitted ( astro - ph/0001383 ) smartt , s. j. , & rolleston , w. r. 1997 , , 481 , l47 steinmetz , m. , & mller , e. 1994 , , 281 , l97 thielemann , f. k. , nomoto , k. , & hashimoto , m. 1993 , in origin and evolution of the elements , ed .n. prantzos , e. vangoni - flam , & m. cass ( cambridge university press ) , 299 tissera , p. b. , lambas , d. g. , & abadi , m. g. 1997 , , 286 , 384 tissera , p. b. 2000 , , 534 , 636 tissera , p. b. , mosconi , m. b. , cora , s. a. , & lambas , d. g. 2000 , in preparation woosley , s. e. , & weaver , t. a. , 1995 , , 101 , 181

Abstract: we report first results of an implementation of a chemical model in a cosmological code , based on the smoothed particle hydrodynamics ( sph ) technique . we show that chemical sph simulations are a promising tool to provide clues for the understanding of the chemical properties of galaxies in relation to their formation and evolution in a cosmological framework .   # 1_#1 _ # 1_#1 _ =  # 1 1.25 in .125 in .25 in


enormous amount of information produced by everyone in their everyday life is characteristic of our modern society . in recent years , there has been a significant change in the pattern of how information is exchanged .previously information was largely produced by few big entities ( e.g. news sites , entertainment organizations , etc . ) and downloaded by individuals .more recently , individual users have been transformed into hubs themselves and are now a large source of information ; an always - connected entity .this changes the structure of the network and increases traffic demands .additionally new services , in particular video streaming , put an increasing load on today s infrastructure , thus taking it to its limits and forcing us to find a solution to expand the throughput capacity and meet the increasing demand . a large fraction of this load is carried by optical fiber networks , which form the backbone of the internet and other communication networks . as technologyapproaches the physical limits of single mode fiber and the capacity reaches its practical limits of the order of / s , a capacity crunch is imminent  .a candidate technology to avoid such a scenario and to keep up the trend of roughly ten - fold increase in capacity every four years , is space division multiplexing ( sdm )  .sdm utilizes space as the final remaining degree of freedom , by tightly packing several spatial channels into a single fiber .this can be done using multiple cores in multicore fibers ( mcf ) or multiple modes in multimode fibers ( mmf ) . in mmf ,orthogonal modes are coupled due to fiber imperfections or twisting and bending .crosstalk between modes can not be avoided in long - haul transmission . in mcf ,crosstalk levels can be kept sufficiently low to be negligible only for moderate number of cores .however , it may be beneficial to deliberately introduce crosstalk , as it gives rise to supermodes with higher effective mode area .this implies a reduced impact of nonlinear impairments and enables higher channel densities .. such fibers also exhibit a sublinear scaling of mode group dispersion with distance  .mode scrambling is also introduced by certain types of nearly lossless multiplexers , so - called photonic lanterns  .figure [ xtalk ] illustrates crosstalk in optical fiber .  in all these casesthe original signals have to be disentangled at the receiver .due to similarities with the wireless channel , the mimo technique has been considered for this purpose .however , the two channels also exhibit fundamental differences .for example , the optical channel does not suffer from information loss due to the absence of diffuse scattering .backscattering is also negligible . due to the low in - fiber power loss , the channel matrix is still unitary , albeit with complex random elements , and the channel matrix is then a gue ( gaussian unitary ensemble ) with dyson index .previously  , we considered the outage capacity of the fiber optical mimo channel in the limit of full subchannel mixing and in the absence of mode dependent loss ( mdl ) . in this paper , we move towards a more realistic model of the optical mimo channel by establishing its analogy with a model from mesoscopic physics : the chaotic cavity  .this model is amenable to a random matrix theory analysis and can interpolate from zero to strong mixing between subchannels and includes mdl . to showcase its validitywe compute the channel s mutual information via a saddle point analysis .we give the channel description in section ii . in section iii and sectioniv we present the mathematical analysis according to random matrix and replica theory and provide our numerical results . finally , in section v we conclude .we use upper case letters in bold font to denote matrices , , with entries given by .the superscript denotes the transpose operation , denotes the conjugate transpose and represents the -dimensional identity matrix.   we deal with integrals over real matrix elements .we integrate over the elements of an matrix . the corresponding integral measure is denoted by optical fiber may be viewed as a cavity where optical power may enter and exit from both ends .the output power is related to the input power through ( see fig . [ xtalk ] ) with the scattering matrix     the and submatrices correspond to the reflected and transmitted signals , respectively .in contrast to a general cavity , backscattering in the fiber is negligible .thus we have .in addition , because the two fiber ends are not distinguishable , therefore a signal entering from the left and propagating to the right experiences the same phenomena as a signal entering from the right end and propagates to the left .the fiber exhibits random distributed crosstalk between modes or cores .we assume this mixing to be random over different frequency bands , due to strong delay spread .the situation is analogous to that of a chaotic cavity , which randomly mixes the cavity states .the analytic expression of the scattering matrix for a chaotic cavity reads  : here is the channel hamiltonian and is a matrix containing the coupling constants of the fiber to the outside world .the dimension is as there are incoming states from the left and incoming states from the right , while inside the fiber there are states propagating from left to right and states propagating from right to left .so in case of perfect ( lossless ) leads , .the channel hamiltonian is the offdiagonal sub - matrices vanish due to absence of reflection . are hermitian . because we can write for simplicity . then becomes    but , as , for simplification it is andfinally we have to model the mdl we add the loss matrix  just as for we have . for simplicity we can assume that is a diagonal matrix .    or    we wish to compute the capacity of the optical mimo channel .the mutual information is given by the well - known expression where is the signal strength , is the complex channel matrix where and , are the number transmitted and reflected modes . are and dimensional vectors of the transmitted and received signals , respectively . both are assumed to be zero - mean gaussian .the maximum of the mutual information over the input distribution yields the capacity of the channel .the capacity is the maximum error - free information transmission rate when the channel matrix varies through its whole distribution .we assume to be gaussian distributed .the matrix is a sub - matrix of the matrix . to extract we use two diagonal matrices and with . and which yields so that we can write , to incorporate the idea of a fading channel subject to crosstalk in our analysis , we assume that the channel consists of a random part , plus a deterministic part . thus the final equation becomes here is a parameter controlling the randomness . is a diagonal matrix and corresponds to the line - of - sight component inside the fiber while the gaussian distributed matrix describes the crosstalk .in the previous sections we introduces the channel .next , we will showcase the behavior of the mutual information and we will derive our results .we start from the mutual information eq .  of an optical mimo channel . by introducing eq .  to eq .we obtain - \nonumber\\ -\log\mbox{det}\left[(\mathbf{h}_0+\gamma\mathbf{g})^2+\mathbf{\gamma}^2\right]\bigg\rangle \nonumber\\   = \mathbb{e}\left[\mathcal{i}_1 - \mathcal{i}_2\right ] ,   \label{eq : i }   \end{aligned}\ ] ] where \\ \mathcal{i}_2&= & \log\mbox{det}\left[(\mathbf{h}_0+\gamma\mathbf{g})^2+\mathbf{\gamma}^2\right].\end{aligned}\ ] ]  the generating function of , following , is ^{-\nu}\right\rangle = \left\langle e^{-\nu\mathcal{i}}\right\rangle\nonumber\\ & = & 1-\nu \langle\mathcal{i}\rangle+\frac{\nu^2}{2}\langle \mathcal{i}^2 \rangle+\dots\end{aligned}\ ] ]  so we have we are interested in the mean and the variance of the mutual information. therefore we have to calculate the mean value of both and and also their respective variances and their covariance . in order to set ourselves either to - or -space , we set or , respectively .the and limits in the evaluation of can be interchanged by first taking the former and then the latter without changing the final answer. indeed , the two limits of large number of propagating modes and small are not different from each other .higher terms in the expansion lead to higher terms in the expansion.   using identity [ iden:1 ] ( appendix [ ids ] ) , we can write ^{-\nu _ 1}\right\rangle $ ] , as    using identity [ iden:2 ] setting and , we write all - together we obtain the ensemble average over channel realization for an arbitrary function is the last exponential can be written as in order to evaluate the first term of we will use identity [ iden:3 ] and introduce matrices : the evaluation of the quadratic parts of the exponential is more tricky . this time we will introduce matrices have   [ [ saddle - point - analysis ] ] saddle - point analysis + + + + + + + + + + + + + + + + + + + + +  so , bringing everything together , we have where to consider the vicinity near the saddle - point we rewrite as    where are matrices which represent deviations around the saddle point .  that way, we can use the taylor expansion for of as : with .\end{aligned}\ ] ]  continuing the evaluation of the next term of the taylor expansion , since we are looking for saddle point solution , must be stationary with respect to variations in .therefore and the corresponding saddle - point equations are :    and the second order term is where is a hessian matrix the entries of which can be seen in appendix [ hessianmatrix ] .following , the final outcome for the variance is :    for the calculation of the and the corresponding variance , we follow the same method as above but instead of we only have .   again , using identity 1  \mathbf{x}_1 } \nonumber\\ & -&\frac{1}{2}\mathbf{x}_2^\dagger\left [ \gamma ^2 + \left ( \mathbf{h}_0 + \gamma\mathbf{g } \right)^2 \right]\mathbf{x}_2\bigg)\bigg\rangle.\end{aligned}\ ] ]  following the previous method we have   again , using identity [ iden:2 ] and identity [ iden:3 ] we introduce the ever - helpful matrices , matrices and matrices and following the diagonalizing method we end up to matrix which can be seen in appendix [ amatrix ] .  at the saddlepoint the cross - terms etc are equal to zero .so becomes a block - diagonal matrix and the evaluation of the determinant ( identity [ iden:1 ] ) is just the multiplication of these 2 blocks .the rest of the calculations for the computation of the covariance is straightforward and the hessian matrix for the covariance can be seen in appendix [ hessian_cov_matrix ] .so finally the variance is   order to check our analytical results we have numerically computed mutual information and its cumulative distribution function ( cdf ) .the results are shown in figs .[ fig : cdf ] and [ fig : capacity ] .we compare the gaussian distribution evaluated using the analytical calculations in this paper with results obtained by averaging over a large number of random matrix realizations for .remarkably , we find that in both cases , the discrepancy between the different curves is small .this shows that our analytical results are valid even for the practical relevant case of small number of modes or cores .in this paper we have introduced a novel channel model for the optical mimo channel by relating it to a chaotic cavity .this new modeling approach captures the fundamental properties of the optical mimo channel . using tools from random matrix theory and a saddle point analysis ,we have shown that in the limit of large , the distribution of the mutual information approaches a gaussian .numerically we find that this method is valid even for low .this analytic method gives us the means to analyze the statistics of throughput in the fiber optical mimo channel in the presence of arbitrary level of crosstalk and mdl .the introduced channel model is also amenable to extensions , such as dispersion and nonlinear effects in deterministic and random part respectively .the proofs for the next identities , can be found in .[ ids ]  [ iden:1 ] let be a hermitian , positive , definite square matrix and a complex matrix , then   [ iden:2 ] let , be complex matrices , then   [ iden:3 ] let , be arbitrary complex matrices , where here is assumed to be an arbitrary positive integer .then , where .where \times \nonumber\\ & & \left [ ( \gamma^2 + \gamma t_2)(\mathbf{i } + \gamma r_2)+(\mathbf{h}_0 - \gamma p_2)^2\right]\end{aligned}\ ] ] and are scalars . due to simplification reasons we assumed that the loss is not frequency selective .r. ryf , r .- j .essiambre , s. randel , m. a. mestre , c. schmidt , p. winzer _et al . _ ,`` impulse response analysis of coupled - core 3-core fibers , '' in _european conference and exhibition on optical communication_.1em plus 0.5em minus 0.4emoptical society of america , 2012 , pp .t. a. birks , i. gris - snchez , s. yerolatsitis , s. g. leon - saval , and r. r. thomson , `` the photonic lantern , '' _ adv_ , vol . 7 , no . 2 ,[ online ] .available : http://dx.doi.org/10.1364/aop.7.000107    y. v. fyodorov , d. savin , and h. sommers , `` scattering , reflection and impedance of waves in chaotic and disordered systems with absorption , '' _ journal of physics a : mathematical and general _ , vol .38 , no .49 , p. 10731, 2005 .a. l. moustakas , s. h. simon , and a. m. sengupta , `` mimo capacity through correlated channels in the presence of correlated interferers and noise : a ( not so ) large n analysis , '' _ ieee transactions on information theory _49 , no .25452561 , oct 2003 .

Abstract: technology is moving towards space division multiplexing in optical fiber to keep up the trend in rate increase over time and to avoid an imminent capacity crunch . thus , it is of paramount interest to estimate the potential gains of this approach . as more spatial channels are being packed into a single fiber , the increased crosstalk necessitates the use of mimo to guarantee reliable operation . in this paper , we exploit the analogy between an optical fiber and a model from mesoscopic physics a chaotic cavity to obtain a novel channel model for the optical fiber . the model captures both random distributed crosstalk and mode - dependent loss , which are described within the framework of scattering theory . using tools from replica theory and random matrix theory , we derive the capacity of the fiber optical mimo channel model .  optical fiber transmission , mimo , channel capacity , saddle point analysis , random matrix theory , scattering theory