document: the introduction of new quantum mechanical technologies promises to fundamentally alter the way we communicate . 
   quantum key distribution ( qkd ) , for instance , will allow us to communicate in an intrinsically secure way @xcite . 
   but new quantum communication technologies will require a new telecommunications infrastructure , one which is quantum - enabled . 
   that is , this network must be able to properly accommodate the quantum properties that quantum communications inherently rely on 
   . such a quantum network will contain many novel components , such as quantum memories @xcite , quantum repeaters @xcite , or , most generally , quantum channels . 
   these components must each operate in a strictly quantum way . 
   of course , no technology is perfect , and quantum technologies offer a new set of practical challenges . 
   however , as we have learned from qkd , perfectly ideal devices are not a necessity . by shifting our efforts into classical post - processing of data , 
   we can deal with imperfections in quantum technologies . 
   the question then becomes , how much imperfection can be tolerated before a device is no longer operating in a sufficiently quantum way ? 
   we can enforce a minimal quantum requirement on devices by insisting that they do not act as _ measure and prepare _ 
   channels @xcite ( or , in the parlance of qkd , _ intercept and resend _ channels ) , since communication through such channels is equivalent to classical communication . 
   indeed , this type of channel destroys any quantum correlations in bipartite states when one subsystem is sent through it . 
   of course , this is just the minimum requirement . 
   it is also important to quantify the quantum behaviour , as is done in the field of entanglement measures , or in qkd through the secret key rate . for quantum channels , we can ask , _ how well does the channel preserve quantum correlations in bipartite systems , when only one subsystem passes through it ? _ to study this question , we take a state with well - quantified quantum correlations , send one subsystem through the channel , and examine the output . 
   we then compare the quantum correlations detectable in the output with the input correlations . 
   in fact , as we shall see , we can test for these correlations in a so - called ` prepare and measure ' picture , bypassing the need to use actual bipartite states . 
   a strong quantum channel is one which preserves all or nearly all of the quantum correlations . 
   this idea corresponds to what we shall call the _ 
   quantum throughput_. such a measure would allow us to characterize the suitability of devices for quantum communication tasks . 
   the goal of this work is to illustrate that these ideas about device characterization via quantum throughput can be implemented in a meaningful way . 
   although we will make specific choices regarding device types or quantification measures , the basic idea remains quite general , and our scheme can be extended and adapted to other methods as well .    finally , if we picture a future quantum communications network consisting of many components , it should be evident that any device - testing procedure should be as experimentally practical as possible . ideally , we seek a testing scenario where a finite number of test states and a limited set of measurements are sufficient to understand the quantum throughput . 
   the latter requirement is especially important for optical systems , which are perhaps the most natural choice of carrier for quantum information . in these systems , 
   full tomography is not really a practical option because of the dimension of the hilbert space . 
   we have previously examined quantum correlations in optical devices in a qualitative way @xcite ; in the present contribution , we will extend those results to provide a quantitative picture of optical devices . 
   the rest of this paper is organized as follows . in sec . 
   [ sec : quant ] we outline our quantitative device - testing scheme , focusing mainly on optical systems . 
   we show how to estimate important parameters from homodyne measurements on the output , and how to use these estimates to make quantitative statements about the optical device . in sec . 
   [ sec : results ] , we give the results of this quantification procedure for a wide class of optical channels , and examine the strength of our method . sec . 
   [ sec : conclusion ] summarizes the paper , while appendices [ app : overlapbounds]-[app : offdiagbounds ] provide technical details and derivations . 
   the quantum device testing procedure we employ is the same as the one found in @xcite . 
   this protocol is based on the idea that a truly quantum channel should be distinguishable from those channels where the input quantum state is temporarily converted to classical data before a new quantum state is output , a so - called _ measure and prepare _ channel . 
   measure and prepare channels are also called _ entanglement - breaking _ channels , as the two notions are equivalent @xcite . 
   this provides a hint on how to quantify a channel s quantum throughput , namely by sending part of an entangled state through the channel and determining the amount of entanglement that still remains afterwards .    to this end , imagine we have an entangled state of the form @xmath0\ ] ] where system @xmath1 is a qubit and system @xmath2 is an optical mode . 
   we can assume , without loss of generality , that @xmath3 , so that @xmath4 and @xmath5 denote coherent states of opposite phase . 
   this is an entangled state for all values @xmath6 , as can be seen by calculating the entropy of entanglement . 
   keeping subsystem a isolated , an optical channel can be probed using subsystem b of this state , followed by local projective measurements @xmath7 by alice and homodyne measurements @xmath8 by bob . 
   these expectation values , along with the knowledge of alice s reduced density matrix @xmath9 , can be used to determine just how much of the initial state s entanglement is remaining . 
   of course , states like eq . 
   ( [ eq : initialstate ] ) may be difficult to create and therefore not suited for practical device testing . 
   however , notice that alice s reduced density matrix does not depend on what happens in the optical channel , nor on any of bob s measurement results . 
   her expectation values can be completely determined from the initial state @xmath10 . 
   indeed , alice s measurement results can be thought of as classical registers which merely record which mode state was sent through the device . 
   this observation allows us to move from an entanglement - based ( eb ) picture to an equivalent ` prepare and measure ' ( pm ) scenario @xcite , in which alice s measurements are absorbed into the initial state preparation .    in a pm scenario , 
   we retain full knowledge of @xmath9 , in particular the off - diagonal coherence term @xmath11_{01}={\left\langle{\alpha}\right\vert { { \hspace{-0.1 em}}}{{\hspace{-0.1 em}}}\left . 
   { -\alpha}\right\rangle}$ ] . we must insert this additional information _ by hand _ into the set of expectation values for @xmath12 . 
   this distinguishes the expectation values from data which would come from using just a classical mixture of test states @xmath13 . 
   other than this , the procedure is the same as the eb scenario described above . 
   quantum correlations introduced in this way are referred to as ` effective entanglement . ' using this convenient theoretical trick , the testing protocol can be accomplished simply by probing the channel using a source which prepares one of the two conditional states @xmath14 with equal probability . if the measured expectation values , along with the inserted knowledge of @xmath9 , are not compatible with any separable qubit - mode state , then there is ( effective ) entanglement and the channel is certifiably quantum . exploiting the duality between the pm picture and the eb picture , we can quantify the quantum correlations remaining in the output state through a suitable entanglement measure . in turn , this can be compared to the entanglement of the state in eq . 
   ( [ eq : initialstate ] ) to determine the quantum throughput . 
   our main goal in this work is to give an estimate of the amount of effective entanglement observable in an optical system after transmission through an optical channel . 
   our method is based on the following observation : when the two , initially pure , conditional states @xmath15 pass through the channel , they are subject to loss and noise , and evolve in general to mixed states @xmath16 on the infinite - dimensional mode hilbert space ; however , since we work with coherent states , this change in purity comes only from the noise . 
   thus , for any loss value , if the noise introduced by the channel is not too high , then the output states @xmath17 and @xmath18 will still be nearly pure . in this case , most of the information about the state is still contained in a very small subspace of the full infinite dimensional hilbert space . estimating the ` most significant ' subspace on the mode system can therefore be quite useful . 
   this subspace should contain as much information as possible about both conditional states . 
   additionally , we will concentrate on the simplest non - trivial mode subspace , namely one of dimension 2 . writing the conditional output states @xmath17 and @xmath18 in terms of their eigenvectors , in order of descending eigenvalues , we have @xmath19 the most significant subspace is then the one formed using @xmath20 and @xmath21 as basis vectors . 
   three parameters will be important to identify this subspace : @xmath22 , @xmath23 , and @xmath24 . 
   we will estimate these parameters using homodyne detection . specifically , 
   if @xmath25 is the annihilation operator for the mode at bob s detector , then a balanced homodyne detection scheme allows us to measure the field quadratures , here defined as @xmath26 we will use the mean values @xmath27 and the variances @xmath28 of the quadratures from both conditional states to estimate the three subspace parameters . exactly how this is done will be shown in the next part . with these parameters 
   , we can build a @xmath29 density matrix @xmath30 , which corresponds to the projection of the full qubit - mode state @xmath12 onto the two - qubit subspace spanned by the basis @xmath31 . 
   the idea is now to bound the entanglement of the full state @xmath12 using the entanglement of the projection @xmath30 . 
   for this , we need to exploit the strong monotonicity under local operations and classical communication ( locc ) property found in many entanglement measures . 
   specifically , if we perform a complete set of local measurements on a bipartite state @xmath32 , which yields ( perhaps using some classical communication ) the state @xmath33 with probability @xmath34 , then the strong monotonicity property captures the idea that the entanglement should not increase , on average , under this process . in other words , for a given measure @xmath35 , @xmath36 as the name implies , this is a stronger condition than just monotonicity under locc alone . in our case , 
   the measurement consists of projecting the mode system onto the most significant subspace or onto the orthogonal complement . 
   we denote the former projection by @xmath37 and the latter by @xmath38 . 
   then , if we choose an entanglement measure @xmath35 with the strong monotonicity property , we have @xmath39 where @xmath40 and @xmath41 .    for later practicality purposes 
   , we would like to factor the probabilities through the entanglement measure , so that we work directly with unnormalized states . 
   the unnormalized projected state is thus given by @xmath42 
   . we must be careful to choose an entanglement measure which , in addition to being a strong monotone , can be defined for unnormalized states and which permits a positive prefactor to be absorbed into the state . 
   we will focus on the negativity @xcite in this work , for which this choice is justified . 
   we will not attempt to estimate the second term in eq . 
   ( [ eq : strongmono ] ) coming from the orthogonal projection ; we only note that it is non - negative , so that we have the bound @xmath43    in practice , the projected matrix @xmath30 will not be fully characterized and will contain open parameters . on the other hand , some constraints can be imposed on @xmath30 from our knowledge of the initial conditional states and the homodyne measurement results , as well as natural positivity ( @xmath44 ) and trace constraints @xmath45 for unnormalized @xmath30 ) . as a final step , we must determine the minimal entanglement of @xmath30 compatible with all allowed values of these open parameters , subject to the known constraints . 
   we are left with the final relation @xmath46 which will be used as the basis for calculating bounds on @xmath47 . 
   the next two subsections will cover how to estimate @xmath30 and how we minimize the entanglement over all compatible forms of @xmath30 . 
   the first step in our method requires determining the projection @xmath30 of the full state @xmath12 onto the most - significant subspace . 
   for this , we need to estimate the three parameters @xmath22 , @xmath23 and @xmath24 from the decomposition in eq . 
   ( [ eq : eigendecomp ] ) . in ref . 
   @xcite , which considers the related problem of effective entanglement verification using heterodyne measurements ( i.e. full knowledge of the @xmath48 function ) , several useful formulas for estimating these maximal eigenvalues and overlaps are given . 
   these bounds are later refined in @xcite , where they are used to derive secret key rates for continuous variable quantum key distribution . here 
   , we use these bounds as a starting point to build up a good estimate of the projected state for our quantification scheme . 
   we will roughly follow the notation of @xcite in the following . 
   first , since the conditional output states @xmath49 have unit trace , their maximal eigenvalues can be parameterized by @xmath50 , with @xmath51 $ ] . 
   then eqs . ( 66 ) and ( 69 ) from @xcite give directly the following bound : @xmath52 = : u_j.\ ] ] this bound comes up several times , so it is denoted @xmath53 ( @xmath54 ) to make later equations more readable . 
   importantly , the bound can be calculated using only the measured variances of the conditional states . 
   estimating the overlap @xmath24 is more involved . 
   we need to derive bounds on its magnitude based on our available information . 
   again , we begin with bounds provided in refs . 
   @xcite . with suitable relaxations , 
   their bounds can be put into a specific form which will be more desirable for us later , as we would ultimately like to do a convex optimization . 
   the specific details of this relaxation are straightforward , and are outlined in appendix [ app : overlapbounds ] . 
   we will need an additional parameter , @xmath55 , which can be calculated directly using the measured first moments @xmath56 . 
   defining two coherent states with the same means as the conditional states , @xmath57 the new parameter is given through the overlap of these coherent states , @xmath58 with this definition in place , we can give the relaxed bounds @xmath59 where @xmath60 and @xmath61    having these bounds , obtained purely through homodyne measurements , we can now move on to estimating the elements of the projected density matrix @xmath30 . 
   we can already estimate matrix elements of the form @xmath62 using eq . 
   ( [ eq : noisebound ] ) , but to build @xmath30 we also require bounds on the supplementary elements @xmath63 for @xmath64 . to get these , 
   we first expand @xmath17 into its eigenbasis , eq . 
   ( [ eq : eigendecomp ] ) . then , using the fact that @xmath65 $ ] for any normalized vector @xmath66 , we can easily derive the following bounds on the desired matrix element ( see appendix [ app : suppdiagbounds ] for full details ) : @xmath67 analogous bounds can be given for @xmath68 .    finally , we need to estimate some elements of the off - diagonal blocks of @xmath30 , or else there would be no way to differentiate an entangeld state from a classical mixture of the conditional states . to this end , we label the off - diagonal block of the full density matrix @xmath12 by @xmath69 , so that it is naturally split into the form @xmath70 where the diagonal blocks correspond to the two conditional states . in the pm picture , 
   we hold full knowledge of the alice s reduced density matrix @xmath71 where @xmath72 . each element in eq . 
   ( [ eq : rhoa ] ) is the trace of the corresponding element in eq . 
   ( [ eq : rhoblocks ] ) , so we can enforce the condition @xmath73 . using this as our starting point , and with an appropriate basis choice for system b , we can determine the following off - diagonal bounds which can be incorporated into @xmath30 : @xmath74 details on how to arrive at these inequalities can be found in appendix [ app : offdiagbounds ] . 
   we now have sufficient information to construct a useful estimate of the projected state . to summarize 
   , we have the quantities @xmath55 and @xmath75 , which can be calculated from measurements of the first moments and second moments , respectively . 
   we want to determine @xmath30 , which is the projection of @xmath12 from eq . 
   ( [ eq : rhoblocks ] ) onto the subspace spanned by @xmath31 . 
   we have estimated some of the overlaps of @xmath12 with these basis vectors in eqs . 
   ( [ eq : suppdiagbounds1]-[eq : suppdiagbounds2 ] ) and ( [ eq : offdiagbound1]-[eq : offdiagbound2 ] ) . 
   these estimates depend only on the input parameter @xmath72 and on the output state quantities @xmath76 , @xmath77 , and @xmath78 . 
   this last overlap quantity is itself bounded to a region defined by eqs . 
   ( [ eq : overlapbounds]-[eq : bupper ] ) , which depends only on @xmath76 , @xmath77 and @xmath55 . hence , for a fixed input overlap @xmath79 and a fixed set of homodyne measurement results , we have a parameter region which forms a set of constraints on @xmath30 . 
   this region must be searched to find the minimal entanglement compatible with @xmath30 . 
   we will now move on to address the question of how to find the minimal entanglement compatible with our constraints .      as mentioned earlier 
   , we will choose the negativity as the entanglement measure for demonstrating our method . 
   in principle , we would like to find the minimal entanglement using the methods of semidefinite programming . 
   but we must make some simplifications and relaxations which will allow us to do so . 
   first , we exploit the fact that local unitary operations can not change the quantity of entanglement . therefore , without loss of generality , we can assume that the overlap of the maximal eigenstates is real and positive ( since this can be accomplished by a relative change of phase on subsystem b ) . 
   @xmath80 as well , we can perform local phase changes on subsystem a , which allows us to also make the restriction @xmath81    the other off - diagonal element of interest , @xmath82 , is in general still a complex number . 
   the main problem is that eq . 
   ( [ eq : offdiagbound2 ] ) is a _ non - convex _ constraint on @xmath83 . to use this constraint in a semidefinite program 
   , we have to replace it with a set of convex constraints . 
   we accomplish this by denoting the right - hand side of eq . 
   ( [ eq : offdiagbound2 ] ) as @xmath84 and expanding our constraints to the region @xmath85 this new constraint still non - convex , but we can search for the minimum entanglement independently in each of the four quadrants , where the constraints are convex ( see fig . 
   [ fig : convexregions ] ) , and take the minimum over these four searches . 
   the final result will be a lower bound to the minimum entanglement in the region constrained by eq . 
   ( [ eq : offdiagbound2 ] ) . 
   we can extend this idea further , replacing the inscribed square from fig . 
   [ fig : convexregions ] with any other inscribed polygon . with more sides , we can better approximate the non - convex constraint eq . 
   ( [ eq : offdiagbound2 ] ) , but this will also increase the number of convex subregions which must be searched to find the overall minimum . 
   numerical evidence indicates that the minimum entanglement is often , though not always , found at a point outside the circle . 
   we tested with an inscribed octagon and it was not found to alter the final results significantly . 
   the final hurdle comes from the overlap @xmath24 . 
   since the maximal eigenstates will in general have a non - zero overlap ( indeed , for zero overlap , we will not find any entanglement in @xmath30 ) , we must construct an orthogonal basis in order to explicitly write down a matrix representing @xmath30 . 
   doing so introduces matrix elements that are both linear and quadratic in the overlap @xmath24 . 
   if the overlap is used as a parameter in the semidefinite programming , this non - linear dependence becomes problematic . 
   fortunately , it turns out that to find the minimal entanglement we only need to consider the case where the overlap takes the largest allowed value , i.e. @xmath86 . 
   the reason for this is that , for fixed values of @xmath22 , @xmath23 , and @xmath24 , there always exists a cptp map on the b subsystem which preserves the maximal eigenvalues while making the corresponding overlap larger . 
   such a local map can not increase the entanglement , so indeed the minimal entanglement will be found at @xmath86 . 
   this useful result will be shown in detail elsewhere @xcite . 
   in the previous section , we outlined a method for calculating the effective entanglement in optical systems . 
   this began with the observation that we can get bounds just by looking at the most significant two - qubit subsystem . 
   the remainder of sec . 
   [ sec : quant ] provided the necessary tools to allow us to calculate these bounds efficiently as a semidefinite program . 
   now that all the pieces are in place , we can turn to applying our scheme .    to illustrate our quantification method 
   , we use data corresponding to the action of the optical channel on the field quadratures , which we assume to be symmetric for both signal states and for both quadratures . 
   these symmetry assumptions are made solely to aid the graphical representation of our results , and our method does not rely on them . 
   it is also important to note that , beyond the symmetry , we do not make any assumptions about how the channel works . in the absence of experimental data , we merely parameterize the channel s effect on the first quadrature moments by a loss parameter and on the second moments by the excess noise . specifically , if the means of the two conditional output states are denoted by @xmath87 from eq . 
   ( [ eq : cohmean ] ) , then the loss is parameterized through the transmittivity @xmath88 and the symmetric excess noise ( expressed in shot noise units ) by @xmath89 the input states are characterized entirely by the overlap parameter @xmath90 . 
   the quantification program was carried out using the negativity @xcite , @xmath91 this measure has all the properties demanded by our quantification method , but more importantly , the trace norm @xmath92 of a matrix can be computed efficiently as a semidefinite program @xcite . 
   we have normalized the negativity so that a maximally entangled two - qubit state has @xmath93 . 
   our calculations were done in matlab using the yalmip interface @xcite along with the solver sdpt3 @xcite . 
   our main results are shown in fig . 
   [ fig : mainresults ] , where the minimal negativity of @xmath30 compatible with the initial overlap @xmath90 and excess noise @xmath94 is given , for various values of the transmittivity @xmath95 . 
   this quantity gives a lower bound on the negativity of the full state @xmath12 . 
   the entanglement of the initial state , eq . 
   ( [ eq : initialstate ] ) , is also shown as a function of the initial overlap in fig . 
   ( [ fig : test1 ] ) . for figs . 
   ( [ fig : test2]-[fig : test3 ] ) , the modification @xmath96 is made to eq . 
   ( [ eq : initialstate ] ) for these comparisons . 
   the initial entanglement can be compared with the calculated bounds to help understand the quantum throughput of a device . in the limit of zero excess noise and zero loss , 
   our entanglement bound is tight with the initial entanglement . 
   our bounds are quite high for very low noise , but they become lower as the measurement results get more noisy . at some point , a non - trivial entanglement bound can no longer be given , despite the fact that quantum correlations can still be proven for higher noise values ( cf . 
   @xcite ) . as well , for larger loss values , the tolerance for excess noise is lower , and the region where non - trivial bounds can be given becomes smaller . 
   the exact noise value where our bounds become trivial depends on the initial overlap and on the measured loss , but the highest tolerable excess noise is around 5% of the vacuum for @xmath97 . 
   this shrinks to about 3% for a transmittivity of @xmath98 . 
   though the quantification region is small , it is within the limits of current experimental technology @xcite . 
   some entanglement degradation should be expected as the noise is increased , but , as mentioned earlier , entanglement can still be verified ( though not previously quantified ) under the same testing scenario up to much higher noise values than seen here @xcite . 
   thus , our bounds do not provide the full picture . 
   the weakening of the bounds with higher noise is mainly due to the estimation procedure . 
   certain approximations become cruder ( though still valid ) as the noise increases . 
   first , for higher noise , the conditional states become more mixed , spreading out into more of the infinite - dimensional mode hilbert space . 
   this leads to additional information being lost when we truncate down from @xmath12 to @xmath30 . 
   another problem stems from the bounds we use to estimate @xmath30 . 
   higher noise leads to weaker bounds on the maximal eigenvalues from eq . 
   ( [ eq : noisebound ] ) , which weakens all other inequalities .    to examine the effects of these two approximations 
   , we briefly consider a simple channel where the test state , eq . 
   ( [ eq : initialstate ] ) , is mixed at a @xmath99 beam - splitter with a thermalized vacuum . 
   the first moments reduce by a factor of @xmath100 , and the increased variances of the output optical states can be determined from the mean photon number @xmath101 of the thermal state . for @xmath102 , the conditional output states are displaced thermal states . 
   the reason for studying this channel is that we can _ exactly _ determine the maximal eigenvalues @xmath22 , @xmath23 , and the overlap @xmath24 . 
   this allows us to study our approximations independently , since we decouple the effects of the two - qubit projection from the homodyne parameter estimation ( in practice , of course , our quantification scheme must use both ) . in fig . 
   ( [ fig : comparison ] ) we show the result of the quantification scheme , when this extra information is included . 
   we see that the tolerable excess noise is @xmath103 of the vacuum , more than three times what it would be if we had to estimate the eigenvalues and overlap using homodyne results ( cf 
   ( [ fig : test3 ] ) ) . also included in fig . 
   ( [ fig : comparison ] ) is an entanglement verification curve , obtained using the methods of @xcite . 
   any points with lower noise than this verification curve must come from entangled states . 
   the two - qubit projection is tight to the entanglement verification curve for low overlaps . for higher values , 
   the projection becomes weaker , only working to about half the noise value that the entanglement verification curve reaches . 
   ideally , we want to be able to calculate non - trivial values for the entanglement wherever it be verified . 
   this would give us a true quantitative complement to existing entanglement verification methods . 
   one obvious extension to our method would be to truncate the mode subspace using the two largest eigenstates from each conditional state , or even more . in theory 
   , this would strictly improve the estimates . 
   however , in practice , this will increase the complexity of the quantification calculation , since some simplifying assumptions ( i.e. certain overlaps are real ) may no longer be valid . as well , the number of additional minimizations we have to do , as in our non - convex relaxation of eq . 
   ( [ eq : offdiagbound2 ] ) , increases fourfold with each added dimension . 
   another approach might therefore be necessary to overcome this problem . 
   nevertheless , the quantification scheme outlined here is a useful method for characterizing the degree of quantumness of optical channels , especially when these channels introduce low noise . 
   we have outlined a method for quantifying the effective entanglement in qubit - mode systems using only homodyne measurement results and knowledge of the initial preparation . 
   this quantification method works particularly well if the mode subsystem exhibits low noise . by combining this quantification scheme with a device testing scenario which uses two nonorthogonal test states , 
   one can examine how strongly an optical device or experiment is operating in the quantum domain . 
   our scheme provides a useful tool for understanding the quantum nature of optical devices , especially the question of how well they preserve quantum correlations . 
   in this appendix , we derive the bounds from eqs . 
   ( [ eq : blower]-[eq : bupper ] ) for the absolute value of the overlap of the maximal eigenstates , @xmath104 . from @xcite , we have the following :    _ overlap bounds . 
   _ let the largest eigenvalue of @xmath105 be parameterized by @xmath106 and let the fidelity between the conditional states and the coherent states @xmath107 from eq . 
   ( [ eq : cohmean ] ) be given by @xmath108 and let @xmath58 then the following holds : @xmath109 with @xmath110 and @xmath111    since we can not calculate @xmath112 or @xmath113 in practice , we now modify these bounds from the above form found in @xcite to one involving only the parameters @xmath55 ( calculated from first moments ) and the @xmath53 ( calculated from second moments ) . to do this 
   , we make use only of the obvious inequality @xmath114 from this , we can easily derive the following auxiliary inequalities : @xmath115 it is important to note that the second and third inequalities only hold so long as @xmath116 . for symmetric noise 
   , the value @xmath117 corresponds to @xmath118 , almost twice the vacuum variance . 
   this value is far outside the region where our method gives non - trivial bounds , so it is not an issue . substituting the inequalities ( [ eq : aux1]-[eq : aux3 ] ) into eqs . 
   ( [ eq : oldoverlaplowerbound ] ) and ( [ eq : oldoverlapupperbound ] ) , we arrive at the bounds given in eqs . 
   ( [ eq : blower]-[eq : bupper ] ) . 
   here we aim to bound the quantities @xmath63 for @xmath64 , as found in eqs . 
   ( [ eq : suppdiagbounds1]-[eq : suppdiagbounds2 ] ) . 
   an eigenbasis expansion of @xmath17 leads to @xmath119 a lower bound can be derived in a similar way : @xmath120 the bounds for @xmath121 follow by interchanging indices . 
   this appendix outlines the derivation of the off - diagonal bounds from eqs . 
   ( [ eq : offdiagbound1]-[eq : offdiagbound2 ] ) . 
   we completely know @xmath9 , which constrains that we must have @xmath122 . 
   first , we consider the full density matrix @xmath12 in the basis defined by @xmath123 for system @xmath1 and the eigenbasis of @xmath17 , @xmath124 , for system b. we can still write this in the block form of eq . 
   ( [ eq : rhoblocks ] ) , where we denote the diagonal elements of the block @xmath18 by @xmath125 and the diagonal elements of the block @xmath69 by @xmath126 ( the diagonal elements of @xmath17 are its eigenvalues ) . using the triangle inequality 
   , we have @xmath127 from positivity of @xmath12 , we find @xmath128 and from the cauchy - schwarz inequality , @xmath129 the first sum is just @xmath130 and the second is @xmath131 . now , using the bounds from appendix [ app : suppdiagbounds ] , we get @xmath132 which we can substitute above to obtain @xmath133 replacing @xmath134 with @xmath135 , we are led to the off - diagonal bound @xmath136 by applying the same arguments using the eigenbasis of @xmath18 , we can arrive at an analogous bound for @xmath137 .
summary: quantum communication relies on optical implementations of channels , memories and repeaters . in the absence of perfect devices , 
   a minimum requirement on real - world devices is that they preserve quantum correlations , meaning that they have some thoughput of a quantum mechanical nature . 
   previous work has verified throughput in optical devices while using minimal resources . 
   we extend this approach to the quantitative regime . 
   our method is illustrated in a setting where the input consists of two coherent states while the output is measured by two homodyne measurement settings .