quantum entanglement is nowadays considered a central resource in the field of quantum information and computation .it is therefore crucial to be able to determine whether a quantum state is separable or entangled , which is provably a hard decision problem when it comes to mixed states . in the context of continuous - variable systems , such as bosonic modes or collective atomic spins ( in the limit of large ensembles ) ,a necessary criterion for the separability of any two - mode state has been derived by duan __ and simon , which even turns into a necessary and sufficient criterion in the special case of gaussian states .this criterion results from translating to continuous - variable ( infinite - dimensional ) systems the positive partial transpose ( ppt ) condition , which had been established for finite - dimensional discrete systems . following the notation of duan _ et al . _ , the criterion expresses that if a two - mode state is separable , then its so - called einstein - podolsky - rosen ( epr ) variance complies with the inequality for any real ( nonzero ) , where the operators are functions of the quadratures components and of modes 1 and 2 .thus , if a state violates inequality ( [ delta ] ) for at least one value of , it is entangled   .simon s version of this criterion is based on expressing the partial transposition as a mirror reflection operation , which can be viewed as time reversal on mode 2 .the ppt criterion expresses that following such a reflection , any separable state remains physical ( its density operator is positive semi - definite ) .conversely , an entangled state will be detected if the corresponding reflected state is non - physical ( its density operator admits a negative eigenvalue ) .remarkably , this separability condition becomes necessary _ and _ sufficient in the case of gaussian states and can even be extended to ( +)-mode gaussian states .however , for any other state , this criterion may very often leave entanglement undetected .earlier work has aimed at improving the duan - simon separability criterion for arbitrary states .in particular , walborn _ reported on a separability condition using shannon entropy , which was later extended by huang .shchukin and vogel also derived a hierarchy of inequalities involving higher - order moments of the quadrature components ( the previous criterion only depends on the first- and second - order moments ) . in the present work, we investigate an improvement of the duan - simon separability criterion that enables a stronger entanglement detection for non - gaussian two - mode states by taking into account an additional parameter , namely , the degree of gaussianity .it is natural to expect that a stronger criterion can be obtained with more information on the state , but the additional parameter should be chosen carefully .for example , the purity of the state is not a good candidate , since the duan - simon criterion is necessary and sufficient for every gaussian state of arbitrary purity .as we shall show , the degree of gaussianity is indeed a good choice .  in sec .ii we start by introducing the parameter and providing some of its useful properties ( more details are given in appendix a ) . in sec .iii we introduce a criterion for a better detection of entangled states employing this degree of gaussianity , thus improving the duan - simon criterion for non - gaussian states . in sec .iv , we provide explicit examples of entangled non - gaussian states that are left unnoticed by the duan - simon criterion , demonstrating the advantage of our criterion ( the analytical computation of parameter is detailed in appendix b ) .the examples are produced from non - gaussian states belonging both to the set of classical states ( with positive glauber p - function ) and to the set of genuinely quantum states ( fock states ) , reflecting the general applicability of our method .finally , we conclude and discuss possible extensions of our work in sec .gaussian states play a prominent role in continuous - variable quantum information .however , several protocols necessarily require the use of non - gaussian states , such as entanglement distillation or quantum error correction . with the increasing importance of non - gaussian states , the question of measuringthe gaussian character of a state has naturally arisen .several gaussianity measures have been introduced ( see , e.g. , ) , but we find it more convenient here to use the degree of gaussianity introduced in .consider a two - mode state .its first- and second - order moments , denoted , respectively , by and , are expressed from the vector of quadrature components .the elements of the coherent vector are given by , while the elements of the covariance matrix are defined as .we assume , with no loss of generality , that all states considered in the following have vanishing coherent vectors ( ) since first - order moments are irrelevant as far as entanglement detection is concerned .the degree of gaussianity of state is defined as where is the gaussian state characterized by the covariance matrix of state .one may easily verify that for gaussian states ( note that the converse is not true , as shown in appendix [ annexea ] ) .in addition , the degree of gaussianity is invariant under gaussian unitary transformations , transposition , and partial transposition .the proofs are provided in appendix [ annexea ] .these properties are essential for the derivations in this work .we will also exploit the fact that the knowledge of gives a tighter bound in the uncertainty relations , which in turn translates into a stronger condition for detecting entanglement .let us investigate the separability of an arbitrary two - mode state . as mentioned earlier , the ppt criterion consists in verifying the physicality of the partially transposed state ( which must hold for any separable two - mode state ) .then , an entangled state will be detected if is not physical ( it exhibits a negative eigenvalue ) . applying a partial transposition ( acting on the second mode ) in state space is equivalent to a mirror reflection in phase space .thus , in order to detect entanglement , we need to check the physicality of in phase space , which can be achieved based on the symplectic eigenvalues of its covariance matrix .suppose that the two - mode state has a covariance matrix . according to the williamson theorem, there always exists a unitary transformation mapping the state onto such that the associated symplectic transformation maps onto where is the largest ( smallest ) symplectic eigenvalue of and is the identity matrix . note that if is a gaussian state , then is a tensor product of two thermal states ; otherwise is a two - mode non - gaussian state that has the same covariance matrix .the uncertainty principle implies that the inequality must be respected for any physical state .applying this condition to the partially transposed state , we understand that the entanglement of states is detected whenever the smallest symplectic eigenvalue of is strictly smaller than 1 , which is the core of the duan - simon criterion .let us now introduce our improved criterion . in order to detect the entanglement of state , we apply a partial transposition on the second mode ( which may lead to an unphysical state ) followed by a symplectic transformation , which gives access to the symplectic eigenvalues of the partially transposed state .this is the entanglement analyzing box shown in fig .[ box ] .a key observation is that the inequality boils down to expressing the uncertainty relation for the two modes making at the output of the entanglement analyzing box .indeed , we have and , where ( ) refers to the covariance matrix of the first ( second ) mode of . furthermore , a tighter lower bound on the uncertainty of a mode can be obtained if the degree of gaussianity of this mode is known [ we use definition ( [ gauss ] ) for a single mode ] . combining these elements, we can detect the non - physicality of whenever the lowest symplectic eigenvalue lies under the lower bound corresponding to the degree of gaussianity of , as shown in fig .[ graphkat ] .this lower bound is equal to for , but is strictly larger than for non - gaussian states with .plot of , the threshold ( minimum allowed ) value for , as a function of the degree of gaussianity ( details are given in ) .all physical states lie on or above this curve .note that for , the curve exhibits some discontinuities ( see main text ) . ]hence , we obtain an improved separability criterion that works as follows . after applying the entanglement analyzing box of fig .[ box ] to state , we detect its entanglement if the symplectic eigenvalue of the reduced state is smaller than a bound , which is a function of the degree of gaussianity of . in other words , where is the threshold given by the curve in fig .[ graphkat ] . if , the curve is given by .if , the parametric equations of the curve are given by where and .the latter curve consists of consecutive segments , each corresponding to a binary mixture of nearest - neighbor fock states and . in the examples that we will discuss in sec .iv , the degree of gaussianity always lies in the segment where , which corresponds to   in order to exploit condition ( [ improved - criterion ] ) , the last step is thus to compute the degree of gaussianity as given by eq .( [ gauss ] ) .the analytical computation of is not trivial for an arbitrary two - mode state ( although we give an explicit method for some class of states in appendix b ) , but at least a numerical computation is always feasible based on the wigner function .first , we remark that the denominator of is simply equal to since it corresponds to the purity of a gaussian state [ see eq .( [ denom ] ) in appendix [ annexeb ] ] . to express the numerator of , we use the wigner function of the second mode at the output of the entanglement analyzing box .starting from , namely , the wigner function of the initial two - mode state , we find after partial transposition and then after symplectic transformation , corresponding to a change of variable .finally , we integrate over and to have the wigner function of the second mode , which gives where is the wigner function of the gaussian state with covariance matrix .examples of non - gaussian entangled states generated from fock ( ) or phase - diffused coherent ( pdc ) ( + ) states , which are detected by our improved criterion , but not otherwise . in general , all entangled states detected by the duan - simon criterion lie in the red ( lower ) zone , while the entangled states that are detected by our criterion but remain undetected by the duan - simon criterion lie in the blue ( intermediate ) zone .the white ( upper ) zone contains either separable or undetected entangled states .the curve separating the blue and white zones corresponds to the lower bound on for a fixed degree of gaussianity ( see fig .[ graphkat ] ) . ]  figure [ result ] enables us to visualize how entanglement detection is improved by our method .three distinct zones are represented , delimited by the curve of fig .[ graphkat ] and by the constant line .if a state lies in the red ( lower ) zone , it is an entangled state that is detected by the duan - simon criterion ( ) , hence it is uninteresting for our purposes here .if it lies in the white ( upper ) zone , no conclusion can be made because the partially transposed state is physical .however , interestingly , if it lies in the blue ( intermediate ) zone , we detect entanglement which was otherwise unnoticed .we remark that , since partial transposition and symplectic transformation conserve the gaussian character of a state ( see appendix [ annexea ] ) , if is a gaussian state , then and the reduced states and are also gaussian .then , and we recover the ( necessary and sufficient ) duan - simon separability criterion for gaussian states , as expected .let us also mention that our criterion does not improve entanglement detection when the covariance matrix of is diagonal , since the partially transposed state then necessarily remains physical .for example , the `` noon '' states of the form have a diagonal covariance matrix for .thus , even though those states are always entangled , we can not do any better than the duan - simon criterion , and entanglement is undetected by our criterion .however , there exist many other interesting cases where our method is useful , as shown in the next section .in this section , we apply our criterion to two types of non - gaussian states .those examples have in common that entanglement is not detected on the sole basis of the covariance matrix ( using the duan - simon criterion ) , but is detected exploiting the degree of gaussianity .the first example uses non - gaussian states as generated by the circuit of fig .[ figure ] .the preparation of the states works as follows .initially , we have a fock - diagonal state of covariance matrix in the first mode , where , and the vacuum state in the second mode .both states are processed through a two - mode squeezer ( tms ) of parameter .note that if is a fock state , at this point of the circuit we have a photon - added epr state , which we know is always entangled ( its entanglement is monotonically increasing with ) . of course, if we have the vacuum on both modes , the resulting state is simply an epr state .the next step in the circuit consists in processing each mode of the state through two independent gaussian additive - noise channels and .the variance of the added noise on the first ( second ) mode is denoted by ( ) .this construction ensures that the resulting ( not necessarily gaussian ) state will always be physical provided , , and .the state at the output of this circuit has a covariance matrix where is the third pauli matrix .note that this form for a covariance matrix is actually quite general .indeed , duan _ et al ._ have shown that any covariance matrix of a two - mode state can be transformed into the form by applying local linear unitary bogoliubov operations , i.e. , combinations of squeezing transformations and rotations .these operations do not influence the separability of the state , and are thus always allowed when studying entanglement .the covariance matrix ( [ gammaref ] ) depends on three parameters while the most general form ( [ covmat ] ) has only one additional parameter , which reflects that ( [ gammaref ] ) encompasses a wide class of two - mode gaussian states .the entanglement of the resulting state is now analyzed as depicted by the red box of fig .the resulting state has a covariance matrix of the form of eq .( [ symplectic ] ) and the symplectic eigenvalues of can be expressed as a function of the different parameters characterizing , namely note that this expression is valid regardless of whether is gaussian or not . at this point , using the duan - simon separability criterion ( ignoring whether is gaussian or not ) would detect an entangled state only if .however , we can improve on this by taking into account the degree of gaussianity at the output of the circuit of fig .[ box ] , as explained previously .the calculation of could be done through the computation of the wigner function , but this would require some numerical integrations . in appendix [ annexeb ] , we give a different way of calculating for this specific example .the final expression is not very elegant , but calculations are performed completely analytically .now that we have defined a circuit to generate families of non - gaussian states and detect their entanglement , we will focus on some explicit examples of such states in order to illustrate the usefulness of our improved criterion . in the circuit of fig .[ figure ] , we start with fock - diagonal states , which have a diagonal covariance matrix with variance .we are interested in non - gaussian states and will consider two rather extreme cases of such states .the first case is a single fock state with , the parameter of the covariance matrix being thus .this state has clear quantum features , such as negative parts in the wigner function .our second choice is a non - gaussian mixture of coherent states with a random phase , which can be viewed as `` classical '' .this phase - diffused coherent ( pdc ) state can equivalently be represented as a mixture of fock states following a poisson distribution these two examples for are simple at a theoretical level and may also be implemented experimentally . for the state to be feasible experimentally , we will focus on values of the parameter of the two - mode squeezer that are smaller than 0.8 ( db ) .the values of the noise variances and will be chosen smaller than 2 units of vacuum noise because otherwise the state is necessarily separable ( regardless of whether it is gaussian or not ) .indeed , each mode of can be seen as the output of a classical gaussian additive noise channel and it is known that such a channel is entanglement breaking if ( ) .  in fig .[ result ] we exhibit explicit examples of non - gaussian states that are generated from being either a fock state or a phase - diffused coherent state .the corresponding numerical values of the circuit parameters ( ) are displayed in table [ result_tableau ] .we first choose sets of values of the circuit parameters such that , implying that the duan - simon criterion does not detect entanglement . in this case, entanglement is detected as soon as , so all these example states are proven to be entangled with our improved separability criterion .we then extend our search to larger values of .an entangled state is then detected whenever . since in our examples , the function is given by eq . ( [ nuth ] ) .all points localized in the blue zone are thus examples of non - gaussian entangled states that are detected by our improved separability criterion but not otherwise .we remark that entangled states can be found with both choices of ( either a highly non - classical fock state or a classical mixture of phase - diffused coherent states ) ..[result_tableau ] values of the circuit parameters used to generate the examples of non - gaussian entangled states that are detected by our improved separability criterion .the corresponding values of and are also given .[ cols="^,^,^,^,^,^,^",options="header " , ]   evolution of symplectic eigenvalue and degree of gaussianity when varying the circuit parameters , illustrating how they influence the separability of the states . for each curve ,one of the parameters varies while the others are fixed , starting from and ( at the bottom of the curve ) . increasing achieved by increasing , , and and decreasing .the blue curve is given by . ]interestingly , for all states created with our circuit when gaussian noise is added on the first mode only ( i.e. , ) , we find out that the duan - simon separability criterion becomes necessary and sufficient , even for non - gaussian states . indeed ,if , the symplectic eigenvalue is smaller than 1 for all values of and , hence the state is entangled .in contrast , if , we have an entanglement breaking channel , so we know that the state is necessarily separable .this confirms the validity of our method .finally , fig .[ result_2 ] illustrates how the different circuit parameters influence the separability of the state .starting with a fock state ( or with a phase - diffused coherent state ) with circuit parameters , and ( the lowest points of the curves ) , we see that by varying one of the parameters we can always create entangled states that are unnoticed by the duan - simon criterion .( note that there is no curve to plot corresponding to varying for fock states since can only take odd integer values in this case . )   as a second example , let us consider a squeezed single - photon path - entangled state , i.e. the non - gaussian state created from the circuit of fig .[ path ] .quantum circuit used to prepare a squeezed single - photon path - entangled state .the entanglement of the state is analyzed by the red box ( see fig .[ box ] ) . ]a vacuum and single - photon fock states are both squeezed , with respective squeezing parameters and , and are then coupled with a balanced beam splitter .the wave function of the output ( pure ) state has the form this state is obviously entangled for all values of , but the duan - simon separability criterion detects entanglement only for or .however , similarly as what walborn _ et al . _ have shown using their entropic entanglement criterion , we can detect entanglement for all values of with our improved criterion . let us suppose that . applying the `` entanglement analyzing box '' to this state, we find that where is computed with the help of wigner functions . therefore , according to eq .( [ nuth ] ) , and entanglement is detected if however , we supposed at the beginning that .entanglement is thus always detected .the same analysis can be done if .although the entropic criterion of walborn _ et al ._ also detects the entanglement of this state for all , we believe that our method is easier to apply .in this paper we have introduced a continuous - variable separability criterion exploiting the degree of gaussianity of the state , thereby allowing a stronger detection of two - mode non - gaussian entangled states .our criterion works by verifying the physicality of the symplectic eigenvalues of the partially transposed state in terms of gaussianity - bounded uncertainty relations .we demonstrated the advantages of our method by providing explicit examples of states whose entanglement is detected by our criterion but left undetected by the duan - simon criterion based on the epr variance only .we proposed an optical circuit for creating a family of such states and studied the entanglement detection as a function of the parameters of the circuit .the values of those circuit parameters were chosen so that these example states could be experimentally generated to demonstrate the method .the general applicability of the method is witnessed by the fact that these example states can be generated both from genuinely quantum non - gaussian states ( fock states ) and from classical non - gaussian mixtures of phase - diffused coherent states ( states with a positive p - function ) .we expect that many more examples of entangled states could be found , first by testing different values of the parameters and , second by generalizing the circuit ( for example , at the very beginning of the circuit , one can insert a thermal state instead of the vacuum ) , or simply by devising a new circuit generating other types of non - gaussian states such as those of our second example .  as mentioned in the introduction , a separability condition such as inequality ( [ delta ] ) can not be rewritten with a tighter lower bound that would solely depend on purity .this is because the duan - simon criterion is necessary and sufficient for all gaussian states ( of arbitrary purity ) .hence , the lower bound in inequality ( [ delta ] ) can not be moved upward without being violated by some mixed gaussian states that are known to be separable .however , we expect that our separability criterion may be further improved by taking into account both the degree of gaussianity and purity of the state , and then making use of the purity- and gaussianity - bounded uncertainty relations .this topic is worth further investigation .this work was supported by the f.r.s .- fnrs foundation under project no .t.0199.13 and by the belgian federal iap program under project no .p7/35 photonics .a. h. acknowledges financial support from the f.r.s .- fnrs foundation , and a. m. acknowledges financial support from the ministry of education and science of the republic of kazakhstan ( contract no .339/76 - 2015 ) .the degree of gaussianity defined by eq .  holds for an -mode state ( here , we focus on and ) .it obeys the following properties :  * for a gaussian state this measure obviously gives .however , does not imply that state is necessarily gaussian .let us present some counter examples .consider being the mixture of two fock states : the covariance matrix of , determines a gaussian ( thermal ) state it is then easy to see that =\mathrm{tr}[\rho\rho^g]=\frac{1}{1+\sqrt{2}}.\ ] ] this obviously gives , although is a non - gaussian state .other counter examples of non - gaussian states with may be found among the states of the form given , the real roots of the equation satisfying provide .note that eq .  is a polynomial of degree , therefore , the number of its roots providing counter examples is expected to increase with .* the degree of gaussianity is invariant under gaussian unitary operations .+ _ proof : _ consider an arbitrary state and corresponding gaussian state .a gaussian unitary operator transforming gaussian states to gaussian states transforms to .the same operator similarly transforms the gaussian state to .the later transformation is equivalent to a symplectic transformation of the corresponding covariance matrices . by construction of ,its covariance matrix is also the covariance matrix of .this covariance matrix is transformed by the symplectic transformation into the covariance matrix of .therefore , is the gaussian state corresponding to .then a simple calculation gives us the desired result : where we used the invariance of the trace under cyclic permutations . * the degree of gaussianity is invariant under partial transposition .+ _ proof : _ partial transposition implies sign - flip of one of the two momentum quadratures ( say ) , i.e. one of the arguments of the wigner function describing a two - mode state. then we have + + where at the last step we changed the variables as . * the degree of gaussianity is invariant under transposition .+ _ proof : _ the proof follows the same steps as for the partial transposition , but in this case , we have both , and .this does not change the conclusion .  finally , we would like to comment on the possibility of an experimental estimation of the degree of gaussianity .following , an expression of as a converging series on the radial moments of the wigner function is derived for states with an angular - independent wigner function and covariance matrix of the form : equation ( [ se ] ) is also applicable to states with a phase - dependent wigner function but in this case one needs to employ the phase - averaged quantity instead of .equation ( [ se ] ) gives evidence that the experimental estimation of the degree of gaussianity is a feasible task for states where the higher moments are of decreasing strength ( see , for example , ref .let us show how one can perform the computation of the degree of gaussianity of the reduced one - mode state corresponding to at the output of the `` entanglement analyzing box '' of fig .[ box ] ( this state corresponds to the smallest symplectic eigenvalue ) .the same technique allows computation of the degree of gaussianity of the two - mode state ( see fig . [ figure ] ) as well .  by our conventionthe gaussian state has the same covariance matrix as .then , since it corresponds to the purity of , the denominator in the definition of given by eq .is trivially evaluated as =\frac{1}{\sqrt{\det(\gamma_{\sigma_2})}}=\frac{1}{\nu_-}.\ ] ]  the evaluation of the numerator in eq .is more involved .although after the two - mode squeezer , we have a very simple form of the density matrix of the state , the addition of the gaussian noises makes the density matrix of state ( and so the ones of the reduced states and ) very hard to express in a simple form .state is obtained as a result of the application of gaussian additive noise channels and to the first and second modes of correspondingly \label{rhonoise } \\    & = &  \int dx_1dp_1dx_2dp_2 \,\,\frac{1}{2\pi \eta}e^{-\frac{x_1 ^ 2+p_1 ^ 2}{2\eta}}\frac{1}{2\pi \mu}e^{-\frac{x_2 ^ 2+p_2 ^ 2}{2\mu}}\nonumber\\ & \times & d(x_1,p_1)d(x_2,p_2)\,\rho_{\mathrm{\mathrm{tms}}}\ , d^{\dag}(x_1,p_1)d^{\dag}(x_2,p_2 ) , \nonumber\end{aligned}\ ] ] where is the displacement operator .we will perform calculations avoiding the direct use of the density matrix .instead we will use the following construction .let be the density matrix of the two - mode state that has and as reduced states .then we have the following equivalent representation of the numerator in eq .: &=&\textrm{tr}[\sigma ( \mathds{1}\otimes\sigma_2^g)]\nonumber\\ & = & \lim\limits_{v\rightarrow\infty}\,\frac{v+1}{2}\,\textrm{tr}[\sigma(\rho_{\mathrm{th}}^v\otimes\sigma_2^g ) ] , \label{trick}\end{aligned}\ ] ]  where is a thermal state with a covariance matrix . identity ( [ trick ] ) holds because this state multiplied by tends to when tends to infinity .next we express as a result of the transformation of the initial state by the circuit in fig .[ figure ] : \hspace{-18pt}\\  & \hspace{-18pt}=&\hspace{-8pt } \textrm{tr}\left [ u_s\,t[(\phi_{\eta}\otimes\phi_{\mu})[u_{\lambda}(\rho_{\mathrm{in}}\otimes{|0\rangle}{\langle0|})u_{\lambda}^{\dag}]]u_s^{\dag}\,\,\rho_{\mathrm{th}}^v\otimes\sigma_2^g \right]\nonumber\\ & \hspace{-18pt}= & \hspace{-8pt}\textrm{tr}\left [ ( \rho_{\mathrm{in}}\otimes{|0\rangle}{\langle0|})u_{\lambda}^{\dag}(\phi_{\eta}\otimes\phi_{\mu})[t[u_s^{\dag}(\rho_{\mathrm{th}}^v\otimes\sigma_2^g )  u_s]]u_{\lambda}\right].\nonumber \label{trths2}\end{aligned}\ ] ] here is the final symplectic transformation , describes the action of the two - mode squeezer , is the partial transposition in the second mode , and ( ) denotes the additive gaussian noise channel with the noise variance ( ) being applied to the first ( second ) mode . at the final step we used the invariance of the trace under cyclic permutations and the equivalence of the partial transposition and additive noise channel to their duals with respect to the scalar product of operators defined as  ] is completely determined by its covariance matrix , which is the result of the application of the dual circuit to the covariance matrix of , +\begin{pmatrix}\eta\mathds{1}&&0\\0&&\mu\mathds{1}\end{pmatrix}\bigg)s_{\mathrm{\mathrm{tms}}}^t(-\lambda)$},\ ] ] where represents the symplectic diagonalization that gives the symplectic eigenvalues , is the partial transposition ( which acts on the two - mode covariance matrices as ) and , the two - mode squeezing transformation .  from covariance matrix , we can easily deduce the wigner function of and with its help compute the following trace , which is equal to the trace in eq .: \\  & = & ( 2\pi)^2\int dx_1dp_1dx_2dp_2\,w_{\rho_{\mathrm{in}}}(x_1,p_1)\nonumber\\ &  & \times \,w_{{|0\rangle}}(x_2,p_2)w^*(x_1,p_1,x_2,p_2)\nonumber\\ & = & 2\pi\,c \,\int dx_1dp_1 \ , w_{\rho_{\mathrm{in}}}(x_1,p_1)w_{\mathrm{th}}^m(x_1,p_1).\nonumber\end{aligned}\ ] ] here the normalization factor is obtained by integrating over the variables of the second mode ( it is a simple gaussian integral ) : and by stressing out a new wigner function corresponding to a thermal state of variance .returning to the state space , the computation of the trace can be carried out as follows : \ & =  & c \,\textrm{tr}\big[\sum_n\phi_n{|n\rangle}{\langlen|}\rho_{\mathrm{th}}^m\big]\\  & = & c\,\frac{2}{m+1}\sum_{n=0}^{\infty}\phi_n\big(\frac{m-1}{m+1}\big)^{n}.\nonumber\end{aligned}\ ] ] both parameters , and , depend on .the explicit formulas for and are cumbersome and we do not present it here , however , they allow us to carry out the limit which provides the trace in eq . .together with eq . , this gives us a value for following eq . .inequality ( [ delta ] ) was derived in without using the ppt criterion .however , it is straightforward to derive the inequality by applying the ppt criterion on the parametric form of the uncertainty relations introduced in and thus setting the correspondence with the results in .

Abstract: currently available separability criteria for continuous - variable states are generally based on the covariance matrix of quadrature operators . the well - known separability criterion of duan _ et al . _ [ phys . rev . lett . * 84 * , 2722 ( 2000 ) ] and simon [ phys . rev . lett . * 84 * , 2726 ( 2000 ) ] , for example , gives a necessary and sufficient condition for a two - mode gaussian state to be separable , but leaves many entangled non - gaussian states undetected . here , we introduce an improvement of this criterion that enables a stronger entanglement detection . the improved condition is based on the knowledge of an additional parameter , namely the degree of gaussianity , and exploits a connection with gaussianity - bounded uncertainty relations [ phys . rev . a * 86 * , 030102 ( 2012 ) ] . we exhibit families of non - gaussian entangled states whose entanglement remains undetected by the duan - simon criterion .


many interesting problems in astrophysics and engineering involve evolution of macroscopic plasmas , modeled by the equations of magnetohydrodynamics ( mhd ) . these equations ( )are a system of convection - diffusion equations with the magnetic resistivity and heat conduction playing the role of diffusion .many applications like plasma thrusters for deep space propulsion and electromagnetic pulse devices ( ) involve small ( but non - zero ) values of the magnetic resistivity .hence , the design of efficient numerical methods for the resistive mhd equations is essential for simulating some of the afore mentioned models .  numerical study of the ideal mhd equations ( where magnetic resistivity and other diffusions terms are neglected ) has witnessed considerable progress in recent years and a variety of numerical methods are available ( see for a review of the available literature ) .the design of numerical schemes for the resistive mhd equations has not reached the same stage of maturity as the presence of magnetic resistivity complicates the design of stable methods even further .given the formidable difficulties , study of prototypical sub - models ( that mirror some , but not all of the difficulties of the resistive mhd equations ) can be a useful guide for obtaining robust methods for the resistive mhd equations .  in this paper , we consider the magnetic induction equations with resistivity .recent papers ( ) have pointed out the role that the magnetic induction equation ( without resistivity ) plays in the design of numerical schemes for the ideal ( inviscid ) mhd equations .the induction equations with resistivity can play a similar role for designing stable methods for the resistive mhd equations .our goal in this paper is to design stable and high - order accurate numerical schemes for the magnetic induction equations with resistivity .we start with a brief description of how the equations are derived . in a moving medium , the time rate of change of the magnetic flux across a given surface bounded by curve given by ( see ) : where the unknown denotes the magnetic field , the current density and are the spatial coordinates .the current density is given by : .the parameter denotes the magnetic resistivity , and the ( given ) velocity field .  using faraday s law : stokes theorem , the fact that the electric field in a co - moving frame and we obtain , magnetic monopoles have never been observed in nature . as a consequence , the magnetic fieldis always assumed to be divergence free , i.e. , . using this constraing in, we obtain the system : the above equation is an example of a convection - diffusion equation .the version obtained by taking zero resistivity ( ) in is termed the magnetic induction equation ( ) . a standard way to obtain a bound on the solutions of convection - diffusion equations likeis to use the energy method .however is not symmetrizable . consequently it may not be possible to obtain an energy estimate for this systemon the other hand , is symmetrizable .we use the following vector identity and rewrite in the form , where the matrix is given by introducing the matrix , can also be written in the following form , where for .note that the symmetrized matrices in are diagonal and that the coupling in the equations are through both the lower order source terms and the viscous terms .furthermore , taking the divergence of both sides of we obtain , hence , if , it follows that for .this implies that all the above forms , and are equivalent ( at least for smooth solutions ) .although the magnetic induction equations with resistivity are linear , the coefficients are functions of and . therefore , closed form solutions are not available and we must resort to numerical methods in order to calculate ( approximate ) solutions .consequently , it is important to design efficient numerical methods for these equations .  as mentioned beforethe magnetic induction equations is a sub - model in the resistive mhd equations .hence , design of stable and high - order accurate numerical schemes for the viscous induction equations can lead to robust schemes for the non - linear resistive mhd equations .the presence of the constraint leads to numerical difficulties .small divergence errors may change the nature of results from numerical simulations ( see for details on the role of divergence in ideal mhd codes ) .our approach to treating the constraint follows the method developed in and involves discretizing .tfurthermore , a proper discretization of the symmetric form yields energy estimates .these estimates are vital in proving existence of weak solutions .we will approximate spatial derivatives by second and fourth order sbp ( `` summation by parts '' ) operators .the boundary conditions of both the dirichlet and mixed type are weakly imposed by using a sat ( `` simultaneous approximation term '' ) .this work is an extension of the sbp - sat schemes for the case without resistivity ( ) found in a recent paper .we would like to emphasize that other numerical frameworks like mixed finite elements , discrete duality finite volume or mimetic finite differences might also lead to stable schemes for approximating the induction equations with resistivity .however , we are not aware of any papers that have approximated the resistive induction equations with these approaches .the rest of this paper is organized as follows : in section [ sec : cp ] , we state the energy estimate for the initial - boundary value problem corresponding to in order to motivate the proof of stability for the scheme .this is done for mixed type and dirichlet boundary conditions . in section [ sec :scheme ] , we present the sbp - sat scheme and show its stability with both dirichlet and mixed boundary conditions .numerical experiments are presented in section [ sec : numex ] and conclusions from this paper are drawn in section [ sec : conc ] .for simplicity and notational convenience , we restrict ourselves to two spatial dimensions in the remainder of this paper .extending our results to three dimensions is straightforward .  in two dimensions ,reads ,  \end{gathered}\ ] ] and with and denoting the magnetic and velocity fields respectively . throughout this paper, we consider in a smooth domain .one can extend our results to general piecewise smooth boundaries by a standard procedure .we augment with initial conditions , and dirichlet or mixed boundary conditions with homogeneous boundary data .the dirichlet boundary conditions are given as in order to specify the mixed boundary conditions , we need some notation .let denote the outward pointing unit normal at a point .define for and .furthermore , let denote the part of where the characteristics are incoming , i.e. , with this notation , the mixed boundary conditions read where is a given number .  in order to motivate the complicated calculations required to show stability in the discrete case , we start by explaining how stability is proved in the continuous case .we assume that the solution , , is sufficiently regular for our calculations to make sense .[ thm : contidirichlet ] let be a solution of the problem and with boundary conditions or . there existpositive constants ( depending on and its first derivatives ) and , such that   multiplying by and then integrating in space , we get which implies , so that , from the above relation , we see that applying dirichlet boundary conditions , , and integrating in time gives the required result . for the mixed boundary conditions, we split the boundary into and .this yields rearranging the above relation and applying mixed boundary conditions , remembering that , we get from the above relation , after integrating in time then we have the required result .to simplify the treatment of the boundary terms we let the computational domain be the unit square .a justification for this will be provided at the end of this section .the sbp finite difference schemes for one - dimensional derivative approximations are as follows .let  ] using uniformly distributed grid points for , and , such that .we order a scalar grid function as a column vector   to obtain a compact notation for partial derivatives of a grid function , we use kronecker products .the kronecker product of an matrix and an matrix is defined as the matrix for appropriate matrices , , and , the kronecker product obeys the following rules : using kronecker products , we can define 2-d difference operators .let denote the identity matrix , and define for a smooth function , and similarly .set , define and the corresponding norm .also define , , and .  for a vectorvalued grid function , we use the following notation and so on . in the same spirit , the inner product of vector valued grid functions is defined by .note that the kronecker products is just a tool to facilitate the notation . in the implementation of schemes using the operators in the kronecker products we can think of these as operating in their own dimension , i.e. , on a specific index .thus , to compute , we can view as a field with two indices , and the one - dimensional operator will operate on the first index since it appears in the first position in the kronecker product .the usefulness of summation by parts operators comes from this lemma .[ lem : byparts ] for any grid functions and , we have w\\     \left(v,{\mathfrak{d}}_yw\right)_{{\mathbf{p } } } + \left({\mathfrak{d}}_yv , w\right)_{{\mathbf{p } } } & =     v^t \left[({\mathcal{u}}- { \mathcal{d}})(p_x \otimes i_m)\right ] w.   \end{aligned}\ ] ]  observe that this lemma is the discrete version of the equality   we calculate the second equality is proved similarly .  for a vector valued grid function , we define the discrete analogues of the and the operators by where etc . before we define our numerical schemes , we collect some useful results in a lemma .  [lem : kurlkurl ] { \frak{curl}}\left({\mathbf{v}}\right ) \\    & \qquad - \left(v^2\right)^t    \left [ ( { \mathcal{r}}-{\mathcal{l}})\left ( i_n\otimes p_y\right)\right ]    { \frak{curl}}\left({\mathbf{v}}\right ) .   \end{aligned}\ ] ] if is a grid function , then \left(u\circ{\mathbf{v}}\right )    \\    & \qquad +    \frac{1}{2}\left(u\circ { \mathfrak{d}}_x { \mathbf{v}}- { \mathfrak{d}}_x\left(u\circ { \mathbf{v}}\right ) ,     { \mathbf{v}}\right)_{{\mathbf{p } } } ,    \\    \left({\mathbf{v}},u\circ{\mathfrak{d}}_y{\mathbf{v}}\right)_{{\mathbf{p } } } & = \frac{1}{2 } { \mathbf{v}}^t    [ ( { \mathcal{u}}-{\mathcal{d}})(p_x\otimes i_m ) ] \left(u\circ{\mathbf{v}}\right )    \\    & \qquad +     \frac{1}{2}\left(u\circ { \mathfrak{d}}_y { \mathbf{v}}- { \mathfrak{d}}_y\left(u\circ { \mathbf{v}}\right ) ,     { \mathbf{v}}\right)_{{\mathbf{p}}}.   \end{aligned}\ ] ]  to prove , \left(-{\mathfrak{d}}_{y}v^1 + { \mathfrak{d}}_{x}v^2\right ) \\    &\qquad + \left(v^2\right)^t \left [ ( { \mathcal{r}}-{\mathcal{l}})\left (     i_n\otimes p_y\right)\right]\left ( { \mathfrak{d}}_{y}v^1 - { \mathfrak{d}}_{x}v^2\right)\\    & = \left({\frak{curl}}\left({\mathbf{v}}\right),{\frak{curl}}\left({\mathbf{v}}\right)\right)_{{\mathbf{p } } }    + \left(v^1\right)^t\left[({\mathcal{u}}-{\mathcal{d}})\left(p_x\otimes      i_m\right)\right ] { \frak{curl}}\left({\mathbf{v}}\right )    \\&\qquad  -    \left(v^2\right)^t \left [ ( { \mathcal{r}}-{\mathcal{l}})\left (     i_n\otimes p_y\right)\right ] { \frak{curl}}\left({\mathbf{v}}\right ) .\end{aligned}\ ] ] to show , first note that since is diagonal , .we use lemma [ lem : byparts ] to calculate this shows the first equation in , the second is proved similarly .now we are in a position to state our scheme(s ) . for or we will use the notation for both the grid function defined by the function and for the function itself .similarly , for the boundary values , we use the notation and for both discrete and continuously defined functions .hopefully , it will be apparent from the context what we refer to .the differential equation will be discretized in an obvious manner .we incorporate the boundary conditions by penalizing boundary values away from the desired ones with a term . to this end set ,\ ] ] where , , and are diagonal matrices , with components ordered in the same way as in ( ) ( and similarly for the other penalty matrices ) , to be specified later .furthermore , the following form of the penalty paramaters will be convenient : and similarly for , etc .  with this notation the scheme for the differential equation with boundary values reads while is given .here denotes the matrix   [ thm : stab1 ] let be as solution to with . if the constants in is chosen as  and all other entries are 0 , then where , , for and is a constant depending on , , and their derivative approximations , but not on or . by construction of the sbp operators , where and .set .taking the inner product of and , we get using lemma [ lem : kurlkurl ] we get { \frak{curl}}\left({\mathbf{v}}\right ) +   \epsilon\left(v^2\right)^t \left [ ( { \mathcal{r}}-{\mathcal{l}})\left ( i_n\otimes     p_y\right)\right ] { \frak{curl}}\left({\mathbf{v}}\right )   \\   & \qquad -\left({\mathbf{v}},u^1\circ{\mathfrak{d}}_x { \mathbf{v}}\right)_{{\mathbf{p } } } -   \left({\mathbf{v}},u^2\circ{\mathfrak{d}}_y { \mathbf{v}}\right)_{{\mathbf{p } } } +   \left({\mathbf{v}},c{\mathbf{v}}\right)_{{\mathbf{p } } } +   \left({\mathbf{v}},\mathcal{b}{\mathbf{v}}\right)_{{\mathbf{p}}}\\   & = -\epsilon\left(v^1\right)^t\left[({\mathcal{u}}-{\mathcal{d}})\left(p_x\otimes     i_m\right)\right ] { \frak{curl}}\left({\mathbf{v}}\right ) +   \epsilon\left(v^2\right)^t \left [ ( { \mathcal{r}}-{\mathcal{l}})\left ( i_n\otimes     p_y\right)\right ] { \frak{curl}}\left({\mathbf{v}}\right )   \\   & \qquad - \frac{1}{2 } { \mathbf{v}}^t\left [ ( { \mathcal{r}}-{\mathcal{l}})(i_n\otimes    p_y)\right](u^1\circ { \mathbf{v } } ) - \frac{1}{2 } { \mathbf{v}}^t\left [    ( { \mathcal{u}}-{\mathcal{d}})(p_x\otimes i_m)\right](u^2\circ { \mathbf{v } } )   \\   & \qquad - \frac{1}{2}\left(u^1\circ { \mathfrak{d}}_x { \mathbf{v}}- { \mathfrak{d}}_x(u^1\circ    { \mathbf{v}}),{\mathbf{v}}\right)_{{\mathbf{p } } } - \frac{1}{2}\left(u^2\circ { \mathfrak{d}}_y { \mathbf{v}}-    { \mathfrak{d}}_y(u^2\circ { \mathbf{v}}),{\mathbf{v}}\right)_{{\mathbf{p } } }   \\   & \qquad + \left({\mathbf{v}},c{\mathbf{v}}\right)_{{\mathbf{p } } } +   \left({\mathbf{v}},(\mathcal{b'}+\epsilon \mathcal{b''}){\mathbf{v}}\right)_{{\mathbf{p}}}.  \end{aligned}\ ] ] note that by , for some constant depending on the first derivatives of and . using the conditions ( [ eq : primeconstants ] ) we arrive at { \frak{curl}}\left({\mathbf{v}}\right ) +\epsilon\left(v^2\right)^t \left [ ( { \mathcal{r}}-{\mathcal{l}})\left ( i_n\otimes     p_y\right)\right ] { \frak{curl}}\left({\mathbf{v}}\right )   \\   & \quad    + \epsilon\left({\mathbf{v}},\mathcal{b''}{\mathbf{v}}\right)_{{\mathbf{p}}}.  \end{aligned}\ ] ] next , for any grid function ( with components as in ( ) ) , we have similarly combining this we find that we also compute using for , and for we find   { \frak{curl}}\left({\mathbf{v}}\right ) -\left(v^2\right)^t \left [ ( { \mathcal{r}}-{\mathcal{l}})\left (     i_n\otimes     p_y\right)\right ] { \frak{curl}}\left({\mathbf{v}}\right ) \\   & \quad \hphantom{-\epsilon\bigl ( } + { \mathbf{v}}^t \bigl ( -   \sigma''_{{\mathcal{l}}}(i_n\otimes p_y){\mathcal{l}}- \sigma''_{{\mathcal{r } } } ( i_n\otimes p_y){\mathcal{r}}-\sigma''_{{\mathcal{d } } } ( p_x\otimes i_m){\mathcal{d}}- \sigma''_{{\mathcal{u } } } ( p_x\otimes   i_m){\mathcal{u}}\bigr )   { \mathbf{v}}\bigr)\\   &  = : -\epsilon a\end{aligned}\ ] ] choose the remaining penalty parameters as in ( [ eq : doubleprimeconstants ] ) and write where furthermore , we used the notation and . summing up, we have shown that and the result follows by gronwall s inequality .the scheme for the mixed boundary conditions reads where is the desired value of on the boundary .[ thm : mixedtype ] if is a solution of with , and is chosen so that holds , then where is a constant depending on , and the derivatives of and , but not on or .the proof of this theorem proceeds as the proof of theorem [ thm : stab1 ] .note that we are subtracting the boundary terms coming from , so that we do not need to split .the other terms are estimated as before , and we get the inequality which yields the stability result .we have assumed a constant resistivity co - efficient in the above discussion .however , in many practical applications , the co - efficient of resistivity can vary in space . in such cases , our theoretical resultshold provided that the resistivity co - efficient is uniformly bounded away from zero .the analysis has been carried out on a cartesian equidistant grid on the unit square . however ,this is not a restriction as problems on general domains may be addressed using coordinate transformations .the stability proofs will hold as long as the norm matrices ( ) are diagonal .sbp finite - difference schemes with diagonal matrix have a truncation error of in the interior and near the boundary resulting in a global order of accuracy / convergence rate of .( see for further details . )the resistive magnetic induction equations include diffusive terms .those are discretized by applying the first - derivative operators twice .this results in a truncation error of near the boundary for the diffusive terms .however , thanks to the energy stability of the scheme , the global convergence rate and order of accuracy remains at ( see ) .in this section , we are going to write down the three - dimensional version of the finite difference scheme for the equation . to begin with , we discretize unit cube ^ 3 ] .in this case , the exact solution is a smooth hump ( centered at and decaying exponentially ) rotating about the origin and completing one rotation in time .the hump remains completely inside the domain during the course of the rotation .since the exact solution is known in this case , we use this solution to specify the data for the boundary conditions or .the above setup is simulated using the sbp2 and sbp4 schemes . using dirichlet ormixed boundary conditions led to very similar results .hence , we present results only with the mixed boundary conditions in this case .the time - integration was performed with a second - order runge - kutta method at a number of .the resistivity was used .also we have used the value of the penalty parameters as mentioned in theorem [ thm : stab1 ] , for instance we used and so on .we plot the norm of the magnetic field : , at times ( half rotation ) and ( full rotation ) for both the sbp2 and sbp4 schemes in figure [ fig:1 ] .[ fig:1 ]  as shown in the figure , both and schemes resolve the solution quite well .there are very few noticeable differences between the second and fourth order schemes at this resolution .the shape of the hump is maintained during the rotation .a quantitative view of the results is presented in table [ tab:1 ] where  is the numerical approximation and is the exact reference solution and ..relative percentage errors in for at time and rates of convergence for numerical experiment . [cols="^,>,>,>,>",options="header " , ]we have presented finite difference schemes for the magnetic induction equations with resistivity .these equations arise as a sub model in the resistive mhd equations of plasma physics .we have shown that the symmetric form of the magnetic induction equations with resistivity is well - posed with general initial data and both dirichlet boundary conditions as well as mixed boundary conditions .sbp - sat based finite difference schemes were designed for the initial - boundary - value problem corresponding to the magnetic induction equations with resistivity .these schemes were based on the form and use sbp finite difference operators to approximate spatial derivatives and an sat technique for implementing boundary conditions .the resulting schemes are high - order accurate and shown to be energy stable .the schemes were tested on numerical experiments illustrating both the stability as well as high - order of accuracy . we also the use of the divergence errors as a measure of the accuracy of the solution .the results show that the sbp - sat approach is a promising method to simulate initial boundary value problems for more complicated equations like the resistive mhd equations .for the sake of completeness , here we will present all the sbp operators used in the analysis .we consider second and third order accurate finite difference approximations .

Abstract: in this paper , we design high order accurate and stable finite difference schemes for the initial - boundary value problem , associated with the magnetic induction equation with resistivity . we use summation - by - parts ( sbp ) finite difference operators to approximate spatial derivatives and a simultaneous approximation term ( sat ) technique for implementing boundary conditions . the resulting schemes are shown to be energy stable . various numerical experiments demonstrating both the stability and the high order of accuracy of the schemes are presented .


adaptive beamforming technology has received considerable attention for several decades and found widespread applications in radar , sonar and wireless communications .the optimal linearly constrained minimum variance ( lcmv ) beamformer which minimizes the array output power while maintaining the array response on the direction of the desired signal , is the most well - known beamformer .however , the computation of the inverse of the received data covariance matrix and the requirement of the knowledge of the cross - correlation vector present difficulties for its implementation .  a number of adaptive algorithms have been reported for the implementation of the lcmv beamformer , . among themthe stochastic gradient ( sg ) algorithm , is popular due to its simplicity and low complexity .the drawback of the sg algorithm is that the convergence depends on the eigenvalue spread of the received data covariance matrix. this condition could be worse when the number of elements in the filter is large since it requires a large amount of snapshots to reach the steady - state .reduced - rank signal processing was introduced to provide a way out of this dilemma - .the reduced - rank schemes project the received vector onto a lower dimensional subspace and perform the filter optimization within this subspace .the advantages are their fast convergence properties and enhanced tracking performance when compared with full - rank schemes operating with a large number of parameters in the filter for the beamformer design .the reduced - rank algorithms range from the auxiliary vector filtering ( avf ) , the multistage wiener filter ( mswf ) , to the joint iterative optimization ( jio ) based works reported in , . despite the improved convergence and tracking performance achieved with the existing reduced - rank methods, they have to afford the heavy computational load as a tradeoff .  in this paper , we introduce a new lcmv reduced - rank algorithm based on the jio scheme .the proposed algorithm employs the set - membership filtering ( smf ) technique , to reduce the computational complexity significantly without the convergence speed reduction and the performance loss .the smf specifies a bound on the magnitude of the estimation error ( or the array output ) and uses the data - selective updates to adjust parameters according to this predetermined bound .it involves two steps : information evaluation and ) parameters adaptation .if the parameter update does not occur frequently , and the information evaluation does not require much complexity , the overall computational cost can be saved substantially .the current smf algorithms ( see and the reference therein ) pay attention to the full - rank parameter estimation .considering the fact that the reduced - rank algorithms exhibit superior performance over the full - rank methods , , , it motivates the deployment of the smf mechanism to the reduced - rank scheme to guarantee the good performance with low complexity .we employ the smf technique with the reduced - rank jio scheme for the lcmv beamformer design and develop the sg based algorithm for implementation . compared with the work reported in , , the devised scheme consists of a bank of full - rank adaptive filters , which constitutes the projection matrix , and an adaptive reduced - rank filter that operates at the output of the bank of full - rank filters .it provides an iterative exchange of information between the projection matrix and the reduced - rank filter and thus leads to improved convergence and tracking performance .compared with the jio based method ( with the fixed step sizes ) in , the proposed algorithm uses the smf technique to adjust the step sizes for the updates of the projection matrix and the reduced - rank weight vector , and thus has an attractive tradeoff between the convergence rate and misadjustment .furthermore , a time varying bound is incorporated in the proposed algorithm to avoid the risk of overbounding and underbounding in dynamic scenarios , and to improve the performance with a small number of update .the remaining of this paper is organized as follows : we outline a system model for beamforming and present the problem statement in section 2 .section 3 derives the proposed adaptive reduced - rank algorithm .simulation results are provided and discussed in section 4 , and conclusions are drawn in section 5 .the most important feature of the reduced - rank algorithms is to perform the dimensionality reduction and retain the key information of the original signal in the reduced - rank received vector , which is where denotes the projection matrix that is structured as a bank of full - rank filters ^t\in\mathcal c^{m\times1} ] is followed to produce the output   the main concern left to us is how to effectively design and calculate the projection matrix .the popular reduced - rank schemes include avf , mswf , and jio .the sg type algorithm can be employed in these schemes to estimate .however , it is difficult to set the step size for the existing methods to achieve a satisfactory tradeoff between the fast convergence and the misadjustment in dynamic scenarios . besides , the generation of the projection matrix is still a complicated task that increases the computational cost .in this section , we introduce a new adaptive reduced - rank algorithm based on the jio scheme and employ the smf technique with the time - varying bound to realize the data - selective updates for the beamformer design .it should be remarked that the jio scheme is selected here due to its improved performance and relatively simple implementation over the avf and the mswf ones .   in the existing full - rank smf scheme -, the filter is designed to achieve a predetermined or time - varying bound on the magnitude of the estimation error ( or the array output ) .this bound can be regarded as a constraint on the filter design , which performs the updates for certain received data , namely , the data - selective updates . for the reduced - rank jio scheme with the smf technique ,we need to take both the projection matrix and the reduced - rank weight vector into consideration due to the feature of their joint iterative exchange of information .let represent the set containing all the pairs of for which the corresponding array output at time instant is upper bounded in magnitude by a time - varying bound , yields ,    where denotes the transmitted data of the desired user from , is the set of all possible data pairs ( ) , and the set is referred to as the _ feasibility _ set .the pairs of in the set satisfy the constraint . in practice, can not be traversed allover .a larger space of the data pairs provided by the observations leads to a smaller feasibility set .that is , as the number of data pairs increases , there are fewer pairs of that can be found to satisfy the constraint . under this condition, we define the exact _ membership _ sets to be the intersection of the _ constraint _ sets over the time instants , i.e. , where is the constraint set to provide information for the membership set to construct a set of solutions .it is clear that the feasibility set is a limiting set of the exact membership set in practice .the two sets will be equal if the data pairs traverse completely .the proposed smf scheme introduces the principle of set - membership into reduced - rank signal processing and inherits the advantage of the joint optimization between the projection matrix and the reduced - rank filter .it updates the parameter vectors such that they will always belong to the feasibility set .note that the time - varying bound has to be chosen appropriately in order to devise an effective algorithm and avoid the risk of overbounding or underbounding .the selection of the bound will be given in the following part .in this section , we use the data - selective updates of the proposed smf scheme in the reduced - rank jio based algorithm . for the lcmv beamformer design, the proposed algorithm can be derived according to the minimization of the following cost function \\ & \textrm{subject~to}~~\bar{\boldsymbol w}^h\bar{\boldsymbol a}(\theta_0)=\gamma,~~\textrm{and}~~|\bar{\boldsymbol w}^h\boldsymbol t_r^h{\boldsymbol x}(i)|^2=g^2(i ) , \end{split}\ ] ] where is the reduced - rank received vector defined in ( [ 5 ] ) , is the reduced - rank steering vector of the desired user , and is a coefficient that determines a set of estimates within the constraint set and .the cost function in ( [ 9 ] ) depends on the projection matrix and the reduced - rank filter .the solution of ( [ 9 ] ) should be encompassed in the feasibility set in ( [ 7 ] ) and the pairs of should satisfy the constraint set . in order to solve the optimization problem in ( [ 9 ] ) , using the method of lagrange multipliers and considering the constraint , we have + 2\lambda re\big[\bar{\boldsymbol w}^h\bar{\boldsymbol a}(\theta_0)-\gamma\big]\ ] ] where is the lagrange multiplier and $ ] selects the real part of the quantity .note that the constraint is not included in ( [ 10 ] ) .this is because a point estimate can be obtained from ( [ 10 ] ) whereas the bounded constraint determines a set of .we use the constraint with respect to to determine one solution and to expand it to a hyperplane by using the bounded constraint .we use the sg type algorithm to update the parameter vectors . specifically , assuming is known , computing the _ instantaneous _ gradient of ( [ 10 ] ) with respect to , equating it to a zero matrix and solving for , we have \boldsymbol x(i)\bar{\boldsymbol w}^h(i),\ ] ] where is the step size for the update of the projection matrix and is the corresponding identity matrix .  assuming is known , taking the instantaneous gradient of ( [ 10 ] ) with respect to , equating it to a null vector and solving for , we get \bar{\boldsymbol x}(i)\ ] ] where is the step size for the update of the reduced - rank weight vector .the updates of the projection matrix and the reduced - rank weight vector may spend a long time or terminate suddenly due to the misadjustment if the step sizes can not be set suitably .the second constraint in ( [ 9 ] ) can be used to set a bound on the power of the array output for adjusting the step size to avoid these problems .the predetermined bound , in the smf technique makes contributions to this topic .however , we can not often determine the bound accurately since there is usually insufficient knowledge about the underlying system , especially when the scenario is changing . in such cases , a predetermined bound always has the risk of underbounding or overbounding , both of which result in the performance degradation .it motivates us to introduce a time - varying bound to circumvent the aforementioned problems .  in the proposed algorithm, we use a time - varying bound in the second constraint of ( [ 9 ] ) to offer a good tradeoff between the convergence rate and the misadjustment by adjusting the step size values automatically following the time instant . under this condition, the update is performed only if the constraint can not be satisfied .this scheme provides data - selective updates ( updates with respect to certain snapshots ) for the filter design and thus reduces the computational complexity compared with the existing reduced - rank algorithms ( updates for all the snapshots ) .specifically , by substituting the update equations of ( [ 11 ] ) and ( [ 12 ] ) into the constraint on the time - varying bound , respectively , we obtain {\boldsymbol x}(i)}&\textrm { if}|y(i)|^2\geq\delta^2(i)\\           & \\          0 & \textrm { otherwise}\\          \end{array}\right.\ ] ] and \bar{\boldsymbol          x}(i ) } & \textrm { if}~|y(i)|^2\geq\delta^2(i)\\          & \\0 & \textrm{otherwise}.\\          \end{array}\right.\ ] ] the only challenge left to us now is how to select the time - varying bound in the constraint to make the proposed algorithm work effectively .we introduce a parameter - dependent bound ( pdb ) that was reported in to update the reduced - rank weight vector for capturing the desired user and suppressing the interference and noise ., that is where is a forgetting factor that should be set to guarantee an appropriate time - averaged estimate of the evolutions of the weight vector , which is given by , is the variance of the inner product of the weight vector with that provides information on the evolution of , is a tuning coefficient with  , and is an estimate of the noise power .this time - varying bound provides a smoother evolution of the weight vector trajectory and thus avoids too high or low values of the squared norm of .  until now, we finish the derivation of the proposed algorithm , which is called jio - sm - sg .a summary of the proposed method is given in table [ tab : jio - sm - sg ] , where the initialization procedure is important to start the update . from ( [ 11 ] ) and ( [ 12 ] ) , the projection matrix and the reduced - rank weight vector depend on each other , which provides an iterative exchange of information between each other and thus leads to an improved performance .the smf technique is employed in the proposed algorithm and the time - varying bound is calculated with the incoming of the received vector .it encompasses the pairs of the filter design in the feasibility set that satisfies the constraint .the filter updates are not performed for each time instant but only when this constraint can not reach ( i.e. , data - selective updates ) .the computational complexity is reduced significantly and the convergence rate is further increased compared with the existing reduced - rank methods .the proposed algorithm combines the positive features of the reduced - rank jio scheme and time - varying based smf technique for the implementation of the lcmv beamformer design ..the proposed jio - sm - sg algorithm [ cols= " < " , ]we evaluate the performance of the proposed jio - sm - sg algorithm and compared it with those of the existing methods , i.e. , full - rank sg , full - rank sg with smf , avf , mswf - sg , and reduced - rank jio - sg .we assume that the doa of the desired user is known by the receiver . in each experiment, a total of runs are carried out to obtain the curves .we use the bpsk and set the input snr db and inr db .simulations are performed by an ula containing sensor elements with half - wavelength interelement spacing .  in fig .[ fig : iswcs102_final ] , there are users , including one desired user in the system .the step size values are initialized by .we set , , and the rank .this experiment exhibits that the output sinr values of the existing and the proposed algorithms increase to the steady - state as the increase of the snapshots ( time index ) .the proposed algorithm shows faster convergence and better performance over the existing methods .the step size values are adapted to ensure the fast convergence rate without the risk of misadjustment for the proposed algorithm .due to the data - selective updates , the proposed algorithm could reduce the computational load significantly as it only requires updates ( updates for snapshots ) , which is significantly lower than those of the existing reduced - rank methods ( normally updates for snapshots ) .[ fig : iswcs101_final ] , which includes two experiments , shows the effectiveness of the proposed algorithm with the time - varying bound .the scenario is the same as that in fig .[ fig : iswcs102_final ] .we compare the full - rank ( fig .[ fig : iswcs101_final ] ( a ) ) and proposed ( fig .[ fig : iswcs101_final ] ( b ) ) algorithms with the fixed and time - varying bounds , respectively . in fig .[ fig : iswcs101_final ] ( a ) , the one with the fixed bound shows better performance than that with the time - varying bound but increases the computational cost as a tradeoff ( ) .the algorithms with higher ( ) or lower ( ) bounds exhibit worse convergence and steady - state performance .the same result can be found in fig .[ fig : iswcs101_final ] ( b ) for the proposed algorithm .it indicates that the time - varying bound is capable of improving the performance of the proposed jio - sm - sg algorithm , while realizing the data - selective updates to reduce the computation complexity .in this paper , we employed the time - varying bound smf technique in the reduced - rank jio scheme and developed a new adaptive algorithm for the lcmv beamformer .the proposed algorithm retained the positive feature of the iterative exchange of information between the projection matrix and the reduced - rank weight vector , and used data - selective updates to adjust parameters that satisfy the constraints of the lcmv cost function .the variable step sizes in the proposed algorithm provided a way to circumvent the problem between the convergence and misadjustment .it achieved superior performance , especially in large array scenarios , with relatively low complexity .r. c. de lamare and r. sampaio - neto , low - complexity variable step - size mechanisms for stochastic gradient algorithms in minimum variance cdma receivers , " _ ieee trans . signal processing _ , vol .2302 - 2317 , june 2006 .r. c. de lamare and r. sampaio - neto , adaptive interference suppression for ds - cdma systems based on interpolated fir filters with adaptive interpolators in multipath channels " , _ ieee transactions on vehicular technology _ , vol .56 , no . 6 ,september 2007 .r. c. de lamare , l. wang , and r. fa , adaptive reduced - rank lcmv beamforming algorithms based on joint iterative optimization of filters : design and analysis , " _ elsevier signal processing _ , vol .640 - 652 , feb . 2010 .r. c. de lamare and p. s. r. diniz , set - membership adaptive algorithms based on time - varying error bounds for cdma interference suppression , " _ ieee trans .vehicular technology _ , vol .644 - 654 , feb . 2009 .r. c. de lamare and r. sampaio - neto , reduced - rank adaptive filtering based on joint iterative optimization of adaptive filters , " _ ieee signal processing letters _ , vol .14 no . 12 , december 2007 , pp .980 - 983 .r. c. de lamare and r. sampaio - neto , adaptive reduced - rank processing based on joint and iterative interpolation , decimation and filtering " , _ ieee transactions on signal processing _7 , july 2009 , pp . 2503 - 2514 .r. fa , r. c. de lamare , and l. wang , reduced - rank stap schemes for airborne radar based on switched joint interpolation , decimation and filtering algorithm , " _ ieee transactions on signal processing _ , vol.58 , no.8 , aug .2010 , pp.4182 - 4194 .de lamare , r. sampaio - neto , m. haardt , `` blind adaptive constrained constant - modulus reduced - rank interference suppression algorithms based on interpolation and switched decimation , '' _ ieee trans . on signal processing _ , vol.59 , no.2 , pp.681 - 695 , feb .

Abstract: this paper proposes a new adaptive algorithm for the implementation of the linearly constrained minimum variance ( lcmv ) beamformer . the proposed algorithm utilizes the set - membership filtering ( smf ) framework and the reduced - rank joint iterative optimization ( jio ) scheme . we develop a stochastic gradient ( sg ) based algorithm for the beamformer design . an effective time - varying bound is employed in the proposed method to adjust the step sizes , avoid the misadjustment and the risk of overbounding or underbounding . simulations are performed to show the improved performance of the proposed algorithm in comparison with existing full - rank and reduced - rank methods .