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several criteria to detect non - classicality of quantum states of a harmonic oscillator have been introduced , mostly based on phase - space distributions @xcite , ordered moments @xcite , or on information - theoretic arguments @xcite . at the same time , an ongoing research line addresses the characterization of quantum states according to their gaussian or non - gaussian character @xcite , and a question arises on whether those two different hierarchies are somehow linked each other .    as a matter of fact , if we restrict our attention to pure states , hudson s theorem @xcite establishes that the border between gaussian and non - gaussian states coincides exactly with the one between states with positive and negative wigner functions . however , if we move to mixed states , the situation gets more involved . attempts to extend hudson s theorem have been made , by looking at upper bounds on non - gaussianity measures for mixed states having positive wigner function @xcite . in this framework , by focusing on states with positive wigner function , one can define an additional border between states in the _ gaussian convex hull _ and those in the complementary set of _ quantum non - gaussian states _ , that is , states that can not be expressed as mixtures of gaussian states . the situation is summarized in fig . [ f : poswigner ] : the definition of the gaussian convex hull generalizes the notion of glauber s non - classicality @xcite , with coherent states replaced by generic pure gaussian states , i.e. squeezed coherent states .    quantum non - gaussian states with positive wigner function are not useful for quantum computation @xcite , and are not necessary for entanglement distillation , _ e.g. _ the non - gaussian entangled resources used in @xcite are mixtures of gaussian states . on the other hand , they are of fundamental interest for quantum information and quantum optics . in particular , since no negativity of the wigner function can be detected for optical losses higher than @xmath0 @xcite ( or equivalently , for detector efficiencies below @xmath0 ) criteria able to detect quantum non - gaussianity are needed in order to certify that a _ highly non linear _ process ( such as fock state generation , kerr interaction , photon addition / subtraction operations or conditional photon number detections ) has been implemented in a noisy environment , even if no negativity can be observed in the wigner function . different measures of non - gaussianity for quantum states have been proposed @xcite , but these can not discriminate between quantum non - gaussian states and mixtures of gaussian states . an experimentally friendly criterion for quantum non - gaussianity , based on photon number probabilities , has been introduced @xcite , and then employed in different experimental settings to prove the generation of quantum non - gaussian states , such as heralded single - photon states @xcite , squeezed single - photon states @xcite and fock states from a semiconductor quantum dot @xcite .    ]    in this paper we introduce a family of criteria which are able to detect quantum non - gaussianity for single - mode quantum states of a harmonic oscillator based on the wigner function . as we already pointed out , according to hudson s theorem , the only pure states having a positive wigner function are gaussian states . one can then wonder if any bound exists on the values that the wigner function of convex mixtures of gaussian states can take . by following this intuition we present several bounds on the values of the wigner function for convex mixtures of gaussian states , consequently defining a class of sufficient criteria for quantum non - gaussianity . in the next section we will introduce some notation and the preliminary notions needed for the rest of the paper . in sec . [ s : criteria ] we will prove and discuss our wigner function based criteria for quantum non - gaussianity and in sec . [ s : examples ] we will prove their effectiveness by considering different families of non - gaussian states evolving in a lossy ( gaussian ) channel . we will conclude the paper in sec . [ s : conclusions ] with some remarks . throughout the paper we will use the quantum optical terminology , where excitations of a quantum harmonic oscillator are called photons . all the results can be naturally applied to any bosonic continuous - variable ( cv ) system . we will consider a single mode described by a mode operator @xmath1 , satisfying the commutation relation @xmath2 = \mathbbm{1}$ ] . a quantum state @xmath3 is fully described by its characteristic function @xcite @xmath4(\gamma ) = { { \rm tr}}[\varrho d(\gamma)]\:,\end{aligned}\ ] ] where @xmath5 represents the displacement operator . in addition , the quantum state @xmath3 can be fully described by the fourier transform of the characteristic function , _ i. e. _ the wigner function @xcite @xmath6(\alpha ) = \int \frac{d^2 \gamma}{\pi^2 } e^{\gamma^ * \alpha - \gamma \alpha^ * }   \chi[\varrho](\gamma)\:.\end{aligned}\ ] ] a state is defined to be _ gaussian _ if and only if its wigner function ( or equivalently its characteristic function ) is gaussian . all single - mode gaussian states can be expressed as @xmath7 where @xmath8 , is the squeezing operator , and @xmath9 $ ] is a thermal state ( @xmath10 and @xmath11 ) . pure gaussian states can be written as @xmath12 , and , according to hudson s theorem @xcite , they are the only pure states having a positive wigner function . together with that of a gaussian state , one can define the concept of _ gaussian map _ : a quantum ( completely positive ) map is defined gaussian iff it transforms gaussian states into gaussian states . all unitary gaussian maps can be expressed as @xmath13 , and they correspond to hamiltonian operators @xmath14 at most bilinear in the mode operators . similarly , a generic gaussian map can be decomposed as a gaussian unitary acting on the system plus an ancilla ( the latter prepared in a gaussian state ) , followed by partial tracing over the ancillary mode @xcite . another complete description of a cv quantum state @xmath3 may be given in terms of the so - called @xmath15-function @xmath16(\alpha)$ ] @xcite , defined implicitly via the formula @xmath17(\alpha )     where @xmath18 represents a coherent state . according to glauber a state is _ non - classical _ iff its @xmath15-function is not a proper probability distribution , _ e.g. _ the @xmath15-function is more singular than a dirac - delta function . note that the negativity of the wigner function is a more restrictive definition of non - classicality : there exists non - classical states having a positive wigner function ( _ e.g. _ squeezed states ) , while all the states having a non - positive wigner function are non - classical according to glauber .    in a similar spirit as in glauber s approach to non - classicality , in this paper we study the concept of _ quantum non - gaussian _ states . these are defined as follows . the gaussian convex hull is the set of states @xmath19 where @xmath20 denotes the hilbert space of continuous - variable quantum states , @xmath21 is a proper probability distribution and @xmath22 are pure gaussian states , i.e. , in the single mode case , squeezed coherent states identified by the set of parameters @xmath23 . since gaussian states do not form a convex set , the set in eq .   includes states which are not gaussian . moreover any mixed gaussian state can be written as a weighted sum of pure gaussian states , and hence the set above also includes convex mixtures of mixed gaussian states . the definition of _ quantum non - gaussianity _ naturally follows :    * definition*. _ a quantum state @xmath3 is quantum non - gaussian iff it is not possible to express it as a convex mixture of gaussian states , that is iff @xmath24_.    as illustrated in fig . [ f : poswigner ] , the border here defined dividing quantum non - gaussian states and mixtures of gaussian states falls in between the border dividing classical and non - classical states , and the one which divides states with positive and non - positive wigner functions . the importance of such a further distinction is evident if we note that all states in @xmath25 can be prepared through a combination of gaussian operations and classical randomization . on the contrary , if @xmath26 then a highly _ non - linear _ process ( due to a non - gaussian operation or measurement ) had necessarily taken place in the generation the quantum state @xmath3 . while the negativity of the wigner function is always sufficient to certify it , more elaborated criteria , as those elaborated in this paper , are needed in order to detect such a characteristic when quantum states exhibit a positive wigner function . in order to find criteria for the detection of quantum non - gaussian states , we follow the intuition given by hudson s theorem for pure gaussian states . we will focus on lower bounds on the values taken by the wigner function of states which belong to the gaussian convex hull @xmath27 . in this section we present our main findings as one lemma leading to two final propositions and two additional corollaries . the ` quantum non - gaussianity criteria ' , derived directly from these results , are presented at the end of the section . [ l : boundpure ] for any given pure single - mode gaussian state @xmath28 , the value of the wigner function at the origin of the phase space is bounded from below as @xmath29(0 ) \geq \frac2\pi \exp\ { -2 n ( 1+n)\ } \ : , \label{eq : boundpure}\end{aligned}\ ] ] where @xmath30 .    * proof * a generic pure single - mode gaussian state can be always written as @xmath31 , where @xmath32 and @xmath33 ( @xmath34 ) are two complex numbers . we can thus write the wigner function evaluated in zero as @xmath35&(0)=\frac{2}{\pi}\exp\left\{-2|\alpha|^2 \left[\cosh{2r } \right . \right.\nonumber \\ & \left.\left . -\cos{(2\theta+\phi)}\sinh{2r } \right ]   \right\ } . \label{eq : wiggausszero}\end{aligned}\ ] ] our goal is to minimize the value of the wigner function or equivalently to maximize the function @xmath36 a first maximization is obtained by considering @xmath37 which yields @xmath38 in the last equation we introduced the displacement and squeezing photon numbers , @xmath39 and @xmath40 , and we used the formula @xmath41 . note that these two parameters obey @xmath42 where @xmath43 is the average photon number of the state @xmath44 . we can thus express the right hand side ( rhs ) of eq . ( [ eq : max1 ] ) in terms of @xmath43 and @xmath45 , obtaining @xmath46 for a given average photon number @xmath43 , the above function is maximized with regard to the parameter @xmath45 by choosing @xmath47 and obtaining @xmath48 this leads to @xmath49(0 ) \geq \frac 2\pi \exp\left\{-2n(1+n)\right\ } .\end{aligned}\ ] ] @xmath50    by looking at the proof , we remark that the bound obtained is tight : given a fixed energy @xmath43 , by choosing the phases according to condition ( [ eq : phases ] ) and the squeezing energy according to ( [ eq : optsq ] ) , it is always possible to find a family of pure gaussian states saturating the inequality . in particular the maximization obtained via condition ( [ eq : phases ] ) simply corresponds , at fixed @xmath51 and @xmath45 , to displace the state along the direction of the squeezed quadrature . the condition ( [ eq : optsq ] ) shows that for small values of @xmath43 , the minimum of the wigner function is obtained by using the energy in displacement , while for larger values of @xmath43 , the optimal squeezing fraction @xmath45 tends to an asymptotic value @xmath52 . + let us now generalize the bound obtained to a generic convex mixture of gaussian states . [ p : boundmix ] for any single - mode quantum state @xmath3 which belongs to the gaussian convex hull @xmath27 , the value of the wigner function at the origin is bounded by @xmath6(0 ) \geq \frac2\pi \exp\ { -2 \bar{n } ( 1+\bar{n})\ } \label{eq : boundmix}\end{aligned}\ ] ] where @xmath53 $ ] .    * proof * the multi - index @xmath54 , which labels every gaussian state in the convex mixture @xmath55 , contains the information about the squeezing @xmath56 and displacement @xmath57 . we can then equivalently consider as variables @xmath58 . by exploiting the linearity property of the wigner function we obtain @xmath6(0 ) & = \int d{{\boldsymbol \lambda}}\ : p({{\boldsymbol \lambda } } ) w[|\psi_{\sf g}({{\boldsymbol \lambda}})\rangle\langle\psi_{\sf g}({{\boldsymbol \lambda}})|](0 ) \nonumber \\ & \geq \frac 2\pi   \int d{{\boldsymbol \lambda}}\ : p({{\boldsymbol \lambda } } ) \exp \ { -2n(1+n ) \ } \ : , \label{eq : integral}\end{aligned}\ ] ] where inequality ( [ eq : boundpure ] ) has been used . by defining @xmath59 which is a valid probability distribution with respect to the variable @xmath43 , eq . ( [ eq : integral ] ) becomes@xmath6(0 ) \geq \frac 2\pi \int_0^{\infty } dn \ : \widetilde{p}(n ) \ :   \exp \ { -2n(1+n)\}.\end{aligned}\ ] ] studying the second derivative of @xmath60 we conclude that the function is _ convex _ in the whole _ physical _ region ( _ i.e. _ @xmath61 ) . as a consequence , @xmath62 where @xmath63 $ ] . from the last inequality we obtain straightforwardly the thesis @xmath6(0 ) \geq \frac 2\pi \exp\ { -2 \bar{n } ( 1+\bar{n } ) \ } \:.\end{aligned}\ ] ] @xmath50    the following proposition generalizes the bound obtained above . [ p : bound2mix ] for any single - mode quantum state @xmath64 , and for any given gaussian map @xmath65 ( or alternatively a convex mixture thereof ) , the following inequality holds @xmath66(0 ) \geq \frac2\pi \exp\ { -2 \bar{n}_\mathcal{e } ( 1+\bar{n}_\mathcal{e})\}\ : , \label{eq : bound2mix}\end{aligned}\ ] ] where @xmath67 $ ] .    * proof * given a quantum state @xmath3 which can be expressed as a mixture of gaussian state , and a gaussian map @xmath65 ( or a convex mixture thereof ) , the output state @xmath68 can still be expressed as a mixture of gaussian states . as a consequence we can apply to the state @xmath69 the result in proposition [ p : boundmix ] , obtaining the thesis . @xmath50 +    proposition [ p : bound2mix ] leads to two corollaries that will be used in the rest of the paper .    for any single - mode quantum state @xmath64 , the following inequality holds @xmath6(\beta ) \geq \frac2\pi \exp\ { -2 \bar{n}_\beta ( 1+\bar{n}_\beta)\}\ : , \:\ : \forall \beta \in \mathbbm{c } \ : , \label{eq : bounddisp}\end{aligned}\ ] ] where @xmath70 $ ] .    * proof * the proof is straightforward from proposition [ p : bound2mix ] with the gaussian map @xmath71 . we also use the property of the wigner function @xmath72(\beta ) = w[d^\dag(\beta)\varrho d(\beta)](0 ) \:,\ ] ] and @xmath73 . @xmath50    for any single - mode quantum state @xmath3 belonging to the gaussian convex hull @xmath27 , the following inequality holds @xmath6(0 ) \geq \max_{\xi \in \mathbbm c } \left ( \frac2\pi \exp\ { -2 \bar{n}_\xi ( 1+\bar{n}_\xi)\ } \right)\ : ,   \label{eq : boundsq}\end{aligned}\ ] ] where @xmath74 $ ] .    * proof * the proof follows from proposition [ p : bound2mix ] , by considering the gaussian map @xmath75 . moreover , since the value of the wigner function at the origin is invariant under any squeezing operation , _ i.e. _ @xmath76(0 ) = w[\varrho](0 ) \:,\end{aligned}\ ] ] one can maximize the rhs of inequality ( [ eq : bound2mix ] ) with regard to the squeezing parameter @xmath56 . @xmath50 +    the violation of any of the inequalities presented in the last two propositions and two corollaries provides a _ sufficient _ condition to conclude that a state is quantum non - gaussian . we formalize this by re - expressing the previous results in the form of two criteria for the detection of quantum non - gaussianity . [ c : uno ] let us consider a quantum state @xmath3 and define the quantity @xmath77 = w[\varrho](0 ) - \frac2\pi \exp\{-2 \bar{n}(\bar{n}+1)\ } \ : . \label{eq : delta1}\end{aligned}\ ] ] then , @xmath77 < 0   \:\ : \rightarrow\:\ :   \varrho \notin \mathcal{g } , \nonumber \end{aligned}\ ] ] that is , @xmath3 is quantum non - gaussian . [ c : due ] let us consider a quantum state @xmath3 , a gaussian map @xmath65 ( or a convex mixture thereof ) , and define the quantity @xmath78 = w[\mathcal{e}_{\sf g}(\varrho)](0 ) - \frac2\pi \exp\{-2 \bar{n}_\mathcal{e}(\bar{n}_\mathcal{e}+1)\}\ : . \label{eq : delta2}\end{aligned}\ ] ] then , @xmath79 < 0   \rightarrow \ : \varrho \notin \mathcal{g}. \nonumber \ ] ]    typically , criterion [ c : uno ] can be useful to detect quantum non - gaussianity of phase - invariant states having the minimum of the wigner function at the origin of phase space . on the other hand , criterion [ c : due ] is of broader applicability . to give two paradigmatic examples , the latter criterion can be useful if : ( i ) the minimum of the wigner function is far from the origin , so that one may be able to violate inequality ( [ eq : bounddisp ] ) by considering displacement operations ; ( ii ) the state is not phase - invariant and presents some squeezing , and thus one may be able to violate inequality ( [ eq : boundsq ] ) by using single - mode squeezing operations . in this section we test the effectiveness of our criteria , by applying them to typical quantum states that are of relevance to the quantum optics community . we shall consider pure , non - gaussian states evolving in a lossy channel , and test their quantum non - gaussianity after such evolution . specifically , we focus on the family of quantum channels associated to the markovian master equation @xmath80 the resulting time evolution , characterized by the parameter @xmath81 , models both the incoherent loss of photons in a dissipative zero temperature environment , and inefficient detectors with an efficiency parameter @xmath82 . the evolved state @xmath83 can be equivalently derived by considering the action of a beam splitter with reflectivity @xmath84 , which couples the system to an ancillary mode prepared in a vacuum state . the corresponding average photon number reads @xmath85 = ( 1-\epsilon)\ : \bar{n}_0 \:,\end{aligned}\ ] ] where @xmath86 $ ] is the initial average photon number . + it is well known that , for @xmath87 ( _ i.e. _ for detector efficiencies @xmath88 ) , no negativity of the wigner function can be observed . we will focus then on the violation of our criteria for larger values of @xmath84 , which ensures that the evolved states have a positive wigner function . + notice that the quantum map @xmath89 is a gaussian map . as a consequence , by combining the divisibility property of the map ( inherited from the markovian structure of eq .  ) and criterion [ c : due ] , if a violation is observed for a given loss parameter @xmath90 , then the state is quantum non - gaussian for any lower value @xmath91 @xcite . for this reason we will focus on the maximum values of the loss parameter @xmath84 for which a violation of the bounds is observed , _ i.e. _ @xmath92 & = \max\ { \epsilon \ : : \delta_1[{\mathcal{e}_\epsilon}(\varrho ) ] \leq 0 \ } \ : , \\ { \epsilon_{\sf max}^{(2)}}[\varrho ] & = \max\ { \epsilon \ : : \exists\mathcal{e}_{\sf g}\text { s.t . } \ \delta_2[{\mathcal{e}_\epsilon}(\varrho),\mathcal{e}_{\sf g } ] \leq 0 \ } \:.\end{aligned}\ ] ] + in what follows , we start by focusing on criterion [ c : uno ] , and thus we will look for negative values of the non - gaussianity indicator @xmath93 $ ] defined in eq . ( [ eq : delta1 ] ) . we will consider different families of states , namely fock states , photon - added coherent states and photon - subtracted squeezed states . in section [ s : violation2 ] , we will study how to improve the results obtained , by considering the second criterion and thus by studying the non - gaussianity indicator @xmath94 $ ] . let us start by considering fock states @xmath95 , that is the eigenstates of the number operator : @xmath96 . a fock state evolved in a lossy channel can be written as a mixture of fock states as @xmath97 with @xmath98 we recall here that the wigner function at the origin is proportional to the expectation value of the parity operator @xmath99 , that is @xmath6 ( 0 )   = \frac2\pi { { \rm tr}}[\varrho \pi ] = \frac2\pi \left(p_{\sf even } - p_{\sf odd } \right)\ : , \label{eq : evenodd}\end{aligned}\ ] ] where @xmath100 ( @xmath101 ) represents the probability of detecting an even ( odd ) number of photons . by using eq . ( [ eq : fockev ] ) one obtains @xmath102(0 ) = \frac2\pi ( 2\epsilon -1)^m \:,\end{aligned}\ ] ] and thus the non - gaussianity indicator reads @xmath103 =   \frac2\pi\left\ { ( 2\epsilon -1)^m - e^{-2(1-\epsilon ) m [ ( 1-\epsilon ) m + 1 ] } \right\ } \:.\end{aligned}\ ] ]    $ ] for the first three fock states : red - dotted line : @xmath104 ; green - dashed line : @xmath105 ; blue - solid line : @xmath106 . + ( right ) maximum value of the noise parameter @xmath107 such that the bound ( [ eq : boundmix ] ) is violated for the state @xmath108 , as a function of the fock number @xmath109 . [ f : fock1],title="fig : " ] $ ] for the first three fock states : red - dotted line : @xmath104 ; green - dashed line : @xmath105 ; blue - solid line : @xmath106 . + ( right ) maximum value of the noise parameter @xmath107 such that the bound ( [ eq : boundmix ] ) is violated for the state @xmath108 , as a function of the fock number @xmath109 . [ f : fock1],title="fig : " ]    the behavior of @xmath110 $ ] as a function of @xmath84 for the first three fock states is plotted in fig . [ f : fock1 ] ( left panel ) . one can observe that the criterion works really well for the fock state @xmath111 , which is proven to be quantum non - gaussian for all values of @xmath112 . for the fock states @xmath113 and @xmath114 , a non monotonous behavior of @xmath115 is observed as a function of the loss parameter . still , negative values of the non - gaussian indicator are observed in the region of interest @xmath116 . however , the maximum value of the noise parameter @xmath117 decreases monotonically as a function of @xmath109 , as shown in fig . [ f : fock1 ] ( right panel ) . by increasing the fock number @xmath109 , it settles to the asymptotic value @xmath118 . as one would expect by looking at the bound in eq . ( [ eq : boundmix ] ) , for high values of the average photon number , the criterion becomes practically equivalent to the detection of negativity of the wigner function , and thus the maximum noise corresponds to @xmath119 . a _ photon - added coherent _ ( pac ) state is defined as @xmath120 the operation of photon - addition has been implemented in different contexts @xcite , and in particular non - gaussianity and non - classicality of pac states have been investigated in @xcite . + being the non - gaussianity indicator @xmath93 $ ] phase insensitive , we can consider @xmath121 without loss of generality . the average photon number can be easily calculated obtaining @xmath122 while its wigner function reads @xmath123 ( \lambda)&=\frac{2}{\pi}\frac{e^{-2(\alpha-\lambda)(\alpha-\lambda^*)}}{1+\alpha^2}\times\\ & \left(-1+\alpha^2 + 4|\lambda|^2 - 2\alpha(\lambda+\lambda^*)\right)\;.\end{aligned}\ ] ] the wigner function of the state after the loss channel @xmath124 can be evaluated by means of the formula @xmath125(\lambda)=\int \!\!d^2 \lambda ' k_\epsilon(\lambda , \lambda')w[\varrho](\lambda^\prime)\ ; , \label{eq : wigev}\end{aligned}\ ] ] where @xmath126    $ ] for pac states as a function of @xmath84 and for different values of @xmath57 : red - dotted line : @xmath127 ; green - dashed line : @xmath128 ; blue - solid line : @xmath129 . + ( right ) maximum value of the noise parameter @xmath107 such that the bound ( [ eq : boundmix ] ) is violated for the state @xmath124 , as a function of the parameter @xmath57 . [ f : pacs1],title="fig : " ] $ ] for pac states as a function of @xmath84 and for different values of @xmath57 : red - dotted line : @xmath127 ; green - dashed line : @xmath128 ; blue - solid line : @xmath129 . + ( right ) maximum value of the noise parameter @xmath107 such that the bound ( [ eq : boundmix ] ) is violated for the state @xmath124 , as a function of the parameter @xmath57 . [ f : pacs1],title="fig : " ]    the non - gaussianity indicator @xmath130 can then be straightforwardly evaluated and is plotted in fig . [ f : pacs1 ] ( left ) as a function of @xmath84 for different values of @xmath57 . we note that negative values of the indicator can be observed in an interval for the noise parameter @xmath84 , which decreases with the increase of @xmath57 . we can explain this feature by noting that , as @xmath57 decreases , the pac state approaches the fock state @xmath131 : as a consequence its quantum non - gaussianity can be more easily detected via criterion [ c : uno ] , in particular due to the minimum value of the wigner function approaching the origin of the phase space . we plotted in fig . [ f : pacs1 ] ( right ) the maximum value @xmath117 at which the violation of the bound is observed as a function of @xmath57 . similarly to fock states , we observe that by increasing the energy this value tends to the asymptotic value @xmath118 . let us consider now another important class of non - gaussian states that can be engineered with current technology . the _ photon - subtracted squeezed _ ( pss ) states are defined as @xmath132 for low values of squeezing , these states approximate the _ schrdinger kitten _ states , that is , superpositions of coherent states @xmath133 with opposite phase and small amplitude ( @xmath134 ) @xcite . the generation of this kind of states has been demonstrated experimentally @xcite , and it relies on performing conditional photon number measurements . + without loss of generality we shall consider a real squeezing parameter @xmath135 ; the corresponding average photon number of a pss state reads @xmath136 while its wigner function is @xmath137(\lambda ) & = - \frac2\pi e^{-2|\lambda|^2 \cosh{2 r } +   ( \lambda^2 + \lambda^{*2 } ) \sinh{2 r } } \times \nonumber \\ & \left[1 - 4|\lambda|^2 \cosh{2 r } + 2 ( \lambda^2 + \lambda^{*2 } ) \sinh{2 r }   \right ] \:.\end{aligned}\ ] ] as for the pac states , the wigner function of the evolved state can be evaluated by means of eq . ( [ eq : wigev ] ) and the non - gaussianity indicator @xmath138 $ ] can be evaluated accordingly . its behavior as a function of @xmath84 and for different values of the squeezing factor @xmath139 is plotted in fig . [ f : psss1 ] ( left ) . $ ] for pss states as a function of @xmath84 and for different values of @xmath139 : red - dotted line : @xmath140 ; green - dashed line : @xmath141 ; blue - solid line : @xmath142 . + ( right ) maximum value of the noise parameter @xmath107 such that the bound ( [ eq : boundmix ] ) is violated for the state @xmath143 , as a function of the initial squeezing parameter @xmath139 . [ f : psss1],title="fig : " ] $ ] for pss states as a function of @xmath84 and for different values of @xmath139 : red - dotted line : @xmath140 ; green - dashed line : @xmath141 ; blue - solid line : @xmath142 . + ( right ) maximum value of the noise parameter @xmath107 such that the bound ( [ eq : boundmix ] ) is violated for the state @xmath143 , as a function of the initial squeezing parameter @xmath139 . [ f : psss1],title="fig : " ]    in the right panel of fig . [ f : psss1 ] we plot the maximum noise parameter @xmath117 as a function of the squeezing parameter @xmath139 , observing the same behavior obtained for fock and pac states : the value of @xmath117 decreases monotonically with the energy of the state , approaching the asymptotic value @xmath118 .      we will now show how the second criterion , which is based on the violation of the inequality ( [ eq : bound2mix ] ) , can be exploited in order to improve the results shown in the previous section . since in this case one can optimize the procedure over an additional gaussian channel , in general one has @xmath144 . the simplest gaussian maps that one can consider are displacement and squeezing operations ; correspondingly we are going to seek violation of the bounds described by eqs . ( [ eq : bounddisp ] ) and ( [ eq : boundsq ] ) . as anticipated in sec . [ s : criteria ] , these new criteria are useful for states which are not phase invariant : the paradigmatic examples are states _ displaced _ in the phase - space , that is , having the minimum of the wigner function outside the origin , or states that exhibit squeezing in a certain quadrature . due to this fact , the bounds based on eqs . ( [ eq : bounddisp ] ) and ( [ eq : boundsq ] ) can not help in optimizing the results we obtained for fock states . we will focus then on the other classes of states we introduced , that is pac and pss states .     for @xmath145 . the minimum of the wigner function is not at the origin of the phase - space , and the state has non - zero first moments . [ f : wigpac0 ] ]      by looking at the pac state wigner function in fig . [ f : wigpac0 ] , one observes that its minimum is not at the origin of the phase space . moreover , these states have non - zero first moments , implying that one can decrease their average photon number by applying an appropriate displacement . both observations suggest that it is possible to decrease the value the quantum non - gaussianity indicator defined in eq . ( [ eq : delta2 ] ) , @xmath146 \ : , \label{eq : deltapac}\end{aligned}\ ] ] by means of a displacement operation @xmath147 . to evaluate @xmath148 according to eq . ( [ eq : deltapac ] ) one has simply to evaluate the wigner function of the state @xmath149 in a displaced point in the phase space , _ i.e. _ @xmath150(-\beta)$ ] , and its average photon number @xmath151 where @xmath152 , and for @xmath153 , @xmath154 our goal is then to minimize @xmath148 over the possible displacement parameters @xmath155 . +     as a function of the additional displacement parameter @xmath155 , for @xmath156 and for different values of the initial parameter @xmath57 : : red - dotted line : @xmath127 ; green - dashed line : @xmath128 ; blue - solid line : @xmath129 . + ( right ) optimized non - gaussianity indicator @xmath157 as a function of @xmath84 and for different values of @xmath57 , where the displacement parameter @xmath158 has been chosen as in eq . ( [ eq : betaopt ] ) : @xmath127 ; green - dashed line : @xmath128 ; blue - solid line : @xmath129 . [ f : pacs2],title="fig : " ]   as a function of the additional displacement parameter @xmath155 , for @xmath156 and for different values of the initial parameter @xmath57 : : red - dotted line : @xmath127 ; green - dashed line : @xmath128 ; blue - solid line : @xmath129 . + ( right ) optimized non - gaussianity indicator @xmath157 as a function of @xmath84 and for different values of @xmath57 , where the displacement parameter @xmath158 has been chosen as in eq . ( [ eq : betaopt ] ) : @xmath127 ; green - dashed line : @xmath128 ; blue - solid line : @xmath129 . [ f : pacs2],title="fig : " ]    in fig . [ f : pacs2 ] ( left ) we plot @xmath148 as a function of @xmath155 for different values of the coherent state parameter @xmath57 and for @xmath156 . we observe that , while for @xmath159 the bound is not always violated , it is possible to find values such that @xmath160 and thus prove that the state is quantum non - gaussian . unfortunately the optimal value @xmath158 , which minimizes @xmath148 , can not be obtained analytically . however we observed that for large values of @xmath84 and for @xmath161 one can approximate it as @xmath162 the behavior of @xmath157 as a function of @xmath84 shown in fig . [ f : pacs2 ] ( right ) , for different values of @xmath57 and fixing @xmath158 as in eq .  . if we compare this with fig . [ f : pacs1 ] , not only we observe an improvement in our capacity to witness quantum non - gaussianity for these states , but we also see that @xmath157 remains negative for all values of @xmath84 . indeed , numerical investigations seem to suggest that @xmath163 for all the possible values of @xmath57 : we indeed conjecture that any initial pac state remains quantum non - gaussian during the lossy evolution induced by eq .  , and that this feature can be captured by our second criterion . however , as one can observe from fig . [ f : pacs2 ] ( right ) , the non - gaussianity indicator approaches zero quite fast with both @xmath57 and @xmath84 , and thus it may be more challenging to detect its negativity in an actual experiment for states with a high average photon number and for large losses .       for @xmath141 . the minimum of the wigner function is at the origin of the phase - space , and the state exhibits squeezing in one of the quadratures . [ f : wigpss0 ] ]    like pac states inherit a displacement in phase space from the initial coherent states , pss states inherit squeezing , as we can observe by looking at the wigner function in fig . [ f : wigpss0 ] . this motivates us to make use of corollary [ c : due ] , and thus optimize the non - gaussianity indicator in eq . ( [ eq : delta2 ] ) as @xmath164 \ : , \label{eq : deltapss}\end{aligned}\ ] ] that is by considering an additional squeezing operation @xmath165 on the evolved state @xmath166 . as pointed out in the proof of inequality ( [ eq : boundsq ] ) , the wigner function at the origin is invariant under squeezing operations . hence , the optimal value @xmath167 that minimizes @xmath168 coincides with the value which minimizes the average photon number of @xmath169 , @xmath170 + \nu_s^2\,,\end{aligned}\ ] ] where @xmath171 , @xmath172 and for an initial pss state ( with a real squeezing parameter @xmath139 ) , @xmath173     as a function of the additional squeezing parameter @xmath174 , for @xmath175 and for different values of the initial parameter @xmath139 : red - dotted line : @xmath140 ; green - dashed line : @xmath141 ; blue - solid line : @xmath142 . + ( right ) optimized non - gaussianity indicator @xmath176 as a function of @xmath84 and for different values of @xmath139 , where the squeezing parameter is given by : red - dotted line : @xmath140 ; green - dashed line : @xmath141 ; blue - solid line : @xmath142 . [ f : psss2],title="fig : " ]   as a function of the additional squeezing parameter @xmath174 , for @xmath175 and for different values of the initial parameter @xmath139 : red - dotted line : @xmath140 ; green - dashed line : @xmath141 ; blue - solid line : @xmath142 . + ( right ) optimized non - gaussianity indicator @xmath176 as a function of @xmath84 and for different values of @xmath139 , where the squeezing parameter is given by : red - dotted line : @xmath140 ; green - dashed line : @xmath141 ; blue - solid line : @xmath142 . [ f : psss2],title="fig : " ]    the behavior of @xmath168 as a function of the additional squeezing @xmath174 is plotted in fig . [ f : psss2 ] . as we observed in the previous case , the optimised criterion works in cases where the bound ( [ eq : boundmix ] ) ( corresponding to @xmath177 ) was not violated . + moreover the optimal squeezing value can be evaluated analytically , yielding @xmath178     and @xmath107 , obtained respectively by means of the optimized and not - optimized criteria , for the state @xmath143 , as a function of the initial squeezing parameter @xmath139 . red - dotted line : @xmath179 ; blue - solid line : @xmath117 . [ f : psss3 ] ]    the optimized quantum non - gaussianity indicator @xmath176 is plotted in fig . [ f : psss2 ] ( right ) , where we observe that negative values are obtained for large values of losses . however , while for pac states we had evidence that the maximum value of losses is @xmath163 for all the possible initial states , this is no longer true for pss states . the behavior of @xmath179 as a function of @xmath139 is plotted in fig . [ f : psss3 ] , together with the previously obtained @xmath117 . we can notice the big improvement in our detection capability , obtained by exploiting corollary  [ c : due ] ; however for large values of @xmath139 we still observe that @xmath179 decreases towards the same limiting value @xmath180 . moreover , as it can be observed in fig . [ f : psss2 ] ( right ) , the indicator @xmath181 approaches zero by increasing @xmath139 and @xmath84 , and thus also in this case it can become challenging to witness quantum non - gaussianity with our methods , in experiments with large values of the initial squeezing @xmath139 and large losses . we have presented a set of criteria to detect quantum non - gaussian states , that is , states that can not be expressed as mixtures of gaussian states . the first criterion is based on seeking the violation of a lower bound for the values that the wigner function can take at the origin , depending only on the average photon number of the state . to verify the effectiveness of the criterion , we considered the evolution of non - gaussian pure states in a lossy gaussian channel , looking for the maximum value of the noise where such bound is violated . we observed that the criterion works well , detecting quantum non - gaussianity in the non - trivial region of the noise parameters where no negativity of the wigner function can be observed . we have also shown how the criterion can be generalized and improved , by optimising over additional gaussian operations applied to the states of interest . notice that in a possible experimental implementation one does not need to perform such additional gaussian operations , such as displacement or squeezing , in the actual experiment . indeed , it suffices to use the data obtained on the state itself , and then apply suitable post - processing to evaluate the optimized non - gaussianity indicator . our criterion , which expresses a sufficient condition for quantum non - gaussianity , shares some similarities with hudson s theorem for pure gaussian states , in the sense that it establishes a relationship between the concept of gaussianity ( combined with classical mixing ) , and the possible values that a wigner function can take . the successful implementation of our criteria corresponds to the measurement of the wigner function at the origin of the phase space which , in turn , corresponds to the ( photon ) parity of the state under investigation . this may be obtained with current technology by direct parity measurement @xcite , or by reconstruction of the photon distribution either by tomographic reconstruction or by the on / off method @xcite . when the criterion is satisfied , one can confirm that the quantum state at disposal has been generated by means of a highly non - linear process , even in the cases where , perhaps due to inefficient detectors or other types of noise , negativity of the wigner function can not be detected . mgg , tt and msk thank radim filip for discussions . mgap thanks vittorio giovannetti for discussions . mgg acknowledges support from uk epsrc ( ep / i026436/1 ) . t.t . and m.s.k . acknowledge support from the nprp 4- 426 554 - 1 - 084 from qatar national research fund . so and mgap acknowledge support from miur ( firb `` lichis '' no . rbfr10yq3h ) . 99 e. p. wigner , phys . rev . * 40 * , 749 ( 1932 ) r. glauber , phys . rev . * 131 * , 2766 ( 1963 ) . e. c. g. sudarshan , phys . * 10 * , 277 ( 1963 ) . c. t. lee , phys . a * 44 * , r2775 ( 1991 ) . c. t. lee , phys . a * 45 * , 6586 ( 1992 ) . arvind , n. mukunda , r. simon , phys.rev . a * 56 * , 5042 ( 1997 ) . arvind , n. mukunda , r. simon , j. phys a * 31 * : 565 ( 1998 ) . m. a. marchiolli , v. s. bagnato , y. guimaraes , b. baseia , phys . a * 279 * , 294 ( 2001 ) . m. g. a. paris , phys . a * 289 * , 167 ( 2001 ) . dodonov , j. opt . b * 4 * , r1 , ( 2002 ) . a. kenfack , k. zyczkowski , j. opt . b * 6 * , 396 ( 2004 ) . e. v. shchukin , w. vogel , phys . rev . a * 72 * , 043808 ( 2005 ) . t. kiesel , w. vogel , v. parigi , a. zavatta , m. bellini , phys . rev . a * 78 * , 021804 r ( 2008 ) . w. vogel , phys . lett . * 100 * , 013605 ( 2008 ) . r. simon , phys . . lett . * 84 * , 2726 ( 2000 ) . lu - ming duan , g. giedke , j.i . cirac , p. zoller , phys . * 84 * , 2722 ( 2000 ) . p. marian , t. a. marian , h. scutaru , phys . lett . * 88 * , 153601 ( 2002 ) . p. giorda , m. g. a. paris , phys . lett . * 105 * , 020503 ( 2010 ) ; 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we introduce a family of criteria to detect quantum non - gaussian states of a harmonic oscillator , that is , quantum states that can not be expressed as a convex mixture of gaussian states . 
 in particular we prove that , for convex mixtures of gaussian states , the value of the wigner function at the origin of phase space is bounded from below by a non - zero positive quantity , which is a function only of the average number of excitations ( photons ) of the state . as a consequence , 
 if this bound is violated then the quantum state must be quantum non - gaussian . 
 we show that this criterion can be further generalized by considering additional gaussian operations on the state under examination . 
 we then apply these criteria to various non - gaussian states evolving in a noisy gaussian channel , proving that the bounds are violated for high values of losses , and thus also for states characterized by a positive wigner function .