Sufficient condition for a quantum state to be genuinely quantum non-Gaussian

We show that the expectation value of the operator  ˆ ≡ exp ( − c x ˆ 2 ) + exp ( − c p ˆ 2 ) defined by the position and momentum operators x ˆ and p ˆ with a positive parameter c can serve as a tool to identify quantum non-Gaussian states, that is states that cannot be represented as a mixture of Gaussian states. Our condition can be readily tested employing a highly efficient homodyne detection which unlike quantum-state tomography requires the measurements of only two orthogonal quadratures. We demonstrate that our method is even able to detect quantum non-Gaussian states with positive–definite Wigner functions. This situation cannot be addressed in terms of the negativity of the phase-space distribution. Moreover, we demonstrate that our condition can characterize quantum non-Gaussianity for the class of superposition states consisting of a vacuum and integer multiples of four photons under more than 50 % signal attenuation.


Introduction
Gaussian states play a major role in the field of quantum information with continuous variables [1]. They are not only readily accessible in experiment, for example in the form of squeezed states, but also allow a compact description in theory [2]. On the other hand, non-Gaussian states and non-Gaussian operations are known to provide an advantage in quantum information processing such as the distillation of quantum entanglement [3] and enhanced performance in quantum teleportation [4]. They are even a critical requirement for some quantum tasks such as universal quantum computation [5], nonlocality tests [6] and entanglement distillation [7].
In this regard, the problem of distinguishing between Gaussian and non-Gaussian states has recently attracted considerable attention, in particular, detecting a state of genuine quantum non-Gaussian character that cannot be represented by a mixture of Gaussian states. Quantum non-Gaussianity is in a sense a stronger notion than nonclassicality as the latter refers to states that cannot be represented by a mixture of coherent states, which are strictly a subset of the whole Gaussian states. Nonclassicality and quantum non-Gaussianity provide crucial resources for quantum information processing. For instance, it is known that every single-mode nonclassical state, when mixed with a vacuum state at a beam splitter, becomes a quantum entangled state [8]. This fact also implies that a single-mode quantum non-Gaussian state can be turned to a useful non-Gaussian entangled state by passive transformations.
A well-known criterion [9] for a pure state to be a Gaussian one is the positivity of the corresponding Wigner function [10] at any point of phase space. However, this elementary but fundamental criterion of Gaussianity is not useful in practice as the states we usually deal with are contaminated with noise and thus are mixed ones.
Instead, one may look into the statistics of photon-number distributions to obtain a practically useful criterion on quantum non-Gaussianity. In particular, Filip and Mišta, established [11] a Gaussian bound P G (n) on the photon-number probability for each n=1, 2, K. Provided a given state shows a probability P(n) larger than the Gaussian bound, it demonstrates [11] that the state cannot be expressed as a mixture of Gaussian states.
This approach was further developed to identify the non-Gaussianity of a noisy single-photon state including multi-mode contributions to the signal as well, which requires detection of two output-photons without resolving photon numbers after a beam-splitting operation. This technique was experimentally realized to demonstrate quantum non-Gaussianity of an imperfect single-photon state possessing even a positive Wigner function [12] and to study the resilience of quantum non-Gaussianity under dissipation [13]. Moreover, [14] introduced an experimental criterion to identify quantum non-Gaussianity of multi-photon states of light.
Later, Genoni and co-workers [15] suggested another criterion based on the value of the Wigner function at the origin of phase space addressing the number-parity of the state, which was further extended to the generalized quasi-distributions including the Husimi-Q function [16].
More recently, Park et al established a Bell-type test in phase space to address nonclassicality and quantum non-Gaussianity for single-mode fields [17], while another efficient formalism was also developed to adopt the marginal distributions of the Wigner function [18].
In contrast to the previous methods, which mostly rely on the particle nature, i.e. photon-number statistics, we here present a formalism to adopt continuous observables such as the quadrature amplitudes of a light field or position and momentum of a massive particle, to identify quantum non-Gaussianity. Our method provides not only a fundamentally different approach but also a practically efficient tool to test quantum non-Gaussianity.
In particular, the homodyne detection to measure quadrature amplitudes in quantum optics, while requiring some conditions such as mode-matching with the local oscillator field and the narrow-band filtering of the fields to be measured, is highly efficient achieving efficiency around or above 90%. It can thus provide a practical advantage over photon-counting measurement with lower efficiency [15,16], e.g. the detection of negativity in phase space requiring the measurement of photon-number parity can be challenging [15].
As an illustration, we show that our condition is useful to detect quantum non-Gaussianity for the class of states A recent proposal suggested to use the generalized cat-state as a component of logical qubits dynamically protected under photon loss for universal quantum computation, together with its experimental scheme [19]. Furthermore, such a cat-state mixed with a vacuum using a beam splitter produces a two-mode state that yields a useful resource for quantum metrology, e.g. enhanced sensitivity in phase estimation [20].

In a nutshell
We first briefly describe our condition on quantum non-Gaussianity. In particular, we consider the mean value  (3) with respect to all quantum states from a set a represented by the density operator a r .
We first maximize the mean value, equation (3), over all Gaussian states, that is we establish the value of can formulate a sufficient condition to detect quantum non-Gaussianity as follows.
Theorem. Let r be the density operator of a quantum state. If there exists a constant c 0 > , for which then r describes a quantum non-Gaussian state that cannot be represented by a mixture of Gaussian states.
Here we emphasize three points: (i) the word 'sufficient' indicates that every state corresponding to a point caught between the two lines F G =F G (c) and

Measurement scheme
Next we briefly discuss questions related to an experimental test of our condition. We start by outlining a method to measure the expectation value  á ñ . Then we address the choice of c which is intimately connected to the fact that our condition is sufficient. Since  is the sum of two operators, cx exp 2 -(ˆ) and cp exp , 2  -á ñ (ˆ)ˆis the sum of the two corresponding expectation values. Thus, we do not have to measure both operators simultaneously, but it suffices to obtain cx exp 2 á -ñ (ˆ) and cp exp 2 á -ñ (ˆ) separately and finally add them. In quantum optics, homodyne detection [21] is a well-established and highly efficient technique to measure the quadrature amplitudes. They readily facilitate the evaluation of the expectation values of the two orthogonal quadratures x and p. Suppose we obtain a data set of values {x 1 , x 2 , L, x N } and {p 1 , p 2 , L, p N } from each measurement. Then, we can establish the expectation values as In matter-wave optics, the operator cx exp 2 -(ˆ) can be realized by applying a purely imaginary optical potential [22][23][24][25]. The measurement of Here n denotes the number operator, n x p 1 2 2 2 º + -(ˆˆ) . Figure 1 also indicates the importance of the choice of c. Indeed, a value 1.1  á ñ = confirms a non-Gaussian state in the range of 2c9. On the other hand, it is inconclusive if 0<c2. In this domain the underlying state can still be non-Gaussian but it is not recognized by  á ñ . Likewise, a value of 1  á ñ < does not necessarily imply a Gaussian state. Only when the value  á ñ constructed from experimental data for a fixed c is in the shaded red domain of figure 1, our condition given by equation (6) can identify a quantum non-Gaussian state. In this sense it is a sufficient condition.

Outline
Our article is organized as follows. In section 2 we introduce a geometrical interpretation of the mean value, equation (3), of the operator  defined by equation (4) as an overlap integral in phase space. We then find in section 3 exact analytical expressions for the optimal Gaussian state and the corresponding function representation rather than phase space functions. We also analytically consider the superposition of two squeezed Gaussian states and show that its mean value F sG =F sG (c) is very close to that of the exact optimal state. In section 5, we apply our condition to the class of states C n 4 n n 0 n 4 y ñ = å ñ ¥ = | | under dissipation and demonstrate that our condition is powerful in detecting those states even if the attenuation is larger than 50%. Finally, in section 6 we conclude and provide an outlook.
In order to keep our article self-contained but focused on the central ideas, we present lengthy calculations in appendices. We find in appendix A the mean value given by equation (3) for a general Gaussian state and obtain an analytical formula for the function F G =F G (c). Appendix B presents approximate but analytical results for the function F F c H H = ( ) and the optimal quantum state. We conclude in appendix D by describing the model of the dissipation of a quantum state via interaction with a reservoir at zero temperature. In appendix C we derive the asymptotic behavior of the maximum mean value F sG = F sG (c) for the coherent superposition of two squeezed Gaussian states.

Optimization of state means maximization of phase space overlap
In the preceding section we have defined the optimization problem for the operator  . We now use the Wigner function to cast the relevant expectation value into an overlap integral which provides us with a guide to the optimization. For this purpose we express the mean value, equation (3), as the overlap integral [10] between the Weyl-Wigner symbol corresponding to the operator c  ( ), equation (4), and the Wigner function W x p , r ( ) of the density operator r . This representation has the advantage to provide us with a geometrical point of view on the maximization procedure. Indeed, maximizing the mean value , which occurs when the Wigner function of a given state matches To underline this idea, we show in figure 2 contour-plots of the phase space representation x p , (11), and the Wigner function W G =W G (x, p) of an arbitrary Gaussian state. Hence, in this case the optimization problem reduces to finding the appropriate parameters of W G to maximize the overlap.

Gaussian states
In this section we pursue this phase space approach to find the lower border Motivated by these prerequisites, we start with the wave function We are thus left with only a single parameter to vary, that is the squeezing parameter s=s(c)0, which we determine by maximizing the overlap integral, equation (10), given by

After a direct integration, we obtain
f c s c s cs , 1 The symmetry of the Weyl-Wigner symbol x p , W  ( ), equation (11), with respect to the exchange of the xand p-coordinates reflects itself in equation (15) in the symmetry relation Therefore, the two optimal values s(c) and s −1 (c) give rise to the same value of F G (c).
and note the identity s c s c for any non-negative c, in complete agreement with the symmetry relation, equation (16).
In the domain c2, the two solutions s + and s − coincide, that is s ± =1, and describe the rotationally symmetric vacuum state. Indeed, for small values of c, the Weyl-Wigner symbol x p , figure 3(a), is very broad and there is no preferred orientation of the Wigner function, resulting in the fact that the vacuum state, presented in figure 3(b), is maximizing the overlap integral, equation (10).
At c=2 a bifurcation occurs and for c>2 two solutions emerge, corresponding to squeezed states in  (11) (left), and the Wigner function W G =W G (x, p) of an arbitrary Gaussian state (right) darker colors represent higher values. We maximize the overlap integral, equation (10), by varying the parameters of W G , such as its displacement, purity, orientation, and squeezing. Obviously the optimal overlap arises when W G is located at the origin and its axes are aligned with the axes of phase space. Hence, the optimization problem reduces to finding the optimal value of the width of the Gaussian. the operator  , depicted in the left column of figure 3, becomes more localized along the x-and p-axes of the phase space. In order to maximize the overlap integral, equation (10), the Wigner function W x p , needs to be concentrated along either x-or p-axis resulting in squeezed states. The case of a state squeezed in x-direction, corresponding to s + , is depicted in figure 3(e).
When we substitute the optimal squeezing parameter s ± (c) given by equation (17) This result defines the lower border in the condition equation (6) of quantum non-Gaussianity and is presented in figure 1 by the solid line. For any Gaussian state, the mean value c  á ñ ( ) has to be on, or below this line. From equation (19) we note the asymptotic behavior in the two limits c = 1 and c?1. Hence, the curve F F c G G = ( ) starts from F G =2 and first decays linearly with c. However, for large values of c it approaches unity from above decaying with 1/(2c 2 ).
We conclude by emphasizing that while we have obtained the Gaussian bound, equation (19), for single pure Gaussian states, it is actually the maximum for all incoherent mixtures In this case the corresponding mean value  º -+ -(ˆ) (ˆ) represented in phase space, together with the transition from rotational to star symmetry for small and large values of c, respectively. Here we depict the Weyl-Wigner symbol x p , (11), of the operator c  ( ), equation (4), (left column), the Gaussian Wigner function W W x p , , equation (13), of the optimal Gaussian state (middle column), and the Wigner function W W x p , H H = ( ) of the optimal non-Gaussian state (right column), for the two values c=0.5 (first row) and c=4 (second row). For small values of c the optimal state is mainly determined by the dominant maximum of W  at the origin. For larger values of c such as c=4 the Gaussian Wigner function can only accommodate one of the ridges of W  along the axes of phase space. In contrast, the optimal state represented by W H fits both.
As we have shown above and in appendix A, there exist only the two pure optimal states, s s y y ñá

Non-Gaussian states
In section 3 we have found the lower border F G =F G (c) of the condition, equation (6), corresponding to the optimal Gaussian state. In this section we calculate the upper border F F c H H = ( ) corresponding to the optimal quantum state. Central to our approach is the eigenvalue equation of the operator  , given by equation (4). Since  is hermitian the eigenstates n y ñ | corresponding to the eigenvalues λ n are complete and orthonormal, that is We start by showing that the problem of finding F H is equivalent to finding the largest eigenvalue of  . Then, we discuss (i) an exact but numerical solution of the variational problem representing  in the basis of energy eigenstates of a harmonic oscillator, and (ii) an approximate but analytical approach considering a superposition of two squeezed states. Moreover, an analysis of the asymptotic behavior of the numerical results based on the eigenvalue equation, equation (24), in position representation is presented in appendix B. We demonstrate that our approximate result is in excellent agreement with the exact one.

Largest eigenvalue determines optimal value
We employ the eigenstates n y ñ | of  together with equations (24) l of the operator  . Indeed, the eigenvalues of an hermitian operator describe all possible measurement outcomes. The largest outcome is thus equal to the largest eigenvalue of the operator, which is assumed with certainty if the considered state is the eigenstate corresponding to this eigenvalue.
In the derivation of equation (28) we have assumed that n 0 l is non-degenerate. When n 0 l is degenerate the maximum mean value is not only achieved for a single eigenstate of the operator  , but for any arbitrary mixture of all eigenstates corresponding to the largest eigenvalue n 0 l . However, for finding n 0 l , which is equivalent to the upper border F F c H H = ( ) of the condition, equation (6), it is sufficient to only obtain one of the eigenstates.

Largest eigenvalue: numerical approach via Fock states
In order to find numerically the largest eigenvalue n 0 l of the operator  given by equation (4) for m+n=2, 4, 6, K. Here Γ and 2 F 1 denote the Gamma function and the hypergeometric function [28], respectively. In the case m+n=1, 3, 5, ... the matrix element vanishes. Additionally, the elements satisfying n−m=2, 6, 10, K vanish due to the first term in equation (33).
Taking into account the first 30 000 functions u n , we have numerically obtained the maximum mean value F c H ( ) as the largest eigenvalue of the matrix mn  , equation (33). The function F F c H H = ( ) indicated by dots in figure 1 determines the upper border, which cannot be exceeded by any normalized quantum state, be it a mixed or pure one.

Near-optimal non-Gaussian states: superposition of two squeezed states
Inspired by the analysis of section 3 we now find analytically a near-optimal non-Gaussian state to achieve F c H ( ). In the case of Gaussian states we have shown in section 3 that there exist two solutions s(c) and s −1 (c) for the squeezing parameter s leading to the same maximal mean value F G (c), given by equation (19). With this idea in mind we may optimize the overlap integral, equation is the Wigner function of the single squeezed Gaussian state given by equation (13).
We note that for c 1  the optimal value F sG displays the same behavior as the optimal Gaussian in equation (20). However, for c ? 1 it decays as 1/c which is slower than the decay 1/2c 2 given by equation (20) corresponding to the single Gaussian.

Applicability of our condition to practical states
In the previous sections we have found the two borders, F G and F H , to formulate our condition, equation (6).
Here we highlight the features of our condition and illustrate its power as well as its limitations by applying our condition to several non-Gaussian states. In particular, we consider (i) the superposition of Gaussian states sG y introduced in equation (34) and (ii) the superposition y ñ a | of four coherent states, defined by equation (2) as our examples under a dissipative channel.

Superposition of Gaussian states
We emphasize that our condition is a sufficient one. Indeed, the state sG y , equation (34), given by a superposition of two squeezed states with squeezing parameters s=3 and s=1/3, is non-Gaussian. The corresponding mean value  á ñ is represented by a blue line in figure 6. For values of c  3.5 the blue curve drops Although the dots follow the solid curve rather closely the relative error δF increases rapidly for increasing c indicating that ψ sG is not an eigenfunction of  . We refer to appendix B for a more extensive analysis.
below the Gaussian one and our condition fails to observe the non-Gaussianity of this superposition. Therefore, we are able to detect quantum non-Gaussianity for this state only by using the parameter c in the range 0c  3.5.  Here we depict the two borders of our condition, F G (black line) and F H (red dots), together with the mean value f sG , equation (37), before dissipation (η=1, blue line) and after dissipation (η=0.5, orange line). The dissipation leads to a decrease in the mean value and henceforth also decreases our ability to detect quantum non-Gaussianity, however even for η=0.5 we note a positive gap between the orange and the black curves around c≅2. This gap means we are still able to detect the non-Gaussianity of the state sG y , even though half of the intensity has dissipated. This figure also displays the sufficiency of our condition: indeed, the blue curve which corresponds to the non-dissipated version of this superposition, can still be located in the Gaussian domain. For example, when c>3.5 the mean value f sG falls below the line F G . Nevertheless, the state is non-Gaussian.
One way to experimentally detect the non-Gaussianity of a given state is to obtain its Wigner function via tomography [29]. Specifically, negative values of the corresponding Wigner function immediately indicate that the state is quantum non-Gaussian. In general, however, any measurement is subject to losses 9 which reduce the ability to detect negative values of the Wigner function. Indeed, if the attenuation of a state is higher than 50%, i.e. a dissipation parameter η<0.5, it is experimentally impossible to detect any negativity via tomography or number-parity measurement. In contrast, our condition does not require η>0.5 to detect the non-Gaussianity of a state. This is illustrated in figure 6, where the blue and orange lines display the mean values corresponding to the superposition sG y with s=3, without (η=1) and with (η=0.5) dissipation, respectively. Considering losses of 1−η=0.5, we are still able to detect the non-Gaussian nature of the state, as the orange line exceeds the black curve F F c G G = ( ) in the region around c=2. The minimal value of η (corresponding to maximal losses 1−η) for which this happens is found to be around η min ≅0.401. This demonstrates that our condition works in a regime where most conventional methods fail. It is worth noting however that η min strongly depends on the state under consideration and it might also lie above 0.5.

Superposition of coherent states
Next we test our condition to detect a superposition e e e 4 1 of four coherent states with  a being a normalization constant. In figure 7(a) we show the contour plot of the corresponding Wigner function W α (x, p) for the case of α=1. This generalized Schrödinger-cat state was recently suggested as a logical qubit dynamically protected under photon loss thus suitable for fault-tolerant quantum computation [19]. In addition, it was found to be a useful resource state for enhanced quantum phase estimation [20].
For each state y ñ a | undergoing dissipation, we maximize the ratio of F c c  y y º á ñ a a a ( ) |ˆ( )| to the Gaussian bound F G (c) over the parameter 0<c10 and display the maximum value of F α /F G as a function of α in figure 7(b). For the ideal case (η=1), the quantum non-Gaussianity is successfully detected for 0.18α1.4. On the other hand, with a non-zero dissipation (0<η<1), this detection range shrinks with decreasing η. Nevertheless our condition is capable of detecting those states even below η=0.5. In particular, quantum non-Gaussianity is successfully detected around η<0.3 for certain values of α of the superposition states y ñ a | , as depicted in figure 7(c).

Discussion
Note that the two discussed examples belong to the class of states n 4 y ñ | , equation (1), consisting of a superposition of the vacuum and an integer multiple of four photons. We have numerically tested other states as well and found that our condition is able to detect quantum non-Gaussianity broadly for this class.
Naturally, the efficiency of our condition is crucially determined by the state of interest. Given another class of states one might think of variations of the operator  , e.g. changing the functional form from Gaussian to non-Gaussian including polynomials, to build up the condition which is then particularly suited for detecting the non-Gaussianity of those states, but less useful for other ones. We may not anticipate that there is a single best operator for all states, however a more extensive analysis goes beyond the scope of the present article.

Conclusions and outlook
In this article we have proposed a sufficient condition to distinguish quantum non-Gaussian states from a mixture of Gaussian ones. Our condition relies on the expectation value of an operator  consisting of the sum of two exponentials containing the squares of the appropriately scaled coordinate and momentum operators which are both multiplied by a parameter c.
In classical physics it is possible to maximize the expectation value of x 2 and p 2 simultaneously by a phase space distribution function representing the underlying state which consists of the product of two delta functions centered around the origin. This choice leads to the value 2 cl  á ñ = . However, in quantum mechanics a state has to take up a non-zero area in phase space. This fundamental restriction leads to a suppression of  á ñ below the classical value 2 cl  á ñ = . Indeed, for small values of c the operator  reduces to the difference of the classical result and the product of c with the familiar Hamiltonian of a harmonic oscillator. Due to this difference, the optimal state maximizing  á ñ is the lowest state of the oscillator, that is the vacuum state.
However, for larger values of c the Weyl-Wigner symbol is concentrated mainly along the x-and p-axis in phase space. As a result, the optimal state when expressed by the corresponding Wigner function also has to reflect this symmetry. Indeed, we have found a star symmetry of these optimal states combined with a narrow peak at the origin. The latter contribution is the consequence of the sum of the operators cx exp 2 -(ˆ) and cp exp 2 -(ˆ) leading to a maximum in the Weyl-Wigner symbol of  at the origin. In this sense it is a remnant of the optimal state, i.e. the vacuum state, for small values of c.
It is also interesting to note that there is a transition of the Weyl-Wigner symbol of  from a rotational symmetry, at least in the neighborhood of the origin, for small values of c, to a star structure along the axes of phase space for larger values of c. As a result, the decay of the expectation value of  changes from one linear in c to one linear in 1/c. Moreover, for Gaussian states a bifurcation occurs since a single Gaussian can accommodate either the x-or the p-axis of phase space, but not both. It is for this reason that a superposition of two appropriately squeezed Gaussian states can provide an excellent approximation to the optimal state. In particular, it also leads to an interference peak at the origin capturing the overlap of the two Gaussians in the Weyl-Wigner symbol of  at the origin. Despite the close similarity between ψ sG and the exact wave function H y solving the optimization problem, there are differences and ψ sG is not a rigorous eigenfunction of  . We have applied our condition to the class of states n 4 y ñ | and demonstrated its power by detecting those states even under strong dissipation of the signal below η=0.5. Our condition can be efficiently tested, e.g. using highly efficient homodyne detection in quantum optics, which readily achieves a detection efficiency above 90%. Our current approach can be further extended to a different functional form of the test operator  to be adapted to other class of non-Gaussian states, which will be investigated in future.
We conclude by noting that the eigenvalue problem of  is fundamentally different from the familiar Schrödinger equation since for large values of c the operator cp exp 2 -(ˆ) involves infinitely many derivatives larger than two. Indeed, in this case the eigenvalue equation takes the form of an integral equation and we have presented here approximate but analytical expressions for the largest eigenvalue and the corresponding eigenfunction. However, it is interesting to study in this formulation the behavior of all eigenvalues as a function of c. Indeed, for large values of c they display a rather curious behavior: the largest eigenvalue separates from all other ones which asymptotically cluster around unity. Unfortunately, this analysis goes beyond the scope of the present article and has to be postponed to a future publication [30].
of the operator c  ( ), equation (4) is the vector of the phase space coordinates, that is of the position x and the momentum p.
With the help of the relation for a n-dimensional Gaussian integral with a matrix  and a vector b, we can perform the integration in equation

A.2. Variation of Gaussian
In the next steps we obtain the maximum value of the function f G , given by equation (A.13), by varying the five parameters x 0 , p 0 , σ 11 , σ 12 , and det s.

Displacements.
The dependence of f G on the displacement parameters where H  is a normalization constant. When we compare F H given by equation (B.4) to the corresponding expression, equation (40), for the non-Gaussian state ψ sG we note that for c  ¥ both decay with the inverse of c. However, they differ by their decay constants. Whereas for F sG the slope is determined by unity, for F H it is given by π/2. This difference stands out most clearly in figure B1 where we display the different scalings of the functions F F c sG sG = ( ) and F F c H H = ( ) together with the numerical result. Moreover, this difference corresponding to the relative error δF shown in figure 4 indicates that ψ sG is not an eigenfunction of  . This feature is apparent from figure B2 where we show the corresponding wave functions in position representation. We note slight differences between ψ sG and the optimal eigenfunction H y of  . Figure B1. Comparison between the asymptotic behavior of the optimal values F sG (solid blue line) and F H (red dots) of the expectation value  á ñ for the superposition of two Gaussians and the optimal wave function, respectively. To emphasize the different asymptotic scalings of F sG and F H we display the function c F c  Figure B2. Comparison between the wave functions ψ sG and H y in position space for the optimal superposition of two Gaussians (solid blue line), and the optimal eigenstate (red dots) of the operator  in the domain −100x100 for c=25. In addition, we show the asymptotic approximation FH y (solid green line), given by equation (B.5), of the optimal eigenstate. In order to bring out the characteristic features of these wave functions on the different length scales we have used for the x-axis a highly nonlinear scale which stretches the domain around the origin but compresses the asymptotic behavior. Likewise, the vertical axis is nonlinear. We note that all three curves are very similar and display a narrow peak resting on a low pedestal with rapidly decaying wings-not unlike a mesa. Despite the similarity, ψ sG differs slightly from the exact eigenfunction H y and its asymptotic approximation FH y . is determined [33] by the matrix elements ρ nm (1) ≡ ρ nm (η=1) corresponding to no dissipation, that is η=1.
Here, n! is the factorial of n.