Distance between exceptional points and diabolic points and its implication for the response strength of non-Hermitian systems

Exceptional points are non-Hermitian degeneracies in open quantum and wave systems at which not only eigenenergies but also the corresponding eigenstates coalesce. This is in strong contrast to degeneracies known from conservative systems, so-called diabolic points, at which only eigenenergies degenerate. Here we connect these two kinds of degeneracies by introducing the concept of the distance of a given exceptional point in matrix space to the set of diabolic points. We prove that this distance determines an upper bound for the response strength of a non-Hermitian system with this exceptional point. A small distance therefore implies a weak spectral response to perturbations and a weak intensity response to excitations. This finding has profound consequences for physical realizations of exceptional points that rely on perturbing a diabolic point. Moreover, we exploit this concept to analyze the limitations of the spectral response strength in passive systems. A number of optical and photonics systems are investigated to illustrate the theory.

The sensing applications of EPs (a review can be found in Ref. [33]) rely on the strong spectral response to perturbations. A system with an EP n shows an energy (frequency) splitting proportional to the nth root of the perturbation strength ε [3], which for sufficiently small perturbations is larger than the linear scaling near a DP. In Ref. [34] it was shown that the spectral response to perturbations can be characterized by the so-called spectral response strength ξ. A large ξ not only indicates a large spectral response to generic perturbations but also a large intensity response to excitations and a large dynamic response to an initial deviation from the EP eigenstate.
One convenient way to create an EP is to start with a DP and to perturb it appropriately [5,35,36]. For instance, optical modes of ideal whispering-gallery microcavities naturally come as degenerate pairs of clockwise and counterclockwise traveling waves. Such a DP can be converted into an EP by * jan.wiersig@ovgu.de weak local perturbations [13,37,38] or weak boundary deformations [39,40] without adding extra parasitic radiation losses. Another example is the efficient transfer of excitations between energy levels, which is usually done as rapid adiabatic passage through a DP. Weakly perturbing it maps this scheme to one based on encircling an EP, which has been demonstrated experimentally using microwave waveguides [41]. Another example where EPs originating from perturbed DPs have been observed is systems of interacting fermions [42]. Finally, DPs naturally appear in periodic systems as Dirac or Weyl points in the band structure. These too can be used to create EPs, which has been done for photonic crystals [43][44][45][46], Dirac superconductors [47], optically biaxial crystals [14], and for a Hubbard model [48].
The aim of this paper is to introduce the concept of the distance ∆ between a given EP n and the set of DPs of the same order n. This distance quantifies the difference of the n × n Hamiltonians each exhibiting one of the two different kinds of degeneracies. We show that ∆ is an upper bound for the spectral response strength ξ associated with the EP; see Fig. 1. The important conclusion is that EPs generated by a small perturbation of a DP necessarily exhibit only a weak spectral response to perturbations, which renders a number of applications difficult. A key result of this paper is that while a system with an EP which is close to a DP shows only a weak spectral response to perturbations, a system having an EP with a large distance can exhibit a strong spectral response to perturbations.
The outline of the paper is as follows. Section II provides some mathematical preliminaries that are needed for the sub-arXiv:2205.15685v2 [physics.optics] 12 Sep 2022 sequent sections. Section III introduces the notion of the distance between a given Hamiltonian and the set of DPs. In Sec. IV the relation to the spectral response strength is explored. Section V investigates the relevant aspects for passive systems. Various examples are considered in Sec. VI. A summary is given in Sec. VII.

II. MATRIX NORMS AND DISTANCES
The aim of this section is to review some aspects of linear algebra required in this paper.

A. Matrix norms
A matrix norm ||Â|| is defined as a mapping of a matrixÂ to the nonnegative real numbers, see, e.g., Ref. [49], with the properties for all matricesÂ andB and α ∈ C. The nonnegativity of ||·|| follows from the first two items. Important examples of matrix norms are the Frobenius norm with the trace Tr and the spectral norm We adopt the common but slightly confusing notation ||·|| 2 both for the spectral matrix norm [in the left-hand side (LHS) of Eq. (6)] and the vector 2-norm ||ψ|| 2 = ψ|ψ of a vector |ψ based on the usual inner product in complex vector space [in the right-hand side (RHS) of Eq. (6)]. Both matrix norms (5) and (6) share the important property of unitary invariance, i.e., for all matricesÂ and all unitary matricesÛ andV . Moreover, both matrix norms (5) and (6) are compatible with the vector 2-norm, i.e., for all matricesÂ and vectors |ψ . The following inequality holds for all matricesÂ with equality for rank-1 matrices.
The calculation of the Frobenius norm (5) is particularly easy where A ij are the matrix elements ofÂ in any orthonormal basis. Also beneficial for us is that the Frobenius norm can be derived from an inner product, The inner product is the Frobenius inner product, see, e.g., Ref. [49], of two matricesÂ andB,

B. Matrix distances
The distance between any two matricesÂ andB can be measured with the distance function (see, e.g., Ref. [50]) with arbitrary matrix norm. It is easy to show that this distance function fulfills the usual axioms of a metric: It is important to mention that the assumed unitary invariance of the matrix norm (7) transfers to the distance function (13).

III. DISTANCE BETWEEN A GIVEN HAMILTONIAN AND THE SET OF DPS
In the mathematical literature several matrix nearness problems had been studied, as reviewed in Ref. [51]. For example, one asks the question of how close a given nonnormal ma-trixÂ is to the set of normal matrices and which normal ma-trixB minimizes the distance function such as in Eq. (13). A rather difficult task is to find the nearest matrix having at least two equal eigenvalues; see, e.g., Ref. [52]. Our problem is related but simpler.
We consider an in general nonnormal n × n-matrixĤ and a non-Hermitian but normal n × n-matrixĤ DP = E DP 1 with a DP of order n with complex-valued eigenvalue E DP and n×nidentity matrix 1. Note thatĤ DP is in the above form in any orthonormal basis. Using the distance function (13) we define the distance betweenĤ and a given DP by d(Ĥ, E DP 1). The distance betweenĤ and the set of DPs of order n is then given by Introducing the traceless part of the Hamiltonian with the mean energyĒ := TrĤ/n ∈ C, we write where we have exploited 1, 1 F = n, 1,Ĥ F = TrĤ = 0, and correspondingly Ĥ , 1 F = 0. Hence we get Clearly, this expression is minimal for E DP =Ē and the min-imizingĤ DP = E DP 1 is unique. Plugging this into Eq. (17) gives the first result As the Frobenius norm is unitarily invariant, we can evaluate the matrix norm in Eq. (22) in an orthonormal basis of our choice. From Eq. (10) we can infer that the physical interpretation of the distance (22) is that of an Euclidean distance defined on an n × n complex matrix space. The matrix space is usually of higher dimension than the parameter space, i.e., the space spanned by the physical parameters relevant for a given system. However, this disadvantage is outweighed by the geometric tools provided by linear algebra, such as norms and inner products.
A small distance ∆(Ĥ) means a minor change of the matrix elements ofĤ compared withĤ DP . This translates to a small detuning of the parameters of the system away from the DP.

A. Hamiltonian without an EP
For a nonnormal HamiltonianĤ that does not have an EP, one can express the distance to the set of DPs in terms of the biorthogonal basis of the Hamiltonian with the right eigenstates |R j and the left eigenstates |L j ; see, e.g., Ref. [53]. With L j |R l = 0 if j = l and the normalization L j |R j = 1 for all j, the biorthogonal expansion and Tr (|R j L l |) = L l |R j it is straightforward to show with Eq. (22) that with the n × n matrix This matrix is known as the nonorthogonality overlap matrix [53][54][55]. Its diagonal elements are the Petermann factors of the eigenstates [56,57], a measure of nonorthogonality. In Ref. [31] the expression ||Ĥ || F (without relating it to a distance function) has been used to calculate the Petermann factors for the special case of n = 2. From Eq. (25), we learn that the distance of the Hamilto-nianĤ to its nearest DP depends on the distance of its eigenvalues to the DP eigenvalue E DP =Ē and the nonorthogonality of its eigenstates. To emphasize, we do not advocate the use of Eq. (25) for calculating ∆(Ĥ). It is much easier to directly evaluate Eq. (22). Moreover, Eq. (25) is not valid for H having an EP. At the EP the Petermann factors diverge and the expansion (24) is not valid.

B. Hamiltonian with an EP
Next we consider the more interesting case of an n × n-HamiltonianĤ =Ĥ EP having an EP of order n with complexvalued eigenvalue E EP (frequency ω EP for optical systems). It follows fromĒ = E EP that the traceless part of the Hamiltonian [Eq. (18)] equals the operator The matrixN is nilpotent of index n, i.e.,N n = 0 but N n−1 = 0; see Refs. [3,34,58]. From Eq. (22) it follows for an n × n-Hamiltonian with an EP n the important result The distance of an EP to the set of DPs in Eq. (28) can be related to Henrici's departure from normality [59] where λ j (Â) are the eigenvalues of the matrixÂ. Clearly, D(Â) = 0 only ifÂ is nonnormal. In our caseÂ =N from Eq. (27) has only zero eigenvalues as the matrix is nilpotent. As a consequence, the distance of a given EP to the set of DPs in Eq. (28) can be expressed by the departure ofN from normality, It is known that Henrici's departure from normality is an upper bound for the distance of a matrix to the set of normal matrices [51]. The quantity ∆(Ĥ EP ) is therefore also an upper bound of the distance of the matrixĤ EP to the set of normal matrices.
From the physics perspective we can consider the departure ofN from normality and the distance of an EP to the set of DPs in many cases as a vague measure of the experimental effort in realizing the EP when starting at or near the minimizing DP. We come back to this point later when discussing some examples in Sec. VI.

IV. RELATION TO THE SPECTRAL RESPONSE STRENGTH
In Ref. [34] the spectral response strength of a system with an EP of order n has been determined to be The spectral and Frobenius norms give in this particular case the same result sinceN n−1 has rank 1, which is a consequence of the nilpotency ofN . The spectral response strength is unitarily invariant. It describes the response of the system to perturbations,Ĥ withĤ 0 =Ĥ EP . The spectral response strength shows up as a factor in the bound of the eigenvalue splittings where higher orders in the perturbation strength ε are ignored. The scaling of the splittings |E j − E EP | with the nth root of ε expresses the enhanced sensitivity of the EP with respect to perturbations. It is important to mention that ξ also bounds the intensity response to excitations [34]: where the vector |ψ ∝ e −iωt is the long-time asymptotics of the equation of motion with the excitation power P ≥ 0, the excitation frequency ω ∈ R, and a generic excitation vector |p normalized to unity. Comparing Eq. (28) with Eq. (31) reveals that, in the special case of an EP 2 : Hence the spectral response strength associated with the EP 2 and its distance to the set of DPs in the Frobenius norm is the same. This is different for higher-order EPs. With the submultiplicativity (4), we could use to derive an upper bound for ξ determined by ∆(Ĥ EP ). However, a more stringent bound can be obtained by taking advantage of the nilpotency ofN . For a nilpotent matrixN with index n > 2 holds [60] ||N k || F ≤ γ n,k ||N || k F (38) with k = 1, . . . , n − 1 and positive numbers γ n,k given by For k = n − 1 we get This inequality, valid forN being nilpotent of index n > 2, provides a much sharper bound than the inequality (37). Combining the inequality (40) with Eqs. (28) and (31) we arrive at our next crucial result The spectral response strength associated with the EP is therefore bounded by its distance to the set of DPs in the Frobenius norm. The important implication is: if the EP is generated by a small perturbation of a DP then the spectral response strength of the resulting EP is weak. Clearly, this has a profound impact on applications such as EP-based sensing. It is important to mention that this statement remains true if the DP which is (experimentally) used to generate the EP is not the minimizing DP. A non-minimizing DP has a larger distance ∆ > ∆ to the EP but the inequality (41) is clearly also valid with ∆ replaced by ∆ . Note that, here, small perturbation means small compared with a perturbation needed to create another EP (of the same order) out of a DP. That one has a potentially larger spectral response strength than the first EP, as illustrated in Fig. 1.
Another implication is that an EP which, under parameter variation, approaches a DP, has a vanishing response strength. In this sense a DP is an EP with zero response strength.
Equation (41) can also be seen as an estimate for the spectral response strength ξ. The calculation of ∆(Ĥ EP ) is considerably simpler and faster.
In the context of the response strength of a system with an EP the physical interpretation of the nearest DP is that of a reference. To discuss this, consider the spectral response to a perturbation εĤ 1 . For EP-based sensors it is common to compare with a related DP which is chosen rather ad hoc by removing coupling terms in the Hamiltonian; see, e.g., Ref. [26]. In contrast to the inequality (33) the response of a system with a DP to a perturbation is (see the Appendix for a derivation) With the condition E DP = E EP it is easy to show that the maximum splitting of the DP in inequality (42) equals the maximum splitting of the EP in inequality (33), cf. Fig. 2, if assuming that the perturbation is still small enough such that the inequality (33) is valid. This elementary calculation reveals that the critical perturbation strength ε c below which the maximum energy splitting related to the EP exceeds that of the reference DP (which is the nearest DP in matrix space here) is determined by ξ. It is mentioned that for the special case of n = 2, the distance ∆(Ĥ EP ) can also be related to the eigenstate response strength ζ defined in Ref. [34]. It measures the perturbationinduced change of the eigenvectors with respect to the EP eigenvector. For an EP 2 the simple relation ζ = 1/ξ holds. With Eq. (36) follows ζ = 1/∆(Ĥ EP ). No such relation is known for n > 2.

V. PASSIVE SYSTEMS
In Ref. [34] it had been shown for the special cases n = 2 and n = 3 that there is an upper bound for the spectral response strength ξ in passive (no gain) systems. This upper bound limits the applicability of the given EP in the sensing setting. In this section we derive an upper bound for the distance ∆ for arbitrary n. With the inequality (41) this in turn generalizes the upper bound for ξ in Ref. [34].
For passive systems the Hermitian decay operator is positive semidefinite; see, e.g., Refs. [55,61]. With Eq. (27) we writeN where we have introduced the real number Considering the fact that bothN andN † are traceless because of their nilpotency [49] one gets For a positive semidefinite matrix TrΓ ≥ 0 and hence β ≥ 0.
Taking the Frobenius norm on both sides of Eq. (45) gives In the following we assume thatN is a strictly upper triangular matrix. This can always be achieved by a unitary transformation of the HamiltonianĤ EP which does not change the Frobenius norm; see Eq. (7). According to the Schur theorem (see, e.g., Ref. [60]), a unitary transformation exists that transforms the matrixN defined in Eq. (27) to the sum of a diagonal matrix and a strictly upper triangular matrix. The former matrix is identical to zero because of the nilpotency ofN . Obviously, ifN is a strictly upper triangular matrix thenN † is a strictly lower triangular matrix. With Eq. (10) it follows for the LHS of Eq. (48) With Eq. (5) and the Hermiticity ofΓ we can rewrite Eq. (48) then as The positive semidefiniteness ofΓ restricts the maximum value of the RHS of Eq. (50). It is attained whenΓ is a rank-1 matrix. In this case,Γ has only one nonzero eigenvalue, which, according to Eq. (47), must be nβ. With Eqs. (28) and (46) the important result follows: This inequality valid for passive systems shows that the distance of a given EP to the set of DPs is bounded essentially by the decay rate at the EP. This bound can be transferred to the spectral response strength via the inequality (41). It follows the next important result For n = 2 the bound in Ref. [34] for passive systems is recovered. For n = 3 a slightly lower bound is obtained here. This improved bound is consistent with the numerical data in Ref. [34]. To emphasis, Eq. (52) is valid for all n ≥ 2. It is therefore a generalization of the results proven in Ref. [34].
Note that for small perturbation strength ε, the inequality (52) together with the bound of the eigenvalue splittings (33) provide a much stronger statement than Im E j ≤ 0 which must also be fulfilled in passive systems.
The importance of the inequality (52) becomes clear by remarking that 2|Im E EP | is the linewidth of the spectral peak at the EP. Hence the upper bound for ξ in the inequality (52) limits the resolvability of the frequency splittings under perturbation and therefore the performance of a sensor based on such an EP.
The result in the inequality (52) also has significant consequences for the intensity response of passive systems to excitations. Consider the DP-analog of inequality (34) (see the Appendix for a short derivation), We compare the response of the two kinds of degeneracies at resonance, i.e., at the EP [inequality (34)] with ω = Re(E EP ) and at the DP [Eq. (53)] with ω = Re(E DP ). Using the inequality (52) we obtain This inequality tells us how much an EP n can enhance the intensity response to excitation in a passive system if compared with a reference DP n . For n = 2 the maximal enhancement factor is 2 which is consistent with the findings in Ref. [62]. For n = 3 the maximal enhancement factor is 6.

VI. EXAMPLES
In this section we provide several examples to illustrate our approach.

A. Exceptional ring in wave-number space
Our first example stresses the difference between the commonly used parameter space and wave-number space on the one side and the matrix space discussed in this paper on the other side. In Ref. [43] it has been shown that EPs can be created out of a Dirac point in a photonic crystal slab. The finite thickness of the slab introduces radiation losses which are dissimilar for the two involved dipole and quadrupole modes. The effective Hamiltonian to describe this situation iŝ where k = (k x , k y ) is the two-dimensional wave vector, γ ≥ 0 is the loss rate of the dipole mode (the loss of the quadrupole mode can be ignored), ω 0 > 0 is the frequency of the unperturbed Dirac point, and v g > 0 is the group velocity at the Dirac point. The eigenvalues of the Hamiltonian (55) are with the critical wave number k c := γ/(2v g ). In the absence of radiation (γ = 0), Eq. (56) describes the linear dispersion near the Dirac point, ω ± = ω 0 ± v g | k|, the well-known Dirac cones.
If the loss rate γ is nonzero, even if it is very small, the system possesses an EP at | k| = k c . In the two-dimensional wave-number space (k x , k y ) with all other parameters fixed this degeneracy therefore appears as a ring of EPs. In the three-dimensional parameter space (ω 0 , v g | k|, γ) this degeneracy is the two-dimensional plane v g | k| = γ/2. Both the parameter space and the wave-number space are clearly different from the 2 × 2-dimensional complex matrix space where the distance ∆(Ĥ EP ) of the EP to the set of DPs is defined. It is here calculated by inserting the Hamiltonian (55) at the EP and its eigenvalue ω EP = ω 0 − i γ 2 via Eq. (27) into Eq. (28) This result makes intuitive sense because it is here that the loss turns the DP into an EP. A small loss rate γ is sufficient but, according to Eq. (36), the spectral response strength ξ = γ is then also small. To enlarge the spectral response to perturbations one has to increase the loss in the photonic crystal slab by reducing the thickness of the slab which obviously has a limit. One remark is in order. The DP from which the experiments starts isĤ DP = ω 0 1 according to Eq. (55), in the absence of radiation, and for k = 0 at a Dirac point. The minimizing DP is, however,Ĥ DP = ω EP 1 = (ω 0 − i γ 2 )1. The distance ofĤ EP toĤ DP is 3/2γ and therefore slightly larger than ∆(Ĥ EP ), see Eq. (57). The above conclusions are still correct.
Note that γ = 2|Im ω EP |, so that the inequality (52) for passive systems holds here with the equal sign.

B. Unidirectionally coupled pair of PT -symmetric dimers
Our second example is a higher-order EP at which ∆ and ξ are not equal. We consider a unidirectionally coupled pair of parity-time (PT )-symmetric dimers introduced in the context of hierarchical construction of higher-order EPs [63]. The Hamiltonian iŝ Each of the two 2 × 2 subblocks along the diagonal describes a PT -symmetric dimer with real-valued frequency ω 0 , gain/loss coefficient α > 0, and internal coupling coefficient g > 0. These two dimers are unidirectionally coupled with the strength κ > 0. This system can be realized experimentally by two evanescently coupled microrings each with two modes, one traveling clockwise and one counterclockwise [63]. One microring exhibits gain, the other one exhibits an equal amount of loss. The unidirectional coupling can be achieved by evanescently coupling the lossy microring to a semi-infinite waveguide with an end mirror [64]. This introduces a fully asymmetric backscattering [38] between the traveling waves in that microring. For α = g the Hamiltonian (58) possesses an EP 4 with real eigenvalue ω EP = ω 0 . Inserting this eigenvalue and the Hamiltonian (58) via Eq. (27) into Eq. (28) gives The Pythagoras-like appearance of ∆(Ĥ EP ) in this case is obvious. The experimental effort in realizing a significant ∆(Ĥ EP ) starting from uncoupled microrings/resonators is based on implementing either a large internal coupling g or a large unidirectional coupling κ. For the spectral response strength (31) we get It can be easily verified that the inequality (41) is satisfied. Equation (60) tells us that large g or large κ is only a necessary but not a sufficient condition for getting a large response from the system at the EP. For the sufficient condition the product g 2 κ must be large. The bound in the inequality (52) does not apply since the system is not passive.

C. Coupling of optical modes with different angular momenta
The third example is a deformed microdisk cavity [65,66]. The broad range of applications of deformed microcavities is reviewed in Ref. [67]. For generating EPs in this kind of system a weak boundary deformation is sufficient [39,40]. Such a boundary deformation can be expressed in polar coordinates as r(φ) = R+f (φ) with the radius of the unperturbed microdisk R and the deformation function |f (φ)| R. We consider here only deformation functions that preserve a mirror-reflection symmetry. Moreover, we restrict ourselves to two modes. The mode with lower radiation losses is called mode 1. Its complex frequency is ω 1 and its azimuthal mode number is m. Mode 2 has the higher radiation losses, complex frequency ω 2 , and azimuthal mode number p < m. If the frequencies of the two modes are nearly degenerate, i.e., ω 1 ≈ ω 2 , a first-order perturbation theory can be applied leading to the effective non-Hermitian Hamiltonian [40] H = The eigenvalues of this Hamiltonian are the frequencies of the two modes in the deformed microcavity. Here, x i = ω i R/c are the dimensionless complex frequencies of the unperturbed modes and c is the speed of light in vacuum. The Fourier harmonics of the deformation function are given for the even and odd parity as with ε p = 2 if p = 0 and ε p = 1 otherwise. Restricting to the relevant case m, p > 0 the matrix elements A e/o mp and A e/o pm are equal for fixed parity.
Note that the unperturbed system, i.e., the undeformed microdisk, is strictly speaking not at an DP as x 1 = x 2 . However, the distance to a DP is very small, as x 1 ≈ x 2 has been required.
The Hamiltonian (61) has an EP if The eigenvalue of the EP is Plugging this eigenvalue and the Hamiltonian (61) by means of Eq. (27) into Eq. (28) gives the distance to the set of DPs Clearly, a weak deformation (|A e/o mp | 1) implies a small distance ∆(Ĥ EP ). This is turn leads, according to Eq. (36), to a weak spectral response strength ξ to perturbations (e.g. induced by a nanoparticle close to the boundary of the microcavity). The response can therefore be enhanced if the deformation is increased. This, however, often leads to enhanced radiation losses [68]. Also note that if the deformation is too strong then the perturbation theory for weak boundary deformations is no longer valid.

D. Fully asymmetric hopping model
Here, we study an n × n Hamiltonian describing uniform and unidirectional hoppings in a nearest-neighbor tightbinding chain If the complex hopping parameter A is nonzero, the Hamiltonian has an EP n with eigenvalue E EP . The Hamiltonian (67) can be considered as the nonperiodic, fully asymmetric limiting case of the Hatano-Nelson model of a cylindrical superconductor [69]. The spectral response strength has been determined in Ref. [34] to be The distance of this EP to the set of DPs can be easily calculated from Eqs. (28), (27), and (10) yielding With Eq. (68) this gives Interestingly, this is the inequality (41) with the equality sign. Hence, this fully asymmetric hopping model has the largest possible spectral response strength ξ for given distance ∆(Ĥ EP ).

E. Random Hamiltonians at EPs
Finally, we adopt the random-matrix approach invented in Ref. [34] to numerically generate a whole class of examples. To do so, we introduce the n × n matrixĤ EP having an EP n with eigenvalue E EP via a similarity transformation H EP =QĴQ −1 , withĴ being an n × n matrix with an EP n in Jordan normal form andQ is an, in general nonunitary, n × n matrix consisting of complex random numbers with real and imaginary parts being drawn from a uniform distribution on the interval [−1/2, 1/2]. Clearly,Ĥ EP is not completely random, but nevertheless we refer to it as the "random Hamiltonian with an EP".
To numerically confirm the inequality (41) we define the nonnegative quantity which should be less than or equal to unity. Figure 3 shows a histogram resulting from 10 8 realizations of random Hamiltonians for the case of an EP of order n = 4. It can be clearly seen that x ≤ 1 and therefore the upper bound for the response strength ξ given by the inequality (41)  To check the upper bound for the distance ∆ [inequality (51)] and the spectral response strength ξ [inequality (52)] in passive systems, we consider the random HamiltoniansĤ EP with an EP n for several values of n. We define the nonnegative quantities y := ∆(Ĥ EP ) 2n(n − 1)|Im E EP | and which according to the inequalities (51) and (52) are less than or equal to unity. For each value of n, we select from 10 8 realizations of random Hamiltonians those that have a positive semidefinite decay operatorΓ [Eq. (44)]. Figure 4 shows the resulting maximal values of y and z versus the order n. First, one can observe that y max , z max ≤ 1. Hence the inequalities (51) and (52) are indeed fulfilled. Second, from y max ≥ z max we conclude that the inequality (51) gives a sharper bound than the inequality (52). This is understandable as the former bound results directly from the positive semidefiniteness ofΓ, whereas the latter additionally requires the inequality (41).

VII. SUMMARY
We have introduced the concept of the distance ∆ of an n×n Hamiltonian to the set of DPs of order n in matrix space. This concept is mathematical in nature but it has physical relevance for non-Hermitian Hamiltonians with an EP. In this case, ∆ is a vague measure of the experimental effort to create an EP out of a DP of the same order. Interestingly, ∆ determines an upper bound for the spectral response strength ξ of the given system with the EP. This not only provides an easy way to estimate ξ, but, more importantly, it also reveals that the strategy of experimentally implementing an EP by slightly modifying a DP results in an EP exhibiting only a weak response to perturbations. We have related the distance ∆ to Henrici's departure from normality.
For passive systems we have derived an upper bound for the distance ∆. This bound engenders an upper bound for the spectral response strength ξ which constitutes an improvement and generalization of the bound for ξ in Ref. [34] to EPs of arbitrary order.
Analytical and numerical results for various physical examples have been presented which demonstrated the new insights in the physics of EPs provided by the concept introduced in this paper.

ACKNOWLEDGMENTS
Valuable discussions with J. Kullig and R. El-Ganainy are acknowledged.