Bound entangled states fit for robust experimental verification

Preparing and certifying bound entangled states in the laboratory is an intrinsically hard task, due to both the fact that they typically form narrow regions in the state space, and that a certificate requires a tomographic reconstruction of the density matrix. Indeed, the previous experiments that have reported the preparation of a bound entangled state relied on such tomographic reconstruction techniques. However, the reliability of these results crucially depends on the extra assumption of an unbiased reconstruction. We propose an alternative method for certifying the bound entangled character of a quantum state that leads to a rigorous claim within a desired statistical significance, while bypassing a full reconstruction of the state. The method is comprised by a search for bound entangled states that are robust for experimental verification, and a hypothesis test tailored for the detection of bound entanglement that is naturally equipped with a measure of statistical significance. We apply our method to families of states of $3\times 3$ and $4\times 4$ systems, and find that the experimental certification of bound entangled states is well within reach.

Preparing and certifying bound entangled states in the laboratory is an intrinsically hard task, due to both the fact that they typically form narrow regions in state space, and that a certificate requires a tomographic reconstruction of the density matrix. Indeed, the previous experiments that have reported the preparation of a bound entangled state relied on such tomographic reconstruction techniques. However, the reliability of these results crucially depends on the extra assumption of an unbiased reconstruction. We propose an alternative method for certifying the bound entangled character of a quantum state that leads to a rigorous claim within a desired statistical significance, while bypassing a full reconstruction of the state. The method is comprised by a search for bound entangled states that are robust for experimental verification, and a hypothesis test tailored for the detection of bound entanglement that is naturally equipped with a measure of statistical significance. We apply our method to families of states of 3 × 3 and 4 × 4 systems, and find that the experimental certification of bound entangled states is well within reach.

Introduction
To experimentally prepare, characterize and control entangled quantum states is an essential item in the development of quantum-enhanced technologies, but it also serves the indispensable purpose of testing the predictions of entanglement theory in the laboratory. Among the most intriguing features of this theory stands the existence of bound entanglement [1], a form of entanglement that cannot be distilled into singlet states by any protocol that uses only local operations and classical communication. Originally considered as useless for quantum information processing, bound entangled states were later established as a valid resource in the contexts of quantum key distribution [2], entanglement activation [3,4], metrology [5,6], steering [7], and nonlocality [8], and their non-distillability has been linked to irreversibility in thermodynamics [9,10].
Complementing these theoretical achievements, substantial efforts have been devoted to experimentally producing and verifying bound entanglement. The first experimental report on the preparation of a bound entangled state was presented in [11], although the result was disputed [12] and subsequently amended [13]. The state prepared was the four-qubit Smolin state [14], thus it showcases a multipartite instance of bound entanglement, which is fundamentally distinct from the bipartite case: when multiple parties are present, entanglement can still potentially be distilled if two of the parties work together. Further experimental works on multipartite bound entanglement include Refs. [15][16][17][18][19][20]. Examples of experimental bipartite setups are more scarce. In Refs. [21] and [22] bipartite bound entanglement was produced using two-mode continuous-variable Gaussian states, and Ref. [23] focuses on a family of two-qutrit states.
Since the property of non-distillability is experimentally inaccessible in a direct manner, a natural route to verify a state as bound entangled is to do a full tomographic reconstruction of the density matrix from the experimental data [24] and apply the only known computable criterion on it [1]: if an entangled state has positive partial transpose (PPT), then it is non-distillable and therefore bound entangled 1 . However, it has recently been pointed out that widely used reconstruction methods like maximum likelihood and least squares [25,26] inevitably suffer from bias [27,28], caused by imposing a positivity constraint over compatible density matrices. In some cases, the bias can be large enough to drastically change the entanglement properties of the estimator with respect to the true state. In addition to this state of affairs, the variance of the estimator is usually calculated by bootstrapping [29], which only accounts for statistical fluctuations, and can result in a smaller variance than the actual bias of the estimator [28]. In contrast, a direct reconstruction of the state by linear inversion produces an unbiased estimator, but at the cost of admitting unphysical density matrices. Then, the PPT criterion simply looses all meaning.
All the experimental works cited above support their claims on some combination of maximum likelihood or least squares reconstruction and bootstrapping. There exist more informative methods to derive errors from tomographic data, such as credible [30] and confidence [31,32] regions, and also the alternative of using linear inversion in addition to a sufficiently large number of measurements that guarantees physical estimates [33]. Should these methods be applied to the detection of a bound entangled state, more robust results may be generated, although they might come at the expense of being computationally expensive or even intractable [34]. However, 1 It is still an open question whether the PPT criterion completely characterizes bound entanglement, namely whether all non-distillable states are PPT. regardless of the reconstruction method of choice, the problem of experimentally testing bound entanglement is intrinsically challenging. This is so because bound entangled regions of the state space are typically very small in volume. Furthermore, at least for the known cases in lowdimensional systems, bound entangled states are close to both the sets of separable states and distillable entangled states. This translates into the requirement of a highly precise experimental setup, and the deepening of the potential pitfalls of biased tomographic reconstructions.
In light of the above, we consider that the experimental preparation and certification of bound entanglement is still an unsettled issue. In this paper, instead of advocating for a particular tomographic method for detecting bound entanglement or considering the preparation of a specific state, we address the more generic question: Which are the best candidate states for an experimental verification of bound entanglement? In other words, for bipartite systems, we set ourselves to find states that have the largest ball of bound entangled states around them. To this end, we construct simple lower bounds that the radius r of such a ball (in Hilbert-Schmidt distance) has to obey for a given state, and formulate an optimization problem that maximizes r over parametrized families of states. Having a value for the maximum radius, r * , allows us to assert the robustness of the target state at the center of the ball, and in turn gives us an idea of the required number of preparations of the state in a potential experiment. We proceed by designing a χ 2 hypothesis test directly applicable over unprocessed tomographic data that provides a certificate for bound entanglement within some statistical significance. We show that our proposed method, combining the search of a robust candidate state with a statistical analysis through a hypothesis test, makes rigorous bound entanglement verification not only experimentally feasible with current technology, but also computationally cheap.
The paper is structured as follows. First we derive the constraints for the existence of a ball of bound entangled states of radius r around a generic bipartite state of dimension d. Then, we apply our method to two families of states of dimensions 3×3 and 4×4, known to contain bound entanglement, and find robust candidates for its experimental verification. We proceed to devise a hypothesis test for bound entanglement, and test the robustness of the selected candidate states in terms of the necessary number of samples to achieve a statistically significant certification under realistic experimental conditions.

A bound entangled ball
Verifying the bound entangled character of a bipartite state ρ requires, on the one hand, showing that it is entangled, and on the other hand proving that the entanglement of ρ is nondistillable. Non-distillability is usually verified via the (sufficient) PPT condition, which we denote by Γ(ρ) ≥ 0. Throughout this paper, we identify bound entangled states with PPT entangled states. As for verifying that ρ is entangled, there exist several inequivalent criteria. We choose the violation of the computable cross norm or realignment criterion (CCNR) [35], since it is simple and is generally tight enough to detect bound entanglement. The CCNR criterion dictates that, for some local orthonormal basis [with respect to the Hilbert-Schmidt in- The points in the state space that violate CCNR, fulfill PPT, and correspond to physical states (that is, satisfy the positivity condition ρ ≥ 0), thus define a volume of bound entangled density matrices.
Given a bound entangled state ρ that obeys these conditions, we inquire how far we can move away from it while remaining in the bound entangled region. We construct a new state τ = ρ + rX, where r ≥ 0, and X is a traceless Hermitian matrix with bounded Hilbert-Schmidt norm The set of all such matrices forms a Hilbert-Schmidt ball that we denote by B . We then define the set of all states τ as B(ρ, r) := ρ + rB . Note that ρ − τ 2 ≤ r. We can bound the minimum eigenvalue of τ as where X ∞ = λ max (X) is the uniform norm of X. The first inequality holds since, in the extreme case, the eigenvector associated to the minimal eigenvalue of X is aligned with the corresponding one for ρ. For the second inequality we have used that Similarly, since the partial transpose Γ(τ ) does not change the Hilbert-Schmidt ball, Γ(B) = B, we have We can bound the value of the CCNR criterion over τ in a similar fashion. We have where we first applied the triangle inequality, and for the last inequality we used that (we refer to Appendix A for a proof). Our goal is to find a state ρ such that, if we depart from it by a distance r in any direction, the resulting τ still fulfils PPT and violates CCNR, that is, λ min [Γ(τ )] ≥ 0 and R(τ ) 1 > 1. Then, using Eqs. (3) and (4), we search for a state ρ which admits the largest r under the constraints For any admissible r, all states in B are bound entangled. Note that, while the optimization is naturally performed over physical target states ρ, the resulting ball B can well be partly outside of the state space, cf. Fig. 1, as the positivity of all states inside B is not imposed as a constraint.

Symmetric families
The method described in the previous section is completely general, but for the optimization to actually become feasible one has to restrict the free parameters in ρ. We consider two symmetric families of bipartite states which are characterized by few parameters, and nevertheless contain fairly large regions of bound entanglement.
The parameter r is the radius of a Hilbert-Schmidt ball around a target state ρ such that all physical states τ are bound entangled.
The first case that we consider is a family of two-qutrit states of the form where |φ 3 = 2 k=0 |kk / √ 3, the symbol ⊕ denotes addition modulo 3, and a, b, c are real parameters. A similar three-parameter family of two qutrits, known to contain bound entanglement and arise from symmetry conditions, was analyzed in Refs. [36][37][38][39][40][41], and considered in the experiment reported in Ref. [23]. We search for the optimal values of {a, b, c} that parametrize an optimal target state ρ * . This state will admit any r ≤ r * , where r * is the maximum radius compatible with the constraints (6a) and (6b). An exact solution for this optimization can be found (see Appendix B.1 for details). We obtain the optimal parameters a ≈ 0.21289 , b ≈ 0.04834 , and c ≈ 0.21403 , (8) yielding a maximal radius r * ≈ 0.02345. The resulting ρ * is a rank-7 state.
Since the ball B may contain unphysical states, it is in principle possible that our estimate for the maximal r * is not tight. To explore this, we move away from ρ * by a distance r * in suitable directions and evaluate the position of the resulting state with respect to the boundaries of the PPT and the CCNR sets. We provide the details of this analysis in Appendix B.1. As a result, we obtain that our estimate of r * is indeed tight with respect to the PPT boundary, but we observe that it slightly underestimates the distance with respect to the CCNR boundary.
As our second case, we consider the twoququart states that are Bloch-diagonal, i.e., of the form where g k = (σ µ ⊗ σ ν )/2, µ, ν = 0, 1, 2, 3, with σ 0 = 1, σ 1 , σ 2 , σ 3 the Pauli matrices, and the index k enumerates pairs of indices {µ, ν} in lexicographic order. Bound entangled Blochdiagonal states have already been described in Ref. [42]. The optimization over this family of states, despite having more free parameters, is much simpler than in the two-qutrit case discussed above. The reason is that the problem can be reshaped as a linear program over the coefficients x k , and thus it can also be solved exactly (see Appendix B.2 for details). A vertex enumeration of the corresponding feasibility polytope is possible, and leads us to a set of 4224 optimal states achieving a maximal radius r * ≈ 0.0214.
One example of such optimal state, ρ * , is given by coefficients 3, 4, 5, 6, 9, 12, 14, 16} , where x 1 is fixed by normalization. The state ρ * has rank 10, which is the minimal rank among bound entangled states achieving r * that are of the form (9) and are detectable by CCNR. As a byproduct of our analysis, we also obtain that the overall minimal rank for such bound entangled states is 9, albeit with a fairly smaller radius.

Statistical analysis
In this section we put our method for finding optimal target states to work in a practical scenario. That is, for an experiment aiming at the certification of bound entanglement, we design a statis-tical analysis of the experimental data that crucially hinges on knowing the radius of the bound entangled ball around the target state. The idea is to judge whether the data was obtained from a bound entangled state by considering the membership of the preparation to the bound entangled ball. In order to endow the certification with a measure of statistical significance, we design a hypothesis test for this membership problem. Our null hypothesis, H 0 , is that the prepared state is outside the bound entangled ball B(ρ 0 , r 0 ) of radius r 0 centered at the target state ρ 0 . We make the assumption that an instance of experimental data, x, is drawn from a multivariate normal distribution N ( ξ, Σ) with offset ξ and covariance matrix Σ. This is a good approximation for realistic scenarios. Here the vector notation is used over variables that belong to the same space as the experimental data, that is, e.g., the space of frequencies of measurement outcomes. In accordance to H 0 , the offset ξ is the expected value of the data when a state ρ exp is prepared such that ρ 0 − ρ exp 2 ≥ r 0 . The covariance matrix Σ is determined by the particular experimental procedure used. The goal is then to appropriately design a test statistict( x) for the hypothesis H 1 that the prepared state lies within B and hence it is bound entangled. The quantity through which we assess the significance of a hypothesis test is the worst-case probability of accepting H 1 given H 0 , which is formally written as for some threshold value t. When t =t( x), this probability is a p-value for x given H 0 . Now, let us definet where T is a map that takes a density matrix to the expected data (e.g., to a vector of probabilities), thus it is determined by the experimental procedure. Hence,t( x) gives some notion of distance between the standardized versions of the experimental data and the expected data of a perfect measurement performed over the target state. In Appendix C we show that, ift( x) is of the form in Eq. (12), the probability (11) is naturally upper-bounded by where s → q m (s, u) is the cumulative distribution function of the noncentral χ 2 -distribution with m degrees of freedom and noncentrality parameter u, and r 1 is the equivalent distance in the experimental data space of the Hilbert-Schmidt distance r 0 , or, more precisely, where ∆ is any Hermitian matrix with unit trace. Naturally, the value of r 1 scales with r 0 and strongly depends on the experimental procedure, that is, on T and Σ.
In the following, we show how to evaluate the hypothesis test for the two-qutrit state from Eq. (8) as an example of a target state ρ 0 , and assess the necessary experimental requirements for a desired level of significance. For this, we need to make some assumptions about the experimental procedure. We associate the measurement outcomes in the experiment to semidefinite-positive Hermitian operators E k . We consider a complete set of such operators, that is, the real linear span of {E k } is the set of all Hermitian operators. The probability of obtaining the outcome corresponding to E k when measuring the state ρ is given by p k = tr(E k ρ), and we assume that T (ρ) k ≡ p k . The connection between the probabilities p k and the data gathered in the experiment crucially depends on how the experiment is performed. A straightforward theoretical association can be established if one assumes that measurement outcomes correspond to independent Poissonian trials with parameters nT (ρ) k , renormalized by n, where n is the mean total number of events per measurement setting. That is, if we obtain n k events for the outcome k, we use as data x k = n k /n. Note, however, that in a realistic experiment T (ρ) and n are not known exactly. A reasonable alternative is to use the total number of observed events k n k as an estimate of n, but then the connection between p k and x k is more involved. For our examples below we use this latter approach (a detailed discussion is presented in Appendix C).
In order to get specific predictions, we assume that mutually unbiased bases are used as local measurements (refer to Appendix C for an explicit construction) and that the experimental state ρ exp has 5% white noise over the target, that is, ρ exp = 0.95ρ 0 + 0.051/9. This amount of noise still results in a state within B, since ρ 0 − ρ exp 2 ≈ 0.6r 0 , where r 0 ≈ 0.02345 is the optimal radius associated to ρ 0 (cf. Sec-tion 3). Then, one obtains r 2 1 ≈ 0.0664 2 n (see Appendix C for details on how to compute this value). We now set a critical t 0 below which the p-value will be larger than some acceptable threshold p 0 , so that q m (t 2 0 , r 2 1 ) = p 0 . To obtain t 0 , one inverts the equation q m (t 2 , r 2 1 ) = p to get t 2 (p), called the quantile function. Then, t 0 := t(p 0 ). Once t 0 is fixed, we compute the probability p fail that the test fails to certify bound entanglement over data obtained by measuring the prepared state ρ exp , i.e., thatt( ξ) > t 0 given ξ = T (ρ exp ). We can write this probability as where r 2 2 :=t( ξ) 2 ≈ 0.0416 2 n. Note that, while r 1 is used to determine t 0 considering a worst case scenario for a false positive, r 2 is a distance between standardized probabilities given the particular preparation ρ exp . Therefore, r 1 > r 2 means that the testt has a chance to single out the particular experimental preparation as bound entangled from the worst-case state outside B, and hence p fail will decrease with the number of samples n, which is the case of interest. In Fig. 2 we plot p fail as a function of n, for various levels of significance p 0 expressed as multiples of the standard deviation kσ, so that p 0 = 1 − erf(k/ √ 2).
Similarly, we carry out the same analysis for the two-ququart target state specified in Eq. (10). In this case we construct our measurement settings by regarding each ququart as two qubits and performing local Pauli measurements on them. With r 2 1 ≈ 0.0856 2 n, r 2 2 ≈ 0.0469 2 n, and an admixture of 2.5% white noise over the target ρ 0 , we obtain the results shown in Fig. 2. In both cases, for the selected target states of two qutrits and two ququarts given by Eqs. (8) and (10), our analysis shows that their experimental certification as bound entangled states under realistic assumptions is within reach, with around 70000 samples per measurement setting to reach a 3σ level of significance.

Discussion
We have developed a comprehensive method for the experimental characterization of bipartite bound entanglement, from the selection of robust target states to the statistical analysis of the data. Previous experimental works were based on preparing a bound entangled state and inferring its properties from those of the reconstructed density matrix via maximum likelihood or least squares. Unfortunately, such techniques have been shown to produce unreliable results, particularly for entanglement certification [27,28], which casts a shadow of doubt over past experimental demonstrations of bound entanglement. Instead of using a (necessarily) biased reconstruction of the density matrix and assuming that it shares the same properties that the true prepared state, we show that it is feasible to perform a hypothesis test over the unprocessed experimental tomographic data to test for membership of the preparation to a subset of the state space that is guaranteed to only contain bound entangled states. The hypothesis test is naturally equipped with a measure of statistical significance, which is easy to compute. The subset of bound entangled states is specified as a ball of radius r, and the design of the hypothesis test directly depends on this parameter. We have shown, through explicit examples of families of bipartite qutrit states and bipartite ququart states, how the certification of bound entanglement with high statistical significance is well within current experimental capabilities by using our method.
While our statistical analysis of the data in the form of a hypothesis test is a standalone technique applicable to the preparation and measurement of any state, we would like to stress that the value of the radius r has an important effect in its detection power, hence it is worth aiming at target states with maximal r. To show this, we run our analysis for the two-qutrit state prepared in Ref. [23], assuming a noiseless experimental preparation. This state has a ball of bound entangled states around it of radius r ≈ 0.01182. This value is computed as the maximal r that satisfies Eqs. (6a) and (6b). We obtain that the required number of samples for achieving a 3σ level of significance with a failure probability p fail ≈ 10 −3 is roughly twice as much as for the two-qutrit state in Eq. (8) (which admits a radius r ≈ 0.02345) when assuming a noisy preparation. As a comparison, optimizing over the one-parameter Horodecki family [1]-which is part of the family in Eq. (7)-yields a radius r ≈ 0.01681, for the parameters a ≈ 0.28571, b ≈ 0.07931, and c ≈ 0.15879. As a concluding remark, some comments on the generality of our result are in order. If one has prior knowledge of the expected amount and type of experimental noise, this could be incorporated into the statistical analysis by assuming the noisy state as the target. As a result, one should expect a smaller value of the test statistiĉ t( x) for a given set of data x and hence a smaller p-value. However, our calculations for several examples indicate that this advantage does not compensate in general the drawback of having a smaller bound entangled ball. A refinement of our method could also be achieved by taking into account the form of the covariance matrix Σ into the optimization over target states, as we did incorporate it into the design of the hypothesis test. This would generally yield an ellipsoid around the target, instead of a ball, potentially capturing a larger volume of bound entangled states and thus leading to a stronger test. The possible disadvantage is that the bounds (6a) and (6b) will likely be much more complicated (if computable at all), hence the optimization step will be much harder to carry out. We leave the question open of whether such refinement provides a significant reduction in the experimental requirements. A Proofs of Eqs. (2) and (5) Given a target state ρ, we construct the displaced state τ = ρ + rX, where r ≥ 0, and X is a bounded traceless Hermitian matrix fulfilling X 2 ≤ 1. We claim that the minimal eigenvalue of τ can be lower bounded as [cf. Eq. (1)]

Acknowledgments
where for the second inequality we have used that [cf. Eq.
and B = {X| X 2 ≤ 1, tr X = 0, X = X † }. This holds from the following reasoning. Clearly, the maximum is attained for X = diag(x, y 1 , . . . , y d−1 ) with some vector y fulfilling y ∞ ≤ x = − y · e and x 2 + y 2 ≤ 1. Here, e k = 1 for k = 1, . . . , d − 1. Note that the choice of y that allows x to be largest is the uniform vector y = −x e/ e 2 ≡ z, since it minimizes y 2 . The maximization now reduces to which immediately yields the assertion due to e 2 = d − 1.
In a similar fashion, we then obtain a lower bound of the 1-norm of the realigned state τ as [cf.
where we use that [cf. Eq.
This immediately follows from the facts that the 1-norm is bounded by the 2-norm as A 1 ≤ rank(A) A 2 , via e.g. Cauchy-Schwarz inequality, and that R(X) 2 = X 2 ≤ 1. Hence √ d is an upper bound. It remains to show that this bound is attainable. For arbitrary d, we see that tr X = 0 ⇔ [R(X)] 1,1 = 0 (choosing the local basis such that g 1 = 1). Then, R(X) is a constant anti-diagonal matrix.

B Optimally detectable bound entangled states B.1 A state of two-qutrits
The first case that we consider is a family of two-qutrit states of the form where |φ 3 = 2 k=0 |kk / √ 3, the symbol ⊕ denotes addition modulo 3, and a, b, c are real and nonnegative. We denote the dimension of the total Hilbert space by d = 9. There are three distinct eigenvalues for ρ, which satisfy the constraint a + 3(b + c) = 1 .
Then, the three distinct eigenvalues of its partial transpose Γ(ρ) are given by Now, the trace-norm of the realigned matrix R(ρ) is given by Plugging in Eqs. (22) and (23) in the bounds (6a) and (6b) one obtains two inequalities that, together with Eqs. (21) and λ(ρ) ≥ 0, form the set of constraints that a valid target state should obey. The maximization over r such that these constraints hold can be solved exactly, although the analytical form of the optimal parameters is rather involved. Here we give the approximate values a ≈ 0.21289, b ≈ 0.04834, and c ≈ 0.21403 that yield an optimal state ρ * , which admits any r ≤ r * ≈ 0.02345. Note that the optimization is invariant under the interchange b ↔ c, therefore r * is also not affected by it.
As argued in the main text, since we do not require in our optimization that the full ball B is contained in the state space, an analysis of the tightness of our estimate for r * is in order. To do so, we move away from ρ * by a distance r * in suitable directions and evaluate the position of the resulting state with respect to the boundaries of the PPT and the CCNR sets. We first move towards the PPT boundary. To this end, we take a normalized eigenvector |η from the eigenspace of Γ(ρ * ) with minimal eigenvalue and find, for the matrix after the displacementρ = ρ where we wrote Π[X] for [X − tr(X)1/9]/ξ for some appropriate ξ, so that Π[X] 2 = 1. Hence,ρ is effectively sitting on the PPT boundary, it is still detected as bound entangled by the CCNR criterion, and it is slightly outside the space of physical states. This shows that our estimate for r * is tight with respect to PPT. Similarly, we now consider a displacement towards the CCNR boundary. With a singular value decomposition U DV † = R(ρ * ), we letρ = ρ * − r * Π[R −1 (U V † )] and find the values Therefore, it is possible that our state ρ * is not yet optimal for obtaining the largest value r * , since we are still not touching the CCNR boundary.

B.2 States of two-ququarts
We consider two-ququart states that are Bloch-diagonal, i.e., of the form where g k = (σ µ ⊗ σ ν )/2, µ, ν = 0, 1, 2, 3, with σ 0 = 1, σ 1 , σ 2 , σ 3 the Pauli matrices, and the index k enumerates pairs of indices {µ, ν} in lexicographic order. Since tr(ρ) = 1, we get x 1 = 1/4. By making the assumption that |x k | = s for k = 2, . . . , 16, the optimization can be easily carried out analytically. We will see later that the maximum radius r * for the general states (26) is indeed achieved under this assumption. The trace norm of the realignment is Computing the minimal eigenvalue of Γ(ρ) is not so straightforward, as the signs of the coefficients in Eq. Having optimal states with smaller rank can significantly simplify their experimental preparation.
The vertex enumeration of all feasibility polytopes contained in Eq. (31) allows us to go beyond the set of optimal states and characterize states with even smaller ranks, naturally at the expense of sacrificing some volume of bound entangled states around them. With this in mind, we see that the absolute minimal rank for a Bloch-diagonal two-ququart bound entangled state is 9, with associated radius r 9 ≈ 0.0128. An example of such state is given by parameters x α ≈ 0.0999184 , α ∈ {2, 5} , x β ≈ 0.0750408 , β ∈ {3, 4, 6, 10, 14} , x γ ≈ 0.0501632 , γ ∈ {7, 9} ,

C A hypothesis test for bound entanglement
In this appendix we give a proof of Eq. (13) and show how to compute the parameters r 1 , r 2 , and m, needed to reproduce Fig. 2. Let us start with the proof. The χ 2 -distribution arises naturally in a hypothesis test where the hypotheses are defined in terms of bounds on Euclidean distances in a vector space of normal-distributed random variables. To see this, we consider a generic situation where we draw a sample x from an m-variate normal distribution N ( ξ, Σ) with offset ξ and covariance matrix Σ. The offset shall be of the form ξ = Sλ + η with a matrix S and a constant vector η. We denote by p(t, r) the maximal probability thatt( x) ≤ t under the hypothesis µ − λ 2 ≥ r, for some fixed µ. Then, p[t( x), r] is a p-value for this hypothesis under the test statistict( x). By choosingt the following lemma holds: where s → q m (s, u) is the cumulative distribution function of the noncentral χ 2 -distribution with m degrees of freedom and noncentrality parameter u. Here, λ min (X) denotes the smallest eigenvalue of the symmetric matrix X.
Proof. By our assumptions, we have is not necessary since Σ is invertible. This procedure is not optimal. If N is large enough, we can well approximate N by k n k , at the cost of Σ becoming singular. However, after the dimensional reduction introduced above, the reduced covariance matrix is again invertible. The advantage of this approach is that we reduced the dimension m without discarding data, and hence the overall statistical performance is increased. This is due to the fact that r 1 and r 2 in this approach are the same as in the previous one. Note that an analysis using multinomial statistics yields the same reduced covariance matrix and hence the same parameters m, r 1 , and r 2 . For our examples in the main text we use the latter approach and assume N ≡ n for all settings . The parameters used for computing p fail in Fig. 2 are computed to be r 2 1 ≈ 0.0664 2 n and r 2 2 ≈ 0.0416 2 n for the qutrit example, and r 2 1 ≈ 0.0856 2 n and r 2 2 ≈ 0.0469 2 n in the ququart case.