Complete Characterization of Entanglement Embezzlement

Using local operations and classical communication (LOCC), entanglement can be manipulated but not created. However, entanglement can be embezzled. In this work, we completely characterize universal embezzling families and demonstrate how this singles out the original family introduced by van Dam and Hayden. To achieve this, we first give a full characterization of pure to mixed state LOCC-conversions. Then, we introduce a new conversion distance and derive a closed-form expression for it. These results might be of independent interest.


Introduction
Quantum entanglement [1][2][3][4][5][6] describes correlations between different particles with no classical counterpart and has both deep foundational implications [7] and numerous applications in quantum information science [8][9][10][11][12][13][14][15][16][17][18].If two or more parties are far apart, in practice, they are restricted to local operations and classical communication (LOCC) because during transmission over long distances, physical systems will unavoidably interact with an environment and eventually lose the quantum information they carry.In contrast, it is much simpler to exchange classical information that can easily be amplified and protected.The states that can be prepared with LOCC, i.e., the separable ones, are therefore considered free and the ones that cannot, which are exactly the entangled ones, are considered costly or resourceful.
According to this reasoning, entanglement is a resource and thus studied within the framework of quantum resource theories (QRTs) [19,20].In a QRT, a physically motivated restriction such as the one discussed above divides both states and operations into free or resourceful in a consistent manner: Free operations map free states into free states.Once the free operations and states have been fixed, a QRT studies which quantum advantages depend on the resource under consideration and, closely related, how the consumption of resourceful states can help to overcome the restriction.An example is quantum teleportation [11]: By consuming entangled states, LOCC allows to teleport quantum systems and thereby simulate arbitrary operations outside LOCC.
In this sense, consuming entangled states or more generally resourceful states can lead to operational advantages, e.g., in communication scenarios [8][9][10][11][12][13][14][15].Importantly, if a state can be converted with a free operation into another one, then the former is at least as valuable as the latter in any application that only allows for free operations.Answering the question which states can be converted into each other is thus a central question in Elia Zanoni: elia.zanoni@ucalgary.ca

Notation and Preliminaries
In this article, we restrict ourselves to finite-dimensional Hilbert spaces and denote them with capital Latin letters such as A, B. The dimension of a Hilbert space C is denoted by |C|, and the set of density matrices acting on it by D(C).For the set of pure states in D(C), we write PURE (C).Small Greek letters such as ρ and σ denote density matrices, with ψ, φ, and χ reserved for pure states.For ρ, σ ∈ D(C), the trace distance between ρ and σ is defined as 1  2 ∥ρ − σ∥ 1 , where ∥•∥ 1 is the trace norm.Probability vectors are represented by bold small Latin letters, e.g., p, with p x the x-th component of p.The set of probability vectors of length d is denoted by Prob(d), which contains the subset Prob ↓ (d) consisting of all d-dimensional probability vectors with non-increasing entries.The k-th Ky Fan norm of p ∈ Prob ↓ (d) is defined as where [d] is a shorthand notation for {1, . . ., d}.We write p ≻ q if p majorizes q [62][63][64] and ∥p − q∥ 1 for x |p x − q x |.Since this article is concerned with two spatially separated parties, call them Alice and Bob, it is important to make clear which system is under the control of whom: Systems belonging to Alice will always be denoted by A or A ′ , and systems belonging to Bob by B or B ′ .Quantum channels are represented by calligraphic large Latin letters such as M, N , and the set of quantum channels from a bipartite system AB to A ′ B ′ that Alice and Bob can implement if they are restricted to local operations and classical communication is denoted by LOCC(AB → A ′ B ′ ).For bipartite systems AB, we assume w.l.o.g. that |A| = |B| = d (since with LOCC, Alice and Bob can always attach and remove local auxiliary systems).Fixing an orthonormal basis { |x⟩ A } x∈ [d] for A and { |x⟩ B } x∈ [d] for B, under LOCC, every ψ ∈ PURE(AB) is then equivalent to its standard form x √ p x |xx⟩ AB , where p ∈ Prob ↓ (d) are the Schmidt coefficients of ψ (which, w.l.o.g., we will always assume to be ordered non-increasingly from here on).We denote with SR(ψ) the Schmidt rank of a pure bipartite state ψ ∈ PURE(AB), i.e., the number of non-zero Schmidt coefficients of ψ, and use the symbol m |xx⟩ AB for the maximally entangled state on AB, where |A| = |B| = m.Finally, for ρ ∈ D(AB) and σ ∈ D(A ′ B ′ ), we write ρ LOCC − −−− → σ whenever there exists an N ∈ LOCC(AB → A ′ B ′ ) such that σ = N (ρ) and ρ LOCC − −−− → { t z , τ z } whenever there exists a probabilistic LOCC protocol that converts ρ to τ z with probability t z .

State conversions with LOCC
As motivated in the introduction, the question of how entangled states can be interconverted is at the core of the resource theory of entanglement.The exact deterministic LOCC-conversion problem between pure states is solved by Nielsen's Theorem [3]: For ψ, φ ∈ PURE(AB) and p, q ∈ Prob ↓ (|A|) their associated Schmidt vectors, ψ LOCC − −−− → φ if and only if [65] E k (ψ) where E k (ψ) := 1 − ∥p∥ (k) .In Ref. [21], this result was generalized to the case where the target state φ is replaced with an ensemble of states: For ψ, φ 1 , . . ., φ n ∈ PURE(AB), which can be rewritten as We extend this result considering a generic mixed state as target.
Proposition 1.For ψ ∈ PURE(AB) and σ ∈ D(AB), ψ LOCC − −−− → σ if and only if there exists a pure state decomposition { p z , χ z } of σ (i.e., σ = z p z χ z ) such that That this condition is sufficient follows directly from Eq. (4).That it is also necessary follows from the Lo-Popescu Theorem [66] which implies that if ρ LOCC − −−− → σ, then there exists a pure state decomposition { p z , χ z } of σ that satisfies Eq. (4).The details of this proof can be found in Appendix A.
For certain choices of ρ ∈ D(AB) and σ ∈ D(A ′ B ′ ), ρ cannot be converted to σ by LOCC.It is then interesting to determine how well we can approximate σ with ρ and LOCC.To this end, one can associate with every distance D defined on quantum states the conversion distance This conversion distance determines how close to σ, with respect to the distance D, one can convert ρ using only LOCC.A commonly used distance in quantum information is the trace distance T (σ, τ ) = 1 2 ∥σ − τ ∥ 1 .The conversion distance associated with it is This conversion distance has an operational interpretation in terms of a result in state discrimination known as Holevo-Helstrom Theorem [67,68] (see Ref. [69,Theorem 3.4] for a review).Indeed, if a single copy of either σ ∈ D(AB) or N (ρ) ∈ D(AB), with N ∈ LOCC, is given with equal probability, then the maximum probability p max of correctly identifying the given state is bounded by Moreover, for every fixed ρ and σ, there always exists an N ∈ LOCC such that p max is arbitrarily close to this lower bound.In this sense, T (ρ → σ) describes how well we can approximate σ given access to ρ and LOCC.
Another distance used often in quantum information is the purified distance P (σ, τ ) = inf ψσ,ψτ T (ψ σ , ψ τ ) [26,27], where the infimum runs over all purifications ψ σ and ψ τ of σ and τ , respectively.In this case, the conversion distance is analogously defined as On pure states ψ, φ ∈ PURE(AB), we define what we call star conversion distances via where p, q ∈ Prob ↓ (|A|) are the Schmidt coefficients of ψ and φ, respectively, and D(r, q) = D(diag(r), diag(q)).In general D ⋆ (ψ → φ) ̸ = D(ψ → φ).However, we show in Appendix B that in the case of the purified distance, the two conversion distances coincide on pure states.
This, in turn, is useful to show that T (ψ → φ) and T ⋆ (ψ → φ) are topologically equivalent on pure states, which means that if one can approximate φ arbitrarily well with a sequence of states { ψ n } in the sense that lim n→∞ T (ψ n → φ) = 0, then the same is true for T ⋆ , and vice-versa.This is guaranteed by the following result (Appendix C).
We are particularly interested in T ⋆ (ψ → φ) because using tools from approximate majorization [62][63][64][70][71][72][73], one can derive a closed-form expression for it (Appendix C).Theorem 4. Let ψ, φ ∈ PURE(AB) and let p, q ∈ Prob ↓ (|A|) be their corresponding Schmidt coefficients.Then, If ξ ∈ PURE(AB) is separable and ψ, φ are not, This theorem provides, to the best of our knowledge, the first algorithm to compute a conversion distance in LOCC with a finite number of steps.Indeed, if ψ, φ, and their Schmidt coefficients are known, then one can compute T ⋆ (ψ → φ) using a finite memory, and a finite (perhaps very large) number of operations (additions or subtractions).This is not the case for the other conversion distances because they require a minimization over LOCC, which is in general unfeasible.The second part of the theorem shows that any entangled state is more useful in the approximation of all other entangled states than any separable state.
Whilst Theorem 4 is of independent interest, for example in the characterization of entanglement distillation and dilution [74], we discuss in the following how it yields new results concerning the embezzlement of entanglement.When referring to the conversion distance and star conversion distance, we will thus refer to the versions based on the trace distance from here on.

Entanglement embezzlement
As mentioned in the introduction, it is impossible to create additional entanglement with LOCC alone [2,5,19,34].If ρ ∈ D(A ′ B ′ ) is an entangled state, this implies that there cannot exist a ψ ∈ PURE(AB) and a channel N ∈ LOCC(AB → ABA ′ B ′ ) such that N (ψ) = ψ ⊗ ρ, because this would increase the total amount of entanglement between systems AA ′ and BB ′ with respect to any additive entanglement measure.It might however be possible to approximate ψ ⊗ ρ in the sense that T (ψ → ψ ⊗ ρ) ≤ ε for a small ε.In this case, it is hard to distinguish ψ ⊗ ρ from the approximation.By keeping the systems A ′ B ′ , one would thus embezzle entanglement from the owner of ψ -and if one would be able to make ε arbitrarily small, it would be impossible to detect.
In fact, in Ref. [32], van Dam and Hayden showed that it is possible to embezzle any bipartite state σ ∈ D(A ′ B ′ ) arbitrarily well from a family of pure states χ AB n n∈N in the sense that lim n→∞ T (χ n → χ n ⊗ σ) = 0.This implies that an arbitrarily good approximation of any σ can be embezzled from χ n whilst changing χ n arbitrarily little, as long as n is large enough.This motivates the following definition.

Definition 5. A family of pure bipartite states
This definition is very similar to the one provided in Refs.[32,33].The main difference is that these works only consider protocols using local operations (LO), while in this work, we allow for classical communication too.The set of operations that we consider is therefore larger, and as a result, if a family of states is not an embezzling family according to our definition, then it is not an embezzling family in the sense of Refs.[32,33].Surprisingly, the original embezzling family proposed in Ref. [32] is rather unique in a sense that we will specify later, even when we allow for classical communication.Another difference with Refs.[32,33] is that in those works the fidelity was used to quantify the conversion error.Since the fidelity and trace distance are topologically equivalent, this is irrelevant.
The term 'universal' in Definition 5 underlines the property that any bipartite state can be embezzled.Since for every bipartite state σ ∈ D(AB), where A = B = m, it is possible to LOOC-convert the maximally entangled state Φ m into σ (see, e.g., Proposition 1), it is enough to check whether lim n→∞ T (χ n → χ n ⊗ Φ m ) = 0 for all m ∈ N to determine if { χ n } n∈N is a universal embezzling family: This follows for example by combining Lemma 3 and the triangular inequality for the star conversion distance proven in Appendix C.Even simpler, it is in fact equivalent to only require lim n→∞ T (χ n → χ n ⊗ Φ 2 ) = 0 [33, Lemma 2 for the case of embezzlement with LO], because if one can embezzle enough copies of Φ 2 , then one can convert them into Φ m with LOCC (see Appendix D for more details).It is also important to note that technically, one could have required that lim inf n→∞ T (χ n → χ n ⊗ σ) = 0, since this would also allow to embezzle any state arbitrarily well.However, since one can always choose a subfamily, we decided to keep the definition in line with Ref. [33].Lastly, due to Eq. (11), we can replace where The problem of determining if a family of states is a universal embezzling family has therefore been restated as a rather simple optimization problem, which in many cases can be solved numerically or even analytically.By choosing l = 1 in Eq. ( 14) one obtains the following necessary condition for a universal embezzling family.This implies that if a family of pure bipartite states { χ n } n∈N is a universal embezzling family, then lim n→∞ SR(χ n ) = +∞, where SR(χ n ) is the Schmidt rank of χ n .Indeed, considering that the entries of any p (n) sum to one, if the largest entry converges to zero, then the number of non-zero entries must diverge.
When looking for candidates for universally embezzling families, it is common [32,33] to consider families of bipartite states defined by a function f : N → R + , where With this choice, one ensures that the families of states have a common structure, i.e., χ m>n is obtained from χ n by appending additional coefficients and renormalizing.By choosing f (x) = x −1 , one recovers the universal embezzling family introduced by van Dam and Hayden [32].This family is rather unique: If we assume that f has a reasonable asymptotic behavior, specifically that it is asymptotically non-increasing and f (x)/x α is asymptotically monotonic for all α ∈ R, then the family of states { χ n } is an embezzling family if and only if f is asymptotically close to x −1 in the sense that for every ε > 0, As we will discuss now, the assumptions on the asymptotic behavior of f , which we require for technical reasons, are not particularly restrictive.In entanglement theory, it is possible to consider, w.l.o.g., only states with non-increasing Schmidt coefficients.With the first assumption, we require that the family of states, at least asymptotically, has this behavior.
The second assumption rules out functions that are asymptotically monotonic but have oscillating components, for an example of such a function see Appendix G, Eq. (G23).In Ref. [33], it was shown that certain functions that differ from x −1 by logarithmic factors lead to universal embezzling families too.Since these functions satisfy the constraints in Eq. (16), this is in accordance with our findings.In addition, in Appendix G, we also show that many non-decreasing functions do not lead to universal embezzling families (see Propositions G.3 and G.4 for details).
A special case of the functions discussed so far are the functions f α (x) = x α with α ∈ R. For the families of states { χ α n } generated by such functions, we analytically compute the exact value of lim n→∞ T ⋆ (χ α n → χ α n ⊗ Φ m ) for α ≥ −1 and lower and upper bound it for α < −1.The details of the computations can be found in Appendix F and the results for m = 2 are shown in Figure 1.Clearly, the limit of the star conversion distance is zero only for α = −1, showing again the uniqueness of the choice made by van Dam and Hayden amongst the functions f (x) = x α .

Conclusions
In Proposition 1 we provided necessary and sufficient conditions for a deterministic LOCCconversion from a pure bipartite state to a mixed bipartite state.This extends the results already known for pure to pure state LOCC-conversions, whether deterministic or probabilistic [3,21].We then exploited this result to prove the topological equivalence of the newly defined star conversion distance between pure states and the trace conversion distance commonly used in literature.The star conversion distance exhibits a closed formula (Theorem 4).This is remarkable, since the mixed state LOCC-conversion problem is NP-hard [75].
The closed formula in turn allowed us to completely characterize universal embezzling families (see Definition 5 and Refs.[32,33] for an analogous definition for LO) in terms of a simple optimization problem stated in Theorem 6.With this characterization at hand, we discussed the uniqueness of the van Dam and Hayden family [32].For specific families of states generalizing the van Dam and Hayden family, we explicitly evaluated their star conversion distance to maximally entangled states and showed that they are only universally embezzling if they are exactly the van Dam and Hayden family, see Figure 1.Therefore, the van Dam and Hayden embezzling family shows unique properties even for protocols that involve classical communication, as already noticed in Ref. [32].This suggests a direction for future work, namely, to determine whether LOCC embezzlement implies LO embezzlement (the other direction is trivial), and therefore to investigate if classical communication is relevant in entanglement embezzlement or not.It is worth noting that so far, the research on entanglement embezzlement focuses solely on families composed of pure states.A more comprehensive theory of embezzlement that includes families of mixed states is yet to be developed.
Originally introduced in the resource theory of entanglement as a generalization of catalysis [32], embezzlement has recently also been studied in other resource theories including non-uniformity [76,77], coherence [78][79][80][81], and athermality [76,[82][83][84][85].Moreover, fundamental limits for embezzlement have been proved in Ref. [86] and applied to the resource theories mentioned above.The aforementioned resource theories are related to the resource theory of entanglement via majorization, which is the tool that we used to derive the closed formula for the star conversion distance.A natural next step is to investigate whether it is possible to derive a similar formula in these other majorization-based resource theories.However, this task is not trivial: In the resource theory of non-uniformity, the majorization relation is inverted, and the free state is the maximally mixed state, which is fundamentally different from the free states in the resource theory of entanglement; in the resource theory of athermality, state conversion is described by relative majorization, which is a generalization of majorization.It is worth mentioning that in Ref. [84] it has been shown that embezzlement of athermality allows to violate the second law of thermodynamics.However, by using the work distance [84] as conversion distance, or by imposing physical constraints on the catalyst [76], e.g., finite dimension or finite energy expectation value, athermality embezzlement is no longer possible, thus restoring the validity of the second law of thermodynamics.
As described in the introduction, universal embezzling families have been useful in many applications [15,18,[55][56][57][58][59][60][61].Our results provide an easy way to check whether a family is universally embezzling.Moreover, we ruled out large classes of potential candidates that are fundamentally different from the van Dam and Hayden family.The complete characterization of universally embezzling families therefore contributes to a more efficient usage of entanglement in technology.

A Pure to Mixed State Conversions with LOCC
In this section, we prove Proposition 1 of the main text, which we restate for readability.
Proof.We first assume that ψ can be converted into σ with LOCC operations.Then, according to Refs.[66,87], there exists a protocol in which Alice performs a generalized measurement { M z } and Bob performs a unitary transformation U z conditioned on the measurement's outcome such that If Alice and Bob record the outcome of the measurement in a classical system X, the output of the protocol is where This shows that Alice and Bob can convert ψ into the ensemble { p z , χ z } with LOCC operations, which is equivalent to the condition [21,65] min where E k was introduced in the main text and d = |A| = |B| as per our convention.This proves the necessary condition.
For the reverse, we assume that { p z , χ z } is a pure state decomposition of σ that satisfies min [21,65].Alice and Bob can trace out the classical system and they obtain z p z χ z = σ, thus ψ can be converted into σ with LOCC operations.

B Purified Conversion Distance
In this section, we present some results concerning the purified conversion distance [26,27], which are useful to prove the theorems about the star conversion distance presented in the main text.The purified distance between two states ρ, σ ∈ D (AB) is defined as where is the fidelity.The purified distance is a metric and according to Uhlmann's Theorem [23], where T (ρ, σ) = 1 2 ∥ρ − σ∥ 1 is the trace distance and the minimization runs over all purifications ψ and φ of ρ and σ, respectively.Importantly, the purified distance is topologically equivalent to the trace distance [26], We now recall the definition of the purified conversion distance and purified star conversion distance introduced in the main text, where p, q ∈ Prob ↓ (d) are the Schmidt coefficients of ψ and φ, respectively, and P (r, q) = 1 − F 2 (r, q) is the classical version of the purified distance with We next prove that the two purified conversion distances recalled above are equal on pure states.To this end, we need the following Lemma.
Proof.For details about concave functions, see Ref. [88].Here, we are going to use that a twice differentiable function is concave if and only if its Hessian matrix is negative semidefinite.The functions and their Hessian matrices are given by x qxqy vxvy qxqy vxvy Since det H(v x , v y ) = 0, one of the eigenvalues of H(v x , v y ) is 0 and the other is equal to the trace of H(v x , v y ), which is smaller than or equal to zero.This implies that the functions From this follows that f is concave, since is the sum of concave functions.
We are now ready to prove the promised theorem.
Proof.We assume w.l.o.g. that all pure states are in standard form, that is, |ψ⟩ =  B, respectively (see main text for more details).Let p, q ∈ Prob ↓ (d) be the Schmidt coefficients or ψ and φ, respectively.With r ∈ Prob ↓ (d), as a consequence of Nielsen's Theorem [3], r ≻ p if and only if there exists an N ∈ LOCC such that N (ψ) ∈ PURE(AB) has Schmidt coefficients r.We notice that This implies that This inequality follows from the definition of the star purified conversion distance because the minimization in the star conversion distance is done over a smaller set.The non-trivial part is to show that the opposite inequality holds as well.To this end, we want to show that for every mixed state σ such that ψ LOCC − −−− → σ, there exists a pure state χ such that ψ LOCC − −−− → χ and P (σ, φ) ≥ P (χ, φ).It is then sufficient to consider only pure output states for the computation of the purified conversion distance, which implies the reverse inequality.
Let σ = M(ψ) ∈ D(AB), where M ∈ LOCC.Also, let { t z , χ z } be a pure state decomposition of σ that satisfies Eq. (A1) (where the χ z are not necessarily in standard form), and s (z) be the Schmidt coefficient of χ z for every z.Furthermore, for every χ z , let χz be the pure state in standard form that is equal to χ z up to local unitaries, let σ = z t z χz , and define χ ∈ PURE(AB) as the pure bipartite state (in standard form) with Schmidt coefficients s = z t z s (z) .We notice that for all k ∈ where the last inequality follows from the fact that { t z , χ z } satisfies Eq. (A1).This implies that s ≻ p.

C The Star Conversion Distance
In this section, we discuss properties of the star conversion distance based on the trace distance and provide proofs omitted in the main text.We begin by proving that the trace star conversion distance is topologically equivalent to the standard conversion distance defined via the trace distance.
Proof.Using Eq. (B3) and Theorem 2, we obtain where T (p, q) := T (diag(p), diag(q)).Analogously, In the following, we provide the proof of Theorem 4 of the main text, which we restate below for readability.In other words, we derive a closed formula for T ⋆ (ψ → φ) based on the Schmidt coefficients p, q ∈ Prob ↓ (d) of ψ and φ, respectively.Theorem 4. Let ψ, φ ∈ PURE(AB) and let p, q ∈ Prob ↓ (|A|) be their corresponding Schmidt coefficients.Then, If ξ ∈ PURE(AB) is separable and ψ, φ are not, then As shown in Ref. [73], there exist probability vectors q(ε) ∈ B ε q called steepest ε-approximations of q such that q(ε) ≻ q ′ for all q ′ ∈ B ε q .Moreover, these steepest ε-approximations can be constructed explicitly: The components of q(ε) are then given by q(ε) We now show that min First, we notice that if r ⋆ is an optimizer of min r≻p q and therefore q(ε) ≻ r ⋆ .By transitivity, we also have q(ε) ≻ p, which implies that For the reverse inequality, let ε ⋆ be the optimizer of min ε ∈ [0, 1] : q(ε) ≻ p .By definition, q(ε⋆) ≻ p and 1 2 q − q(ε⋆) This shows that We observe that q(ε) This expression is further simplified by noticing that p is a probability vector, and therefore In combination with the minimization in Eq. (C11) follows that To conclude the proof of the first part of the theorem, we observe that for k ≥ SR(ψ AB ), ∥p∥ (k) = 1 and ∥q∥ (k) is non-decreasing with k, thus we can restrict the maximization to k ≤ SR(ψ AB ).
For the second part, we notice that according to Eq. (C11), Furthermore, from e 1 ≻ p and the transitivity of the majorization-relation, it follows that To rule out equality, suppose that and denote with k ⋆ an index that achieves this maximum.Then From this expression follows that ∥p∥ (k⋆) ≤ 1 only if either q = e 1 or if k ⋆ = 1, and therefore p = e 1 .These conditions are in contrast with the assumption that ψ and φ are not separable.As a consequence, equality in Eq. (C16) is unachievable, which proves the second part of the theorem.
Next, we show that the star conversion distance satisfies a triangle inequality.Let ψ, φ, and χ ∈ PURE(AB) and let p, q, and r be their Schmidt coefficients.With the help of Eq. (C14), this implies that Remark.The star conversion distance T ⋆ (ψ → φ) was so far only defined for pure bipartite states belonging to the same Hilbert space.This restriction is easily lifted by noting that one can always add separable auxiliary states such that the Hilbert spaces (or dimensions) match.This can be done in multiple ways: Since in fact we are only interested in the dimension of systems and their spatial separation, a more compact equivalent notation that we will use later can be defined as follows: Denote with m, m ′ > 0 the smallest integers such that md = m ′ d ′ , and with

D Universal Embezzling Families
In the main text, we provided the following definition of universal embezzling families, which we repeat here for readability.
Proof.Clearly 2. and 3. follow from 1. due to the definition of universal embezzling families and the topological equivalence of T and T ⋆ .Moreover, 3. follows from 2. because Φ (D1) By taking the limit n → ∞ on both sides we obtain the desired result.To conclude, we note that 3. implies 1., since for all σ AB , Φ |A| Next, we derive a formula for the conversion distance T ⋆ (χ → χ ⊗ Φ m ) using Eq.(C14) and Eq.(C20).where u (m) = (1/m, . . ., 1/m). is straightforward to see that and by writing k = a k m + b k , where a k = ⌊k/m⌋, Using the closed formula for the star conversion distance given in Theorem 4, we obtain A simplified version of this expression can be used to characterize embezzling families. where Proof.Due to Lemma D.1 and Eq.(D2), { χ n } n∈N is a universal embezzling family if and only if where a k = ⌊k/2⌋ and k = 2a k +b k .First, we prove the necessary condition.Let { χ n } n∈N be a universal embezzling family.We observe that max Taking the limit for n → ∞ in the expression above, we obtain lim n→∞ p and thus 0 = lim For the sufficient condition, we observe that which implies that lim n→∞ p If k is even, then a k = k/2 = a k+1 , and thus If SR(χ n ) is odd, we can thus without loss of generality restrict the maximization to run Assume that this is the case: It then holds that (D16) which vanishes in the limit n → ∞.As a consequence, Eq. (D17) shows that { χ n } n∈N is an embezzling family and concludes the proof.
An important and easy to check necessary condition for a universal embezzling family is given in the following Corollary.

E Regular Embezzling Families of States
Often, families of states are defined in terms of a positive, monotonic, and continuous function.This motivates following definition (compare to Ref. [33]).Definition E.1.A family of states { χ n } n∈N is a regular family if there exists a monotonic function f : N → (0, ∞) such that where Remark.If { χ n } n∈N is a regular family, then there exists a sequence { n j } j∈N such that the family χ n j j∈N is a universal embezzling family if and only if lim inf n→∞ T ⋆ (χ n → χ n ⊗ Φ m ) = 0.However, according to our definition, the family χ n j j∈N is then no longer regular, unless n j = j for all j ∈ N.
For the following proofs, it is beneficial to extend f to a monotonic continuous function on [1, ∞).This can often be done trivially by simply extending the domain of f .An example is the van Dam and Hayden family where f (x) = x −1 .Otherwise, we can extend f by connecting two consecutive points with straight lines.If the function f is multiplied by a constant factor, this does not change the corresponding family of states.Therefore, from now on we assume for simplicity that f (1) = 1.
Let { χ n } n∈N be a regular family of states and let f be the function associated to it.We define the non-increasing functions f n , n ∈ N, by By our extension of f , f n is also naturally extended to a continuous function g(x, y) such that g(x, n) = f n (x) via the definition The (by definition non-increasing) Schmidt coefficients of χ n are therefore given by In the following proposition, we present bounds on the limit of the conversion distance T ⋆ (χ n → χ n ⊗ Φ m ) in terms of the function g(x, y) introduced in Eq. (E3), which will be of use later.
Proposition E.3.Let { χ n } n∈N be a regular family of states, f be the function associated to it, and g be defined as in Eq. (E3).
Proof.For better readability, we divide the proof into steps and use the notation G(x, y) = x 1 g(t, y) dt.
Step 1: Starting from Eq. (D2), we find Now we observe that the last contribution in the expression above vanishes because by construction f n (x) ≤ 1 and Moreover, by assumption, the right-hand sides converge to zero in the limit n → ∞, and this ensures that the limit inferior is additive.As a result, where we used that p (n) (0) = 0.
Step 2: We start with finding bounds for n x=1 f n (x) − n 1 f n (x) dx.To this end we observe that since f n (x) is by construction non-increasing, By subtracting n 1 f n (x) dx from all terms, the desired bounds follow: Step 3: In this step we want to show that lim inf Analogously to the previous step, one can obtain the following bounds for am x=a+1 fn(x) Fn : and after taking the maximum over a ∈ [⌊n/m⌋] and the limit inferior the bounds become lim inf We can rewrite the last expression as where we replaced F n with G(n, n), which we can do according to step 2.
Step 4: Let c n be the value that maximizes max 1≤a≤n/m { G(am, n) − G(a + 1, n) } and let ′ n be the natural number satisfying c Dividing by G(n, n), taking the limit inferior, and arguing as in the previous step, we obtain The reverse inequality follows from the fact that we maximize over a larger set.This proves that lim inf Step 5: Combining the results of the previous steps, and in particular Eq.(E10), Eq. (E18), and Eq.(E21), we have proven that In the above equation, we can replace am a+1 g(x, y) dx with am a g(x, y) dx because the difference of the two integrals is finite and divided by a diverging term.Thus The expression inside the lim inf on the right-hand side of Eq. ( E23) is a function of n ∈ N \ { 1 } and we call it M (n), We extend this function to real numbers y > 1 as follows: Since the natural numbers are a subset of the real numbers, which proves Eq. (E5).Furthermore, Steps 1-4 can be repeated with exactly the same arguments for lim sup, and they imply This proves Eq. (E6).
In the following, we assume that lim y→∞ M (y) exists.This implies From Eq. (E26), Eq. (E27), and Eq.(E28) we obtain Therefore, lim n→∞ M (n) exists and is equal to lim y→∞ M (y).From Eq. (E23) and Eq.(E27) we derive This proves Eq. (E7) and concludes the proof.In addition, thanks to Proposition E.3, the task of determining if a family of states is a universal embezzling family is converted into an optimization problem.It is enough to find the maximum of the differentiable function G y (a) = am a g(x, y) dx on 1 ≤ a ≤ y/m, which is easily done with the help of its derivative mg(am, y) − g(a, y), 1 ≤ a ≤ y/m. (E33)

F Generalization of the van Dam and Hayden Family
The universal embezzling family { χ n } n∈N introduced by van Dam and Hayden consists of the states where H n = n x=1 x −1 is the n-th harmonic number.We generalize this family of states as follows: For every α ∈ R we introduce the family χ , where and x α is the n-th generalized harmonic number.Note that the families of states χ are regular families of states, and the van Dam and Hayden family is recovered for α = −1.The corresponding Schmidt coefficients, as per our convention arranged in non-increasing order, are In the remaining part of this section, we show that the family of state χ is a universal embezzling family if and only if α = −1 and derive bounds on the star conversion distance.
is finite, thus the regular family of states { χ α n } n∈N is not a universal embezzling family (see Corollary E.2).The largest Schmidt coefficient provides a lower bound on the star conversion distance: Using Eq. (C14), we obtain where the inequality followed from choosing k = 1.By taking the limit, we obtain x α is the Riemann Zeta function.To obtain an upper bound, we observe that, since α < −1, The right-hand side is decreasing in a, and thus Taking the limit n → ∞, and considering that the conversion distance is by definition smaller than one, we obtain Case α = −1.This is the van Dam and Hayden family.Analogously to the previous case, we observe that Since H (1) n = H n diverges, by following the same steps, we have Thus, the limit for the conversion distance is zero as expected for the van Dam and Hayden family.
diverges when n → ∞, and we can use the results of Section E.Here g(x, y) = x α , and the derivative with respect to a of the function which is positive in the domain 1 ≤ a ≤ y/m.This implies that the function am a g(x, y) dx is non-decreasing and the maximum is obtained for a = y/m.Due to Proposition E.3, Case α = 0.Here we have We can compute the star conversion distance directly using Eq.(C14) and obtain Case α > 0. Also here H (−α) n diverges for n → ∞ and we can use the results of Section E. The function g(x, y) is defined as g(x, y) = (y + 1 − x) α /y α , because x α is increasing.From this follows Since a max increases linearly with y and , for large enough y, the value a max belongs to the interval [1, y/m] and is a global maximum.Plugging a max into Proposition E.3, we obtain This completes the study of regular families of states defined by f (x) = x α .The star conversion distance vanishes only for α = −1, thus the only universal embezzling family of this form is the one introduced by van Dam and Hayden.Furthermore, exact values for the limit of the star conversion distance for α ≥ −1 and lower and upper bounds for α < −1 were provided.

G Uniqueness of the van Dam and Hayden Embezzling Family
In this section, we provide further results on the uniqueness of the van Dam and Hayden embezzling family.Given a regular family of states { χ n } n∈N , the asymptotic behavior of the function f associated to it is relevant to determine whether { χ n } n∈N is a universal embezzling family or not (e.g., see Corollary E.2).We thus use the notations little-ω and little-o (see, e.g., Ref. [92]) to describe asymptotic relations between two functions Before we state the results about the uniqueness of the van Dam and Hayden embezzling family, we prove the following Lemma, which is a direct consequence of Theorem 2 in Ref. [93], which we restate here to improve readability: Let µ be a measure on the real line R, and let f i , g i (i = 1, 2) be four Borel-measurable functions: R → R such that f 2 ≥ 0 and g 2 ≥ 0, and |f i g j | dµ < ∞ (i, j = 1, 2).If f 1 /f 2 and g 1 /g 2 are monotonic in the same direction, then , where χ S (x) is the characteristic function of the set S. Furthermore, let The sequence of functions { h k } k∈N converges to h(x) and satisfies 0 ≤ h k (x) ≤ 1. Due to the dominated convergence theorem (see, e.g., Ref. [94]), we have The same result holds for g.Since both f (x)/g(x) and h(x)/h k (x) are non-decreasing, we can apply Theorem 2 of Ref. [93] to (G6) After taking the limit k → ∞ on both sides, we obtain b a f (x) dx which finishes the proof.
With this lemma at hand, we are ready to present the promised results concerning the uniqueness of the van Dam and Hayden embezzling family.Theorem G.2. Let f be a positive non-increasing function such that f (x)/x α is asymptotically monotonic for all α ∈ R and let { χ n } n∈N be the regular family of states associated to it (see Definition E.1).Then { χ n } n∈N is a universal embezzling family if and only , where { n j } j∈N is any sequence of natural numbers, is not a universal embezzling family.
Proof.Before we start, we notice that if lim j→∞ n j = J < ∞, then lim j→∞ F n j = J x=1 f (x) < ∞, which implies that the family of states χ n j j∈N is not an embezzling family (see Corollary E.2).In this proof we will thus assume, w.l.o.g., that lim j→∞ n j = ∞.Necessary condition -Let { χ n } n∈N be a universal embezzling family with corresponding function f satisfying the assumptions above.We already proved in Corollary E.2 that if { χ n } n∈N is a universal embezzling family, then ∞ x=1 f (x) = +∞.To show the remainder, let Since lim x→∞ f (x)/x α = 0 for α > 0, we have that R ̸ = ∅.Now we prove by contradiction that also L ̸ = ∅.Let us assume that L is empty.This implies that lim x→∞ f (x)/x α = l α < for all α.If there exists an α such that l α ̸ = 0, then lim x→∞ f (x)/x α−1 = l α (lim x→∞ x) = +∞.Thus, α − 1 ∈ L, and L ̸ = ∅, leading to the desired contradiction.If instead lim x→∞ f (x)/x α = 0 for all α, then lim x→∞ f (x)/x −2 = 0.This implies that f (x)/x −2 converges monotonically to zero for large x, i.e., there exists an N such that f (x) < x −2 for x > N .Thus, According to Corollary E.2, this contradicts the hypothesis that the family under consideration is universally embezzling.We have therefore shown that L ̸ = ∅ ̸ = R.The next step is to prove that inf R = sup L. From the definition of R and L follows that inf R ≥ sup L. Let us assume that inf R > sup L, i.e., that there exists an α ∈ R such that sup L < α < inf R. Since α / ∈ L ∪ R, there exists a positive real number l such that lim x→∞ f (x)/x α = l.Now pick α 1 such that sup L < α 1 < α < inf R. We observe that lim x→∞ f (x)/x α 1 = l(lim x→∞ x α /x α 1 ) = +∞, thus α 1 ∈ L. This is in contradiction to the choice α 1 > sup L. We therefore showed that sup L = inf R.
So far, we have shown that if { χ n } is a universal embezzling family satisfying our assumptions, then there exists a unique α such that f (x) ∈ o(x α+ε ) ∩ ω(x α−ε ) for every ε > 0. What is left to show is that if α ̸ = −1, then the family of states corresponding to f cannot be universally embezzling.From the above discussion, we know that { α > 0 } ⊆ R, thus we can focus on α ≤ 0 and split our discussion into two scenarios, α < −1 and −1 < α ≤ 0.
If α < −1, then there exists an ε > 0 such that α + ε < −1.Since f ∈ o(x α+ε ) and f (x)/x α+ε converges monotonically to zero for large x, there exists an N such that f (x) < x α+ε for x > N .This implies that which, according to Corollary E.2, contradicts the hypothesis that { χ n } is a universal embezzling family.Let { n j } j∈N be any sequence of natural number such that lim j→∞ n j = ∞.Then lim j→∞ This implies, again due to Corollary E.2, that χ n j j∈N is not a universal embezzling family.
If −1 < α ≤ 0, then there exists an ε > 0 such that α − ε > −1.We notice that f (x)/x α−ε diverges to infinity for large x, because f ∈ ω(x α−ε ).This implies that there exists an N such that f (x) ≥ x α−ε for x > N and therefore ∞ x=N f (x) ≥ ∞ x=N x α−ε = ∞, which allows us to use the results of Section E. Since f (x)/x α−ε is non-decreasing for x > N , for all y such that y/m > N , we can use Lemma G.1 to obtain Next we introduce δ := α − ε + 1 > 0 and take on both sides the limit inferior y → ∞, leading to 1 where in the last equality, we used that N 1 f (x) dx is finite, whilst ∞ N f (x) dx diverges.As last step, we observe that according to Proposition E.3, Using the remark after Definition E.1, we obtain that for −1 < α ≤ 0, there are no sequences { n j } j∈N such that χ n j j∈N is a universal embezzling family, contradicting the hypothesis.Once we combine the results for α < −1 and −1 < α ≤ 0, we have that if { χ n } is a universal embezzling family satisfying our assumptions, f (x) ∈ o(x −1+ε )∩ω(x −1−ε ) for all ε > 0. We have also shown that if f / ∈ o(x −1+ε ) ∩ ω(x −1−ε ) for at least one ε > 0, then there are no sequences { n j } j∈N such that the family χ n j j∈N is a universal embezzling family.This concludes the first part of the proof.Sufficient condition -Let f be a function satysfying our assumptions.Since f (x)/x −1 is asymptotically monotonic by assumption, lim x→∞ f (x)/x −1 exists in [0, +∞].A priori, it can be either 0, 0 < l ∈ R, or +∞.If lim x→∞ f (x)/x −1 = l > 0, then by definition of the limit, for any l > ε > 0 there exists an N such that f (x) x −1 − l < ε for all x > N .This is equivalent to Suppose now lim x→∞ f (x)/x −1 = 0. Since f (x)/x −1 is non-increasing for large x, there exists an Ñ such that f (x) < x −1 for all x > Ñ .For any fixed ε > 0, by hypothesis, lim x→∞ f (x)/x −1−ε = +∞.Furthermore, since by assumption ∞ 1 f (x) dx = ∞, while ∞ 1 x −1−ε dx < ∞, there exists an Ñε , such that y 1 f (x) dx ≥ y 1 x −1−ε dx for y > Ñε .
Let us now consider the case lim x→∞ f (x)/x −1 = ∞.From this immediately follows that ∞ x=1 f (x) = +∞, so we can use the results of Section E. Furthermore, there exists an N such that f (x)/x −1 is non-decreasing for x ≥ N .We write f (x) = x −1 h(x) and thus h(x) is non-decreasing for x ≥ N .Computing the derivative of am a f (x) dx for a ≥ N , we find From this follows that Combining Eq. (G17) and Eq.(G20) we get This is true for all ε > 0, thus taking the limit ε → 0 we obtain the desired result This concludes the proof of the sufficient condition.
The result on the uniqueness of the van Dam and Hayden family can be expressed as follows: Any regular family of states { χ n } n∈N satisfying the conditions of Theorem G.2 is a universal embezzling family if and only if f , the function associated to { χ n } n∈N , is asymptotically close to x −1 , the function associated to the van Dam and Hayden family, where asymptotically close means f ∈ ω(x −1−ε ) ∩ o(x −1+ε ) for all ε > 0.
The assumption that f (x)/x α is asymptotically monotonic for every α ∈ R is crucial for our proof and does not follow from the monotonicity of f .There are functions that are non-increasing, but oscillate asymptotically when multiplied by powers of x.An example is the function The last inequality is based on Lemma G.1 and is derived as in the previous cases.Using the remark after Definition E.
Also here the last inequality is due to Lemma G.1 and the family of states is not a universal embezzling family.Furthermore, there are no sequences { n j } j∈N such that χ n j j∈N is a universal embezzling family.

H Asymptotically Regular Families
In the definition of universal embezzling families, Definition 5, and in all the results about embezzlement, only the asymptotic behaviour of a family of states { χ n } n∈N is relevant.This motivates the following definition, which is a generalization of regular families.
Definition H.1.A family of states { χ n } n∈N is called asymptotically regular if there exists an asymptotically monotonic function f : N → (0, ∞) such that where We next show that our results hold for asymptotically regular families too.To this end, we start with the following theorem.Proof.Since f is by assumption asymptotically monotonic, lim x→∞ f (x) exists in the extended domain [0, ∞].We first study the case lim x→∞ f (x) = l, with < l ∈ R. In this case, we choose f (x) = l, which corresponds to a regular family.Moreover, from the definition of f follows that lim x→∞ f (x)/ f (x) = 1.We also notice that for every 0 < ε < l, there exists an 1 = 0 (where p (n) are the Schmidt coefficients of χ n ), and therefore, according to Eq. (D2), The function f is a rescaling of f (x) = x 0 , which we already studied in Section F. This implies that the family { χn } n∈N is equal to the family χ

(H10)
This proves that neither f nor f corresponds to families with universally embezzling subfamilies (and are therefore also not universally embezzling themselves).
We consider now the case lim x→∞ f (x) = 0, thus f is asymptotically non-increasing.Let N be such that f is non-increasing on (N, ∞) and let a = min x∈{ 1,...,N } f (x).Since f is positive, a > 0. Let M > N be such that f (x) < a for every x ≥ M (such M exists because lim x→∞ f (x) = 0).In this case, we define f as Clearly, the family { χn } n∈N associated to it is regular and lim x→∞ f (x)/ f (x) = 1.Furthermore, the ordered Schmidt coefficients satisfy for all x ≥ M p where again a k = ⌊k/m⌋ and b k = k − ma k .Also in this case, we therefore proved that { χ n } n∈N is a universal embezzling family if and only if { χn } n∈N is a universal embezzling family (and the same holds for subfamilies).Lastly, we consider the case when lim x→∞ f (x) = ∞, and f is asymptotically nondecreasing.Analogously to the previous case, let N be such that f is non-decreasing for x ∈ (N, ∞).Let a be the maximum of f (x) for x ∈ { 1, . . ., N }, and let M > N be such that f (x) > a for x ≥ M .Also here, we define f via (H17) This proves the theorem.
Thanks to Theorem H.2, Theorem G.2 also holds for asymptotically regular families.
Corollary H.3. Let f be a positive asymptotically non-increasing function such that f (x)/x α is asymptotically monotonic for all α ∈ R and let { χ n } n∈N be the asymptotically regular family of states associated to f (see Definition H.1). Then { χ n } n∈N is a universal embezzling family if and only if f ω(x −1−ε )∩o(x −1+ε ) for all ε > 0 and ∞ x=1 f (x) = ∞.Furthermore, if f / ∈ ω(x −1−ε ) ∩ o(x −1+ε ) for at least one ε > 0, then χ n j j∈N , where { n j } j∈N is any sequence of natural numbers, is not a universal embezzling family.
For the same reasons, Proposition G.3 and Proposition G.4 also hold for asymptotically regular families.
Corollary H.4. Let f be a positive asymptotically non-decreasing function such that f (x)/x α is asymptotically non-increasing for at least one α > 0. Then the asymptotically regular family of states { χ n } n∈N associated to f is not a universal embezzling family.Furthermore, there are no sequences { n j } j∈N such that χ n j j∈N is a universal embezzling family.
Corollary H.5. Let f be a positive asymptotically non-decreasing function such that f (x)/e kx is asymptotically non-decreasing for at least one k > 0. Then the asymptotically regular family of states { χ n } n∈N associated with f is not a universal embezzling family.Furthermore, there are no sequences { n j } j∈N such that χ n j j∈N is a universal embezzling family.

Figure 1 :
Figure 1: Uniqueness of the van Dam and Hayden embezzling family -The family of states { χ α n } n∈N introduced in the main text is a universal embezzling family if and only if α = −1.This can be seen in the above plot showing the analytically derived exact value of lim n→∞ T ⋆ (χ α n → χ α n ⊗ Φ 2 ) for α ≥ −1 and lower and upper bounds for α < −1.

x√
p x |xx⟩ AB , where p ∈ Prob ↓ (d) and { |x⟩ A } and { |x⟩ B } are fixed bases for A and

Corollary E. 2
combined with Proposition E.3 leads to the following characterization of embezzling families.Corollary E.4.Let { χ n } n∈N be a regular family of states, f be the function associated to it, and g be defined as in Eq. (E3).If lim y→∞ max 1≤a≤y/m { am a g(x, y) dx } y 1 g(x, y) dx (E31) exists, then the family of states { χ n } n∈N is an universal embezzling family if and only if lim n→∞ F n = ∞ and lim y→∞ max 1≤a≤y/m { am a g(x, y) dx } )Thus, max N ≤a≤y/m am a f (x) dx = y y/m f (x) dx and, by following the same steps as in Eq. (G13), we obtainlim sup n→∞ T ⋆ (χ n → χ n ⊗ Φ m ) ≤ lim sup y→∞ max N ≤a≤y/m { am a f (x) dx } us fix ε > 0. Since f (x)/x −1+ε is non-increasing for x > N ε by hypothesis, x −1+ε /f (x) is non-decreasing for x > N ε .By applying Lemma G.1 (for y large enough), we obtain

)
Taking the lim sup on both sides, and adding the finite contributions Nε 1 f (x) dx to the diverging integrals y/m Nε f (x) dx and y Nε f (x) dx, )where a k = ⌊k/m⌋ and b k = k − ma k .The Schmidt coefficients are by definition nonincreasing, i.e., obtained by reordering { f (x)/F n } x∈N .Since there are at most N natural numbers x that do not satisfy the condition l − ε < f (x) < l + ε, there are at most N Schmidt coefficients that do not satisfy l−ε Fn < p(n)x < l+ε Fn .We call the set of indices corresponding to these Schmidt coefficients A (thus |A| ≤ N ) and observe that for any a, b ∈ { N, . . ., n } such that a ≤ b,x=1 f (x) + (l − ε)(n − N ) < n < N x=1 f (x) + (l + ε)(n − N ),we obtain (H7) holds for every ε > 0, we conclude that lim inf n→∞ max k∈{ N,...,n } this result into Eq.(H2), we have lim inf x→∞ T ⋆ (χ n → χ n ⊗ Φ m ) = 1 − 1 m .(H9)

)
The family of states { χn } n∈N is regular, lim x→∞ f (x)/f (x) = 1, and lim x→∞ Fn /F n = 1.The Schmidt coefficients associated to χ n and χn are related by p Eq. (D2) again and the relation between Schmidt coefficients derived in Eq. (H16) we obtain lim inf n→∞ T ⋆ (χ n → χ n ⊗ Φ m ) Thanks to the closed formula in Theorem 4, we obtain the following complete characterization of universal embezzling families (Appendix D).
Definition 5. A family of pure bipartite states { χ n } n∈N is called a universal embezzling family if lim n→∞ T (χ n → χ n ⊗ σ) = 0 for every bipartite finite dimensional state σ.Thanks to the triangle inequality for the star conversion distance proven in Eq. (C18), a family of states { χ n } n∈N is an embezzling family if and only if it can embezzle the state Φ 2 , as already shown in Ref [33, Lemma 2] for the case of embezzling with LO. (cf.Ref. [33, Lemma 2 (LO)]).Let { χ n } n∈N be a family of pure bipartite states.The following three statements are equivalent 1. { χ n } n∈N is a universal embezzling family.2. lim n→∞ T ⋆ we can restate Corollary 7 for regular families.
n = +∞, where F n is defined in Definition E.1.

)
which is non-increasing, but xf (x) oscillates between 1 and ∞.Theorem G.2 does not provide any information about families of states associated to such functions, and it cannot be used to determine whether such families are universally embezzling or not.Next, we prove two related propositions concerning regular families of states associated to non-decreasing functions.Let f be a positive non-decreasing function such that f (x)/x α is asymptotically non-increasing for at least one α > 0. Then the regular family of states { χ n } n∈N associated to f (see Definition E.1) is not a universal embezzling family.Furthermore, there are no sequences { n j } j∈N such that χ n j j∈N is a universal embezzling family.
1, we obtain that { χ n } n∈N is not a universal embezzling Let f be a positive non-decreasing function such that f (x)/e kx is asymptotically non-decreasing for at least one k > 0. Then the regular family of states { χ n } n∈N associated to f is not a universal embezzling family.Furthermore, there are no { n j } j∈N such that χ n j j∈N is a universal embezzling family.
family.Furthermore, there are no sequences { n j } j∈N such that χ n j j∈N is a universal embezzling family.Proposition G.4.
Theorem H.2. Let { χ n } n∈N be an asymptotically regular family and let f be the function associated to it (see Definition H.1). Then one can construct a function f that satisfies 1. f corresponds to a regular family { χn } n∈N (see Definition E.1), 2. lim x→∞ f (x)/f (x) = 1, 3. { χn } n∈N is a universal embezzling family if and only if { χ n } n∈N is a universal embezzling family, 4. { χn } n∈N contains a universal embezzling subfamily if and only if { χ n } n∈N contains a universal embezzling subfamily.