Entanglement transitivity problems

One of the goals of science is to understand the relation between a whole and its parts, as exemplified by the problem of certifying the entanglement of a system from the knowledge of its reduced states. Here, we focus on a different but related question: can a collection of marginal information reveal new marginal information? We answer this affirmatively and show that (non-) entangled marginal states may exhibit (meta)transitivity of entanglement, i.e., implying that a different target marginal must be entangled. By showing that the global n-qubit state compatible with certain two-qubit marginals in a tree form is unique, we prove that transitivity exists for a system involving an arbitrarily large number of qubits. We also completely characterize—in the sense of providing both the necessary and sufficient conditions—when (meta)transitivity can occur in a tripartite scenario when the two-qudit marginals given are either the Werner states or the isotropic states. Our numerical results suggest that in the tripartite scenario, entanglement transitivity is generic among the marginals derived from pure states.

INTRODUCTION Entanglement 1 is a characteristic of quantum theory that profoundly distinguishes it from classical physics. The modern perspective considers entanglement as a resource for information processing tasks, such as quantum computation [2][3][4][5][6] , quantum simulation 7 , and quantum metrology 8 . With the huge effort devoted to scaling up quantum technologies 9 , considerable attention has been given to the study of quantum many-body systems 10,11 , specifically the ability to prepare and manipulate large-scale entanglement in various experimental systems.
The entanglement of the global system, nonetheless, is not always the desired quality of interest. For instance, in scaling up a quantum computer, one may wish to verify that a specific subset of qubits indeed get entangled, but this generally does not follow from the entanglement of the global state (recall, e.g., the Greenberger-Horne-Zeilinger states 37 ). Thus, one requires a more general version of the problem: Given certain reduced states, can we certify the entanglement in some other target (marginal) state? We call this the entanglement transitivity problem (ETP). Since the global system is a legitimate target system, ETPs include the EMP as a special case.
As a concrete example beyond EMPs, one may wonder whether a set of entangled marginals are sufficient to guarantee the entanglement of some other target subsystems. If so, inspired by the work 38 on nonlocality transitivity of post-quantum correlations 39 , we say that such marginals exhibit entanglement transitivity. Indeed, one of the motivations for considering entanglement transitivity is that it is a prerequisite for the nonlocality transitivity of quantum correlations, a problem that has, to our knowledge, remained open.
More generally, one may also wonder whether separable marginals alone, or with some entangled marginals could imply the entanglement of other marginal(s). To distinguish this from the above phenomenon, we say that such marginals exhibit metatransitivity. Note that any instance of metatransitivity with only separable marginals represents a positive answer to the EMP. Here, we show that examples of both types of transitivity can indeed be found. Moreover, we completely characterize when two Werner-state 40 marginals and two isotropic-state 41 marginals may exhibit (meta)transitivity.

Formulation of the entanglement transitivity problems
Let us first stress that in an ETP, the set of given reduced states must be compatible, i.e., giving a positive answer to the quantum marginal problem 42,43 . With some thought, one realizes that the simplest nontrivial ETP involves a three-qubit system where two of the two-qubit marginals are provided. Then, the problem of deciding if the remaining two-qubit marginal can be separable is an ETP different from EMPs.
More generally, for any n-partite system S, an instance of the ETP is defined by specifying a set S ¼ fS i : i ¼ 1; 2; ; kg of k marginal systems S i (each in its respective state σ Si ) and a target system T ∉ S. Here, S is a strict subset of all the 2 n possible combinations of at most n subsystems, i.e., k < 2 n . Then, σ :¼ fσ Si g exhibits entanglement (meta)transitivity in T if for all joint states ρ S compatible with σ, the reduced state ρ T is always entangled while (not) all given σ Si are entangled. Formally, the compatible requirement reads as: tr SnSi ðρ S Þ ¼ σ Si for all S i 2 S where S\S i denotes the complement of S i in the global system S.
Notice that for the problem to be nontrivial, there must be (1) some overlap among the subsystems specified by S i 's, as well as with T, and (2) the global system S cannot be a member of S. However, the target system T may be chosen to be S and if all σ Si are separable, we recover the EMP 36 (see also refs. 19,23 for some strengthened version of the EMP). Hereafter, we focus on ETPs beyond EMPs, albeit some of the discussions below may also find applications in EMPs.
Certification of (meta)transitivity by a linear witness Let WðρÞ be an entanglement witness 26 , i.e., WðρÞ ! 0 for all separable states in T, and WðρÞ < 0 for some entangled states. We can certify the (meta)transitivity of S in T if a negative optimal value is obtained for the following optimization problem: where trðρ S Þ ¼ 1 is implied by the compatibility requirement and " ≽ " denotes matrix positivity. Then, W detects the entanglement in T from the given marginals in S. Consider now a linear entanglement witness, i.e., Wðρ T Þ ¼ tr ρ S ðW T I SnT Þ Â Ã for some Hermitian operator W T , where ρ T ¼ tr SnT ðρ S Þ is the reduced state of ρ in T. In this case, Eq. (1) is a semidefinite program 44 . Interestingly, its dual problem 44 can be seen as the problem of minimizing the total interaction energies among the subsystems S i while ensuring that the global Hamiltonian is non-negative, see Supplementary Note 1.
Hereafter, we focus, for simplicity, on T being a two-body system. Then, a convenient witness is that due to the positivepartial-transpose (PPT) criterion 45,46 , with W T ¼ η Γ T , where η T ≽ 0 and Γ denotes the partial transposition operation. Further minimizing the optimum value of Eq. (1) over all η T such that trðη T Þ ¼ 1 gives an optimum λ * that is provably the smallest eigenvalue of all compatible ρ Γ T (see Supplementary Note 1). Hence, λ * < 0 is a sufficient condition for witnessing the entanglement (meta)transitivity of the given σ in T.
Three remarks are now in order. Firstly, the ETP defined above is straightforwardly generalized to include multiple target systems {T j : j = 1, …, t} with T j ∉ S for all j. A certification of the joint (meta) transitivity is then achieved by certifying each T j separately. Secondly, other entanglement witnesses 26 may be considered. For instance, to certify the entanglement of a two-body ρ T that is PPT 47 , a witness based on the computable cross-norm/ realignment (CCNR) criterion [48][49][50][51] , may be employed. Finally, for a multipartite target system, a witness tailored for detecting the genuine multipartite entanglement in ρ T (see, e.g., Refs. 14,16 ) is surely of interest.
A family of transitivity examples with n qubits As a first illustration, let Ψ þ j i ¼ 1 ffiffi 2 p ð 10 j i þ 01 j iÞ and consider: which is a two-qubit reduced state of Ω n ðγÞ ¼ γ W n j i W n h jþ ð1 À γÞ 0 n j i 0 n h j, i.e., a mixture of 0 n j i and an n-qubit W state W n j i ¼ 1 ffiffi n p P n j¼1 1 j , where 1 j denotes an n-bit string with a 1 in position j and 0 elsewhere. Now, imagine drawing these n qubits as vertices of a tree graph 52 with (n − 1) edges, see Fig. 1, such that every edge corresponds to a pair of qubits in the state ρ n (γ), that is, where S represents the set of edges. Then we prove the following result: Theorem 1. For any tree graph with n vertices that satisfies Eq. (3), Ω n ðγÞ ¼ γ W n j i W n h jþ ð1 À γÞ 0 n j i 0 n h j is the unique global state and all the two-qubit reduced states are ρ n (γ).
The details of its proof can be found in Supplementary Notes 2. Thus, these ρ n (γ) exhibit transitivity for any of the ðnÀ1ÞðnÀ2Þ 2 pairs of qubits that are not linked by an edge. Indeed, the symmetry of Ω n (γ) implies that all its two-qubit marginals are ρ n (γ), and the We should clarify that the transitivity exhibited by ρ n (γ) requires a tree graph only in that it represents the minimal amount of marginal information for the global state to be uniquely determined. Any other n-vertex graph with equivalent marginal information or more leads to the same conclusion.
These examples involve only entangled marginals. Next, we present examples where some of the given marginals are separable. In particular, we provide a complete solution of the ETPs with the input marginals being a Werner state 40 or an isotropic state 41 .

Metatransitivity from Werner state marginals
as Þ be the projection onto the symmetric (antisymmetric) subspace of C d C d . Then we can write qudit Werner states as the oneparameter family 40 To find the (meta) transitivity region for (v AB , v AC ), it suffices to determine the boundary where the largest compatible v BC ¼ 1 2 . These boundaries are found (see Supplementary Note 3) to be the two parabolas Fig. 1 Tree graph. A tree graph is any undirected acyclic graph such that a unique path connects any two vertices. Graph a and b are the only two nonisomorphic trees with (n − 1) edges for n = 4. Graph c is not a tree because it is disconnected and has a cycle. Fig. 2. It also shows the compatible regions of (v AB , v AC ) obtained directly from Ref. 54 , and the desired (shaded) regions exhibiting the (meta)transitivity of these marginals. In particular, the lower-left region corresponds to (a) while the top-left and bottom-right regions correspond to (b) in Fig. 4. Remarkably, these results hold for arbitrary Hilbert space dimension d ≥ 2 (but for d = 2, the lower-left shaded region does not correspond to compatible Werner marginals).

Metatransitivity from isotropic state marginals
An isotropic state 41 is a bipartite density operator in C d C d that is invariant under U U (or U U) transformations for any unitary U 2 U d ; here, U is the complex conjugation of U. We can write qudit isotropic states as a one-parameter family 41 j¼0 j j i j j i and p gives the fully entangled fraction 55,56 of I d ðpÞ.
Consider now a pair of isotropic marginals σ ¼ fI d ðp AB Þ; I d ðp AC Þg as the reduced states of some joint state τ ABC . Then the "twirled" state e τ ABC ¼ R dμ U ðU U UÞτ ABC ðU U UÞ y , which has a Werner state marginal W d (v BC ) in BC, is easily verified to be a valid joint state for the given marginals. As in the case of given Werner states marginals, it suffices to consider e τ ABC in determining the region of (p AB , p AC ) that demonstrates metatransitivity.
To this end, note that two isotropic states and one Werner state with parameters p ! ¼ ðp AB ; p AC ; v BC Þ are compatible iff 54 the vector p ! lies within the convex hull of the origin p ! 0 ¼ ð0; 0; 0Þ and the cone given by , where α ± = p AB ± p AC and β = 2(v BC − 1). To find the metatransitivity region for (p AB , p AC ) we again look for the boundary where the largest compatible v BC ¼ 1 2 , which we show in Supplementary Note 4 to be 4p AB The resulting regions of interest are illustrated for the d = 3 case in Fig. 3, and they correspond to (b) in Fig. 4.

Metatransitivity with only separable marginals
Curiously, none of the infinitely compatible pairs of marginals given above result in the most exotic type of metatransitivity, even though there are known examples where separable marginals imply a global entangled state (see, e.g., refs. 12,24,25,36 ). In the following, we provide examples where the entanglement of a subsystem is implied by only separable marginals. This already occurs in the simplest case of a three-qubit system. Consider the   rank-two mixed state It can be easily checked that the AB and BC marginals of χ ABC are PPT, which suffices 46 to guarantee their separability, while Eq.
(1) with the PPT criterion can be used to confirm that AC is always entangled. Thus, this example corresponds to (c) in Fig. 4.
Likewise, examples exhibiting different kinds of transitivity can also found in higher dimensions (with bound entanglement 47 ) or with more subsystems, see Supplementary Note 5 for details.
(1) with the PPT criterion that these three marginals together imply the entanglement of all the three remaining two-qubit marginals. Thus, this corresponds to (d) in Fig. 4.
At this point, one may think that the entanglement in the AC marginal already follows from the given AB and BC marginals, analogous to the tripartite examples presented above. This is misguided: the CD marginal is essential to force the AC marginal to be entangled. Similarly, the AB marginal is indispensable to guarantee the entanglement of BD. Thus, the current metatransitivity example illustrates a genuine four-party effect that cannot exist in any tripartite scenario. For completeness, an example exhibiting the same four-party effect but where all input twoqubit marginals are entangled is also provided in Supplementary Note 5.
Metatransitivity from marginals of random pure states Naturally, one may wonder how common the phenomena of (meta)transitivity is. Our numerical results based on pure states randomly generated according to the Haar measure suggest that transitivity is generic in the tripartite scenario: for local dimension up to five, all sampled pure states have only non-PPT marginals and demonstrate entanglement transitivity. However, with more subsystems, (meta)transitivity seems rare. For example, among the 10 5 sampled four-qubit states, only about 7.32% show transitivity while about 3.38% show metatransitivity. For a system with even more subsystems or with a higher d, we do not find any example of (meta)transitivity from random sampling (see Table 1 for details).
Next, notice that for the convenience of verification, some explicit examples that we provide actually involve marginals leading to a unique global state. However, uniqueness is not a priori required for entanglement (meta)transitivity. For example, among those quadripartite (meta)transitivity examples found for randomly sampled pure states, >73% of them (see Supplementary Note 5) are not uniquely determined from three of its two-qubit marginals (cf. refs. [57][58][59]. In contrast, most of the tripartite numerical examples found appear to be uniquely determined by two of their two-qudit marginals, a fact that may be of independent interest (see, e.g., refs. 60-63 ).

DISCUSSION
The example involving noisy W-state marginals demonstrate that the transitivity can occur for arbitarily long chain of quantum systems. This leads us to consider metatransitivity with only separable marginals. Beyond the example given above, we present also in Supplementary Note 5 a five-qubit example with four separable marginals and discuss some possibility to extend the chain. For future work, it could be interesting to determine if such exotic metatransitivity examples exist at the two ends of an arbitrarily long chain of multipartite system. For the closely related EMP, we remind that an explicit construction for a state with only two-body separable marginals and an arbitrarily large number of subsystems is known 23 (see also ref. 36 ).  The second column gives the number of pure states sampled N sample in each scenario (n, d). The next two columns list the fraction of states giving (n−1) neighboring two-body marginals that are, respectively, all NPT (i.e., none of which being PPT) and all PPT. The next three columns summarize how generic the phenomenon of (meta)transivitiy is among such states when the target system T lie at the two ends of an n-body chain. We give from left to right, respectively, the fraction among all sampled states exhibiting transitivity (i.e., with only entangled marginals), metatransitivity with only separable marginals, and metatransitivity with mixed marginals. Enclosed in each bracket is the corresponding fraction among samples having the associated kind of marginals. The next two columns summarize the extent to which the (n−1) two-body marginals lead to a unique global pure state. These are expressed in terms of the largest value of the infidelity 1 À F, where F ¼ min ρ S ψ h jρ S ψ j i and ψ j i is the sampled pure state; the three numbers listed in the second last column are, respectively, for ϵ = 10 −8 , 10 −7 , and 10 −6 . The final column shows the fraction of (meta)transitivity examples having a unique global state (with an infidelity threshold set to 10 −6 ). Throughout, we use 1 † to represent a number that differs from 1 by less than 10 −8 .
So far, we have discussed only cases where both the input marginals and the target marginal are for two-body subsystems. If entanglement can be deduced from two-body marginals, it is also deducible from higher-order marginals that include the former from coarse graining. Hence, the consideration of two-body input marginals allows us to focus on the crux of the ETP. As for the target system, we provide-as an illustration-in Supplementary Note 5 an example where the three two-qubit marginals of Fig.  1(b) imply the genuine three-qubit entanglement present in BCD. Evidently, there are many other possibilities to be considered in the future, as entanglement in a multipartite setting is known 1,26 to be far richer.
Our metatransitivity examples also illustrate the disparity between the local compatibility of probability distributions and quantum states. Classically, probability distributions P(A, B) and P(B, C) compatible in P(B) always have a joint distribution P(A, B, C) (this extends to the multipartite case for marginal distributions that form a tree graph 64 ). One may think that the quantum analogue of this is: compatible ρ AB and ρ BC must imply a separable joint state, and hence a separable ρ AC . However, our metatransitivity example (as with nontrivial instances of tripartite EMPs), illustrates that this generalization does not hold. Rather, as we show in Supplementary Note 8, a possible generalization is given by classical-quantum states ρ AB and ρ BC sharing the same diagonal state in B-in this case, metatransitivity can never be established.
Evidently, there are many other possible research directions that one may take from here. For example, as with the W-states, we have also observed transitivity in n ≤ 3 ≤ d ≤ 6 for qudit Dicke states [65][66][67] , which seems to be also uniquely determined by its (n − 1) bipartite marginals. To our knowledge, this uniqueness remains an open problem and, if proven, may allow us to establish examples of transitivity for an arbitrarily high-dimensional quantum state that involves an arbitrary number of particles. From an experimental viewpoint, the construction of witnesses specifically catered for ETPs are surely welcome.
Finally, notice that while ETPs include EMPs as a special case, an ETP may be seen as an instance of the more general resource transitivity problem (CYH, GNT, YCL), where one wishes to certify the resourceful nature of some subsystem based on the information of other subsystems. In turn, the latter can be seen as a special case of the even more general resource marginal problems 68 , where resource theories are naturally incorporated with the marginal problems of quantum states.

Metatransitivity certified using separability criteria
As mentioned before, we can certify the entanglement (meta)transitivity of a given set of marginals in a bipartite target system T by demonstrating the violation of the PPT separability criterion. We can show this by solving the following convex optimization problem: which directly optimizes over the joint state ρ S with marginals σ Si such that the smallest eigenvalue λ of ρ Γ T is maximized. Because a bipartite state that is not PPT is entangled 45,46 , if the optimal λ (denoted by λ ⋆ throughout) is negative, the marginal state in T of all possible joint states ρ S must be entangled.
In the Supplementary Notes, we compute the Lagrange dual problem to Eq. (1) with a linear witness W T . A similar calculation for Eq. (8) shows that it is equivalent to a dual problem with W ¼ η Γ T , where η T being an additional optimization variable subjected to the constraint of η T ≽ 0 and trðη T Þ ¼ 1.
Meanwhile, to certify genuine tripartite entanglement in the target tripartite marginal T, we use a simple criterion introduced in ref. 69 . Consider the density operator ρ AB on C m C n to be an m × m block matrix of n × n matrices ρ (i, j) . Let f ρ AB denote the realigned matrix obtained by transforming each block ρ (i, j) into rows. The CCNR criterion 48,49 dictates that for separable σ AB , k f σ AB k 1 1. Now, let A|BC denote a bipartition of a tripartite system ABC into a bipartite system with parts A and BC. Finally, for any tripartite state ρ ABC on C d C d C d , define Mðρ ABC Þ :¼ 1 3 kρ TA ABC k 1 þ kρ TB ABC k 1 þ kρ TC ABC k 1 À Á Nðρ ABC Þ :¼ 1 3 k g ρ AjBC k 1 þ k g ρ BjCA k 1 þ k g ρ CjAB k 1 ; where T X means a partial transposition with respect to the subsystem X. It was shown in ref. 69 that for any biseparable ρ ABC , we must have maxfMðρ ABC Þ; Nðρ ABC Þg 1 þ 2d 3 : This means that if any of M(ρ ABC ), N(ρ ABC ) is larger than 1þ2d 3 , ρ ABC must be genuinely tripartite entangled.
Therefore in the metatransitivity problem, we can use this, cf. Eq. (8) for the bipartite target system, for detecting genuine tripartite entanglement. This is done by minimizing M and N of the target marginal and taking the larger of the two minima. To this end, note that the minimization of the trace norm can be cast as an SDP 70 . Further details can be found in Supplementary Notes 1.

Certifying the uniqueness of a global compatible (pure) state
A handy way of certifying the (meta)transitivity of marginals fσ Si g known to be compatible with some pure state ψ j i is to show that the global state ρ S compatible with these marginals is unique, i.e., ρ S is necessarily ψ j i ψ h j. This can be achieved by solving the following SDP: The objective function here is the fidelity of ρ S with respect to the pure state ψ j i. If this minimum is 1, then by the property of the Uhlmann-Jozsa fidelity 71 , we know that the only compatible ρ S is indeed given by ψ j i ψ h j.
For the numerical results that show how typical transitivity is for the bipartite marginals of a pure global state, the marginals are obtained from a uniform random n-qudit state, which is obtained by taking the first column of a d n -dimensional Haar-random unitary.

DATA AVAILABILITY
All relevant data supporting the main conclusions and figures of the document are available upon reasonable request. Please refer to Gelo Noel Tabia at gelonoel-tabia@gs.ncku.edu.tw.