Characterizing quantum states via sector lengths

Correlations in multiparticle systems are constrained by restrictions from quantum mechanics. A prominent example for these restrictions are monogamy relations, limiting the amount of entanglement between pairs of particles in a three-particle system. A powerful tool to study correlation constraints is the notion of sector lengths. These quantify, for different $k$, the amount of $k$-partite correlations in a quantum state in a basis-independent manner. We derive tight bounds on the sector lengths in multi-qubit states and highlight applications of these bounds to entanglement detection, monogamy relations and the $n$-representability problem. For the case of two- and three qubits we characterize the possible sector lengths completely and prove a symmetrized version of strong subadditivity for the linear entropy.


Introduction
Correlations between particles are central for many physical phenomena, ranging from phase transitions in condensed matter systems to applications like quantum metrology. These non-local correlations, however, cannot be completely arbitrary as they are subject to restrictions from quantum mechanics. A prominent example for these kind of restrictions concerns entanglement in three-partite systems. Here, a monogamy relation known as the Coffman-Kundu-Wootters-inequality limits the sum of the entanglement between the first and the second party and the entanglement between the first and the third party [1].
A useful concept to describe the correlation structure of quantum states is the so-called sector length [2]. Sector lengths for n-party quantum states are quadratic expressions and quantify, for different k ≤ n, the amount of k-partite correlations in the state. Thus, to any n-qubit state one assigns a tuple (A 1 , . . . A n ) of the sector lengths {A k } and infer properties of the state based on the sector length configuration. Sector lengths are, as all correlation measures, invariant under local unitary transformations [3]. They are expressible in terms of purities of the reduced states of a system, and as such, they can be experimentally characterized by randomized measurements on a single copy of the state [4].
Consequently, sector lengths have been used for many purposes, for example entanglement detection [5], deriving monogamy relations [6], characterizing quantum codes [7] and excluding the existence of certain absolutely maximally entangled states [8]. Furthermore, bounds on k-sector lengths with k < n can be used to find necessary conditions for a set of reduced density matrices of up to k of the parties to be compatible to a global state. This problem is known as the representability problem [9][10][11].
In this paper we first find exact bounds on individual sectors A k for k ∈ {2, 3, n}. Furthermore, we fully classify the set of admissible tuples of sector lengths for two-and three-qubit states by characterizing all bounds on linear combinations of the sector lengths. Interestingly, the admissible sector lengths form a convex polytope that can be characterized by few constraints. One of these constraints can be viewed as a symmetrized version of strong subadditivity (SSA) of the linear entropy.
The paper is structured as follows: First, we will define sector lengths and review known relations between them. Then, we find tight bounds on the individual sectors A 2 , A 3 and A n in n-qubit states. There, we highlight connections to monogamy of entanglement and apply our results to the representability problem and the problem of entanglement detection. Next, we extensively study the cases of two and three qubits. To that end, we describe how to translate between sector lengths, linear entropies and mutual entropies, which are in one-to-one correspondence. We completely characterize the allowed sector length configurations by considering a symmetrized SSA for linear entropies for three qubit systems. While it is known that SSA does not hold in general for the linear entropy [12], we show, using techniques from semidefinite programming (SDP) that the symmetrized version is true for three qubits.

Basic definitions
Consider a quantum state ρ of n qubits. We expand the state in terms of the Bloch basis, i.e., in terms of tensor products of Pauli matrices: where σ (i) a denotes the Pauli operator acting on particle i in direction a ∈ {x, y, z}, padded with identities on the other particles. We group the terms by the number of non-trivial σ-matrices and call the sum of all terms with i matrices P i . As ρ is Hermitian, the coefficients α, β, . . . are real. Note that the only term that is not traceless is the unit operator, thus the normalization 2 −n is chosen such that Tr(ρ) = 1. The positivity of the state is in general hard to ensure and the origin of many of the restrictions investigated in this paper.
As an example, consider the Greenberger-Horne-Zeilinger (GHZ) state of three particles, |GHZ = 1 √ 2 (|000 + |111 ). In terms of Pauli operators, the density matrix reads Here and in the following, we skip the tensor product symbol for better readability and write X, Y and Z for the Pauli matrices σ x , σ y and σ z . Thus, 1ZZ means 1⊗σ z ⊗σ z . In this example, The sector length A k captures the amount of kbody correlations in a state. It is defined as [2] A k (ρ) : where the sum is over all Pauli operators Ξ k acting on k of the parties nontrivially. Using the expansion in Eq. (1), this means that a∈{x,y,z} α 2 i,a is the sum of the squares of the local Bloch vectors, A 2 = i<j a,b{x,y,z} β 2 ij,ab , etc. Note that A 0 = 1 by normalization. As an example, the GHZ state above has sector length configuration (A 1 , A 2 , A 3 ) = (0, 3, 4). We stress that while we used an explicit choice of a basis to define the A i they are invariant under local unitary operations, and as such they are independent of the choice of the local basis.
Considering the set of all quantum states of n parties, we are interested in the tuples (A 1 , . . . , A n ) that are attainable. First, we find tight bounds on the individual sectors. These bounds can always be attained by pure states, as the quantity A i is convex: Thus, we start by listing some basic facts about sector lengths of pure states. In this case, ρ = ρ 2 and therefore n k=0 A k = 2 n . In fact, the sum of all sector lengths is equal to the purity of the state up to a factor of 2 n .
Additionally, there are many relations among the A i for pure states: Choosing a subsystem S ⊂ {1, . . . , n}, one can define the reduced state of particles S, ρ S := TrS(ρ), whereS = {1, . . . , n} \ S. Using the Schmidt decomposition, one can show that Summing this identity over all subsets of size m ≤ n yields an equation for pure states that is expressible in terms of sector lengths [13]: for all integer 0 ≤ m ≤ n, where for m = 0 one obtains the purity equality, i A i = 2 n . The relations M m = 0 are known in the more general context of coding theory as MacWilliams' identities [14]. A subset of n 2 of them are linearly independent equations and allows for the elimination of certain A i if the state is known to be pure.

Bounds on individual sector lengths
We start by proving some bounds on the smallest sector lengths. First of all, it is known that for n-qubit states, which is attained for pure product states like |0 . . . 0 . This is because A 1 (ρ) is given by the sum of all A 1 (ρ i ) of the one-party reduced states ρ i of ρ, corresponding to the squared magnitude of the Bloch vector, which is bounded by one.

Bounds on A 2
While the bound in Eq. (6) is trivial, the tight bounds on A 2 are only known for n = 2 and n = 3 so far. For n = 2, the bound is given by A 2 ≤ 3, as for the purity holds Tr(ρ 2 ) = 2 −2 [1+A 1 (ρ)+A 2 (ρ)] ≤ 1. For n = 3, however, we obtain from M 1 = 0 in Eq. (5) for pure states that A 2 = 3, and therefore by convexity for all states A 2 ≤ 3. We will show here that for n ≥ 3, the bound is given by A 2 ≤ n 2 , using the following Lemma.

Lemma 1.
If for all quantum states ρ of n 0 qubits it holds that A k (ρ) ≤ n0 k , then for all states ρ of n ≥ n 0 qubits, it holds that A k (ρ ) ≤ n k .
Proof. We prove the Lemma by induction over the number of qubits n. Let the statement be true for a fixed n ≥ n 0 and consider a state ρ of n + 1 parties. There are n + 1 different n-party marginal states of ρ, ρj := Tr j (ρ) for j ∈ {1, . . . , n + 1}. For each of them it holds by assumption that A k (ρj) ≤ n k . Every k-body correlation among the parties i 1 , . . . , i k that is present in ρ is also present in the reduced states that contain the parties i 1 , . . . , i k . This is the case for (n + 1 − k) of the (n + 1) different reductions. Thus, The left hand side of this equation is bounded by assumption by (n + 1) n k , thus we have that Note that in Ref. [15] the authors prove a weaker statement of Proposition 2 for the sum of all bipartite correlation terms involving X and Y only, for which the same bound is obtained.
Using the same induction technique and the base case of four qubits, we can prove an even stronger, non-symmetric version of Proposition 2 for n ≥ 4, by summing only those contributions to A 2 that involve correlations with the (arbitrarily chosen) first qubit.

Proposition 3. For all qubit states of n ≥ 4 parties, it holds that
For the proof, see Appendix A.
Proposition 3 states that in a multi-qubit state, the bipartite correlations of a party with any of the other parties, on average cannot exceed one. Note that maximally entangled bipartite reduced states would obey A 2 = 3, and A 2 for two-qubit states is known to be an entanglement monotone [16]. Thus, Propositions 2 and 3 can be seen as monogamy relations limiting the shared entanglement between a party with the rest, and Proposition 3 is in close connection to the Osborne-Verstraete relation [17].
Furthermore, these bounds are useful in the context of the 2-representability problem [9][10][11]. There, one wants to decide whether a set of two-body marginals is compatible with a common global state. While the 1representability problem for qubits is solved (i.e., the same problem with a set of one-body marginals) [9] and yields a polytope of compatible eigenvalues, the k-representability problem for k > 1 is in general hard to decide [18]. However, Proposition 3 can be turned into a set of necessary conditions on the spectra of a set of two-body marginals in order to be compatible: Corollary 4. Let {ρ ij } 1≤i<j≤n denote a set of twoqubit states. If they originate from a common global state, then for the spectra of the matrices it holds that for all 1 ≤ i ≤ n: Proof. Note that for an n-qubit state ρ, Tr(ρ 2 ) = 2 n k=1 λ 2 k , where λ k are the eigenvalues of ρ. Additionally, for the two-body marginal ρ ij , the purity is given by Tr( 2 ). This allows to write A (ij) 2 as a function of purities and thus as a function of eigenvalues, i.e.
where λ (ij) k are the eigenvalues of ρ ij and λ k the eigenvalues of ρ i , ρ j , respectively. Then for each fixed choice of i, the claim follows by using

Bounds on A 3 and higher sectors
Up to here, the results involved two-body correlations only. In this section, we generalize some of the statements to three-body correlations and the sector length A 3 . Recalling the statement of Lemma 1, we know that if for some n 0 ≥ 3, A 3 (ρ) ≤ n0 3 for all ρ of n 0 qubits, then the same bound holds for all n > n 0 as well. The question arises whether such an n 0 exists. [19][20][21]. But for n ≥ 5, the bound holds. To show this, we need to introduce an additional technique, namely the so-called shadow inequalities, based on an inequality found in Ref. [22].
Let M and N be two positive semidefinite Hermitian operators acting on an n-particle space. Then for all T ⊂ {1, . . . , n} [22,23], Sector len. Here,S = {1, . . . , n} \ S and TrS denotes the partial trace of systemsS. Summing over all T with |T | = k yields a set of inequalities B k ≥ 0 for the sector length A i : Choosing M = N = ρ, the right-hand side can be evaluated in terms of the sector lengths to read [24,25] with the Kravchuk polynomials For [26]. Using these inequalities, we are in position to prove the following: Proposition 5. For all qubit states of n ≥ 5, it holds that A 3 ≤ n 3 . For n = 3, the bound is given by A 3 ≤ 4, for n = 4, it is given by A 3 ≤ 8. The bounds are tight.
Proof. For n = 3 and n = 4, we use a linear program that involves the purity M 0 = 0 from Eq. (5) and state inversion inequality B 0 ≥ 0. For n = 3, these two equations read Subtracting the second inequality from the first and using A 1 ≥ 0, we obtain A 3 ≤ 4. The same works for n = 4. For n ≥ 5, we prove the statement for n = 5. By use of Lemma 1, the result will then be true for larger n as well. We can assume that the total state is pure, as convex combinations of pure states will never increase any sector length. Using a linear program involving relations M j = 0 for j = 0, 1, 2 from Eq. (5), B 1 ≥ 0 reduces to A 3 ≤ 10 = 5 3 . Concerning the tightness, consider the GHZ state for n = 3 having A 3 = 4 and the state |χ for n = 4, given above Eq. (11). For n ≥ 5, consider any product state like |0 ⊗n Numerically, a similar statement seems to hold for A 4 for states of at least 8 qubits, but using a linear program, one can show that shadow inequalities are insufficient to show it. Still, we conjecture: Conjecture 6. For all k there exists an n 0 , such that for all n ≥ n 0 , A k ≤ n k holds for states of n-qubits.

Bounds on A n
Finally, we look at the full-body correlations of states, i.e. A n of an n-qubit state. Lower bounds on this quantity can be used to detect entanglement [5,27]. Upper bounds, however, are so far only known for the case of odd n [27]. In that case, combining again the purity M 0 = 0 from Eq. (5) and state inversion inequality B 0 ≥ 0 from Eq. (13) yields For example, the n-partite GHZ state for odd n fulfills A n = 2 n−1 , thus this bound is tight.
For n even, this trick does not work. In this case, the GHZ state fulfills A n = 2 n−1 + 1, which is why it was conjectured in Ref. [27] that this is the upper bound. Here, we show that this is true at least up to n = 10.
For small n, this follows from the shadow inequality B 1 in Eq. (13). Evaluating B 1 ≥ 0 for n = 2 yields which is the well known bound on the two-body correlations in two-qubit states and is compatible with the conjecture. For n = 4, B 1 ≥ 0 yields where we used the result of Proposition 2. For higher n, we observe that for every state ρ, there exists another stateρ = 1 2 (ρ+Y ⊗n ρ T Y ⊗n ) with the same even correlations P 2k and vanishing odd correlations P 2k+1 [28]. Thus, the bounds on an even sector length can be obtained by setting w.l.o.g. the odd correlations to zero, i.e. A 2k+1 = 0.
For n = 6, we investigate B 1 ≥ 0 and B 3 ≥ 0 and combine them to eliminate A 4 . This yields, using Proposition 2 again, For n = 8, we combine B 1 , B 3 and B 5 to yield the bound, for n = 10 we combine B k for k = 1, 3, 5, 7: Purity Theorem 7. For n-qubit states with n ≤ 10, n even, it holds that A n ≤ 2 n−1 + 1. The bound is tight.
If Conjecture 6 is true for k = 4 and n 0 ≤ 12, as numerical calculation indicates, then the same method works for n = 12, n = 14, n = 16 as well.

Application to entanglement detection
Before continuing, we highlight some applications of the bounds found in this section to the detection of entanglement. As mentioned before, sector lengths are convex and invariant under local unitaries, making them useful for entanglement detection [5]. This can be exploited by noticing that for product states where we set A k (ρ) = 0 if k > n.
. Due to convexity of the sector lengths, it follows that A 2 ≤ 1 for all separable states.
For more than two parties, different entanglement structures occur. A multi-partite state ρ is said to be biseperable, iff it can be written as where i p i = 1 and the A i , B i denote some bipartition of the parties, i.e. A i∪ B i = {1, . . . , n}. A state is called genuinely multipartite entangled (GME), iff it is not biseparable. For n = 3, we showed that A 3 ≤ 4, on the other hand, all bi-separable states obey A 3 ≤ 3, as for states Therefore, also in this case, the highest sector length can be used to detect genuine multipartite entanglement.
For n = 4, however, the situation is different: One can show with the same argument as above that biseparable states fulfill A 4 ≤ 9. But, as seen before, A 4 ≤ 9 is already the bound for all states. Thus, A 4 does not allow for detection of genuine multipartite entanglement. However, there is a nontrivial biseparability bound on A 3 of 7, whereas the bound of Proposition 5 due to positivity of the state is given by A 3 ≤ 8. Therefore, not the highest, but the next-tohighest correlations allow for entanglement detection. This already yields an entanglement criterion which can detect states not detectable by known criteria using the sector lengths [5], an example being again the highly entangled state |χ = 1 √ 6 (|0001 + |0010 + |0100 + |1000 + √ 2 |1111 ) with sector length configuration (A 1 , A 2 , A 3 , A 4 ) = (0, 2, 8, 5). Note that it is known that even vanishing highest order correlations do not exclude multipartite entanglement [29][30][31][32]. Finally, let us note that while sector lengths are quadratic expressions in the quantum state, the additional knowledge of similar quantities of higher order, i.e. higher moments, allows for more refined entanglement detection [33].

Bounds on linear combinations of sector lengths
We now turn to the problem of finding bounds on linear combinations of sector lengths. This is related to the question of whether linear constraints are enough to fully characterize the set, meaning that the set of states forms a polytope in the sector length picture. As mentioned before, sector lengths are in one-to-one correspondence with linear entropies and the mutual information for linear entropies. It turns out that some of the obtained inequalities are easier understood in the language of linear entropies.

Translation into entropy inequalities
The linear entropy of a state ρ is defined as S L (ρ) = 2[1 − Tr(ρ 2 )]. As Tr(ρ 2 ), the purity of ρ, is up to a factor equal to the sum of all sector lengths of ρ, we can express S L in terms of sector lengths. We define the sector entropy of sector k by summing over all linear entropies of reduced states of k particles, i.e.

S (k)
which can be inverted to yield Furthermore, it will be useful to define the k-partite linear mutual entropy, For k = 2 and n = 2, it resembles the usual mutual entropy, I Note that the definition is analogous to the mutual information of von Neumann-entropy. However, in the case of linear entropy, the name mutual entropy is preferred, as the quantity is not additive and does not vanish for product states [34]. Table 1 lists the non-trivial bounds on the sector lengths found above, translated into the two other representations.

Characterization of two-and three-qubit states
Using the results above, we can now characterize the allowed values of sector length tuples (A 1 , . . . , A n ) for two-qubit and three-qubit states. It turns out that in both cases the set of admissible values is a convex polytope. This is interesting as the convexity is not trivial, because the sector lengths are nonlinear in the state. In addition, it is surprising that only a finite number of linear constraints corresponding to the surfaces of the polytope is sufficient for a full description. This reminds of a similar polytope for separable states, if variances of collective spin-observables are considered [35].

The case of two qubits
It is easy to verify that in the case of n = 2, pure product states obey A 1 = 2 and A 2 = 1 [see Eq. (21)]. The Bell state |Φ + = 1 √ 2 (|00 + |11 ) obeys A 1 = 0, A 2 = 3. The purity bound Tr(ρ 2 ) ≤ 1 translates into 1 + A 1 + A 2 ≤ 4. By superposing a pure product state and the Bell state, one can obtain pure states with A 1 ∈ [0, 2] and A 2 = 3 − A 1 . Exceeding the value of 2 for A 1 is impossible due to the bound A 1 ≤ n from Eq. (6).
However, the state inversion bound B 0 ≥ 0 from Eq. (13) yields another bound on A 1 and A 2 due

The case of three qubits
While all the bounds in the case of two qubits are known, the case of three qubits shows an interesting new result that is connected to strong subadditivity of linear entropy. We start by collecting all inequalities that we know: The state inversion bound B 0 ≥ 0 from Eq. (13), the bound A 1 ≤ 3, the shadow inequality B 1 ≥ 0 and the bound from Proposition 2 yield a set of four inequalities, from which the bound Tr(ρ 2 ) ≤ 1 can be obtained using a linear program. These inequalities define a polytope in the three-dimensional space of tuples However, as numerical search indicates, these bounds are not tight. As it turns out, there is a single additional linear constraint replacing B 1 ≥ 0.

Theorem 8. For 3-qubit states, it holds that
The proof is given in Appendix B and uses a semidefinite program for a relaxed version of the problem. The polytope defined by Eqs. (26)(27)(28) is displayed in Fig. 2 and Fig. 3 in the Appendix.
It remains to show that the obtained polytope is tight by showing the existence of states on the boundary. In fact, it suffices to find states on the yellow and the blue surface in Fig. 2, corresponding to the state inversion bound 1 − A 1 + A 2 − A 3 ≥ 0 and the bound A 1 + A 2 ≤ 3(1 + A 3 ) from Theorem 8. This follows from the observation that for every state ρ, also the state inversionρ := Y ⊗n ρ T Y ⊗n is a proper state, with the same coefficients in the Bloch decomposition up to a minus sign for all coefficients of an odd number of Pauli operators [28]. Thus, the family ρ(p) = pρ + (1 − p)ρ corresponds to states with sector lengths ((1 − 2p) , yielding a family of states lying on a straight line connecting a point in the polytope with the point (0, A 2 , 0) on the red dashed A 2 -axis with the same value of A 2 . Therefore, states filling the yellow and the blue surface and their straight-line connections to the A 2 -axis fill the whole polytope.
We find and list these boundary states explicitly in Appendix D, where we also display the net of the polytope.

Connection to strong subadditivity
Theorem 8 is closely related to strong subadditivity (SSA). One formulation of SSA for the specially chosen particle B is S(ρ ABC ) + S(ρ B ) ≤ S(ρ AB ) + S(ρ BC ). However, it holds for the von Neumannentropy only and fails to hold for the linear entropy, a counterexample being the state |Φ + Φ + |⊗ 1 2 . Nevertheless, summing SSA over all particles to symmetrize it, yields or in our language,

3S
(3) This is, using the correspondence in Eq. (24), equivalent to the statement of Theorem 8. Thus, linear entropy for three-qubit states obeys a symmetric SSA, which implies that usual SSA holds for at least one choice of special particle. Another formulation in terms of mutual entropies yields the inequality I L . We state the full set of restrictions for n = 2 and n = 3 in all three representations in Table 2.
Finally, note that the statement of Theorem 8 can be generalized to states of more particles using the same induction trick as in the proof of Lemma 1. We get: Corollary 9. For n-qubit states with n ≥ 3, it holds that I L . In terms of sector lengths, the bound reads Using a linear program, it is evident that this equation is stronger than the shadow inequalities from Eq. (13). As this bound is complementary to the bound A 3 ≤ n 3 , we list it as well in Table 1.

Conclusions
We showed how to combine methods from quantum mechanics, coding theory and semidefinite programming to obtain strict bounds on linear combinations of sector lengths for multi-qubit systems. As a result, we obtained a full characterization of the allowed tuples of sector lengths for n ≤ 3, where for n = 3 one of the constraints is related to a symmetrized version of strong subadditivity of linear entropies. Our results can be understood in the language of entropy inequalities and monogamy relations, they can also be used in the context of entanglement detection and the representability problem.
Our results highlight several problems for further research. First of all, the natural question of a complete characterization of sector bounds for n ≥ 4, but also for higher-dimensional systems beyond qubits arises. The notion of sector lengths can be extended to higher-dimensional states as well, and many of the techniques like state inversion can be generalized. This has been used in the past to obtain some bounds [23,36], however, a complete characterization is still out of reach. Interestingly, we found that for n ≤ 3, the allowed region of sector bounds turned out to be a polytope, perfectly described by few linear constraints. The reason for this remains elusive and deserves further attention, as it may yield deep insight into the complicated structure of the positivity constraints. It might well be that this is a feature exclusive to qubit systems, or systems of few particles only. Apart from a similar characterization of higher-dimensional states of more parties, a deeper understanding of the associated entropy inequalities is crucial. For instance, the question of whether the inequality holds for other entropies is relevant.
To augment the proof to the case of n > 4, we consider all n−1 3 subsets of four of the parties containing the first one, i.e., for n = 5 we would consider the sets {1, 2, 3, 4}, {1, 2, 3, 5}, {1, 2, 4, 5} and {1, 3, 4, 5}. For each of these subsets, the inequality for four parties holds. Summing these inequalities yields, on the one hand, an upper bound of 3 n−1 3 . On the other hand, we obtain each of the two-body correlations A 2 (ρ 1j ) exactly n−2 2 times. Dividing both sides by this factor proves the claim.

B Proof of Theorem 8
In this section, we prove the symmetric strong subadditivity for three-qubit states.

Theorem 8. For 3-qubit states, it holds that
where ρ Tij is the partial transpose of ρ w.r.t. systems i and j. This map can be seen as a sum of partial state inversions of subsystems of size two, flipping the sign of the Pauli matrices of that particular subsystems. Using the Bloch decomposition, it can easily be seen that Tr(ρρ ) = 1 . Note that the map defined above is not positive, however, we will show that Tr(ρρ ) ≥ 0 for all ρ, yielding the claim.
To that end, we consider the Choi matrix η of the map, given by |ii [39,40]. The map can be reconstructed via M (ρ) = 2 3 Tr A [(ρ T ⊗ 1)η]. Thus, the quantity in question can be written in terms of the Choi matrix as Tr(ρρ ) = 2 3 Tr[(ρ ⊗ ρ)η T A ]. As M is not positive, η is not positive as well, and one can directly calculate that η T A has a single negative eigenvalue of −3/2. Nevertheless, it is positive for symmetric product states ρ ⊗ ρ. To see this, we use an SDP to minimize Tr(ση T A ) over symmetric states σ and trying to enforce the product structure on σ using some relaxations of this property.
To begin with, the matrix η T A can be written in Bloch decomposition as Note that due to the special symmetric form of the basis elements, the matrix can also be written as a combination of local flip operators. This allows to write the matrix also in terms of projectors onto the symmetric and antisymmetric subspaces. This representation of the problem is explained in more detail in Appendix C.
The matrix η T A exhibits many symmetries; it is symmetric under the exchange of the first three and the second three parties. Also, it is symmetric under any permutation among the first three parties, if the same permutation is applied to the second three parties as well. Furthermore, it is invariant under single qubit local unitaries V 11V 11 for V ∈ {X, Y, Z, Π, T, H} where Π = diag(1, i), T = diag(1, exp(iπ/4)) and H being the Hadamard gate.
All of these symmetries do not alter the product structure of ρ ⊗ ρ and can therefore be imposed for the optimal state as well.
Apart from the symmetries, we can try to impose the product structure of σ. However, this is a non-linear constraint and thus not exactly tractable by an SDP. Nevertheless, we find a set of linear constraints that brings us close enough to the set of product states to prove the claim.
First of all, product states are separable by definition and must have a positive partial transpose, i.e. σ T A ≥ 0 [41]. Next, using the positivity of Breuer-Hall maps, for separable states σ and skew symmetric unitaries U , i.e., U T = −U , it holds that σ BH = Tr 4,5,6 (σ) [42,43]. It turns out that the choice of U = Y Y Y is suitable in our case.
As a last constraint, for product states, A ⊗ A ρ⊗ρ = A 2 ρ ≥ 0 for all three-qubit observables A. Here, we consider the special choice of A = X11. For product states, it should hold that A ⊗ A σ = A ⊗ 111 2 σ , as σ is symmetric as noted before. To make this constraint linear, note that for Pauli observables, | A | ≤ 1. Thus, A ⊗ A σ ≤ | A ⊗ 111 σ |. Now, there are two possibilities. Either, the optimal state obeys A ⊗ 111 σ ≥ 0 or A ⊗ 111 σ ≤ 0. Therefore, we run the SDP twice, once with the constraint A ⊗ A σ ≤ A ⊗ 111 σ and once with A ⊗ A σ ≤ − A ⊗ 111 σ .
To summarize, we run the following SDP: subject to σ ≥ 0, Here, the symmetry constraint means both, symmetric under exchange of the first three with the last three parties, as well as symmetric under exchange among the first three and the same exchange among the last three parties. The last three constraints are the linear approximations of the product structure, where the ± in the last constraint means that we run the SDP once for each choice. Both cases yield a minimal trace of zero, proving the claim.
The method presented here can also be used to prove bounds for arbitrary linear combinations k c k A k . In this case, one has to choose η T A = k c k Ξ k Ξ k ⊗ Ξ k , where the inner sum iterates over all Pauli operators Ξ k acting on k of the parties nontrivially, as well as choosing appropriate relaxations of the product structure.

C Representation using symmetric and antisymmetric subspaces
As noted in Appendix B, finding bounds on linear combinations of sector lengths is equivalent to solving a quadratic program to find min ρ Tr[(ρ (A) ⊗ ρ (B) )η] with η = k c k Ξ k Ξ k . Due to the special symmetric form, it is possible to express η in terms of local flip operators F = 1 2 j=0,x,y,z σ j ⊗ σ j , which in turn can be written in their eigenbasis with the eigenvectors given by the projectors Π − = |Ψ − Ψ − | and Π + = |Ψ + Ψ + | + |Φ − Φ − | + |Φ + Φ + | onto the antisymmetric and symmetric subspace, respectively. Here, |Ψ ± and |Φ ± denote the usual Bell states. In this representation, the linear combination of sector lengths can be expressed as . The prefactorsc are connect to the prefactors c. This representation was used before to find entanglement witnesses and monotones [44][45][46]. In these references, the authors restrict themselves toc i1...in ≥ 0 to ensure positivity. As we have seen in Appendix B, this approach is too restrictive, as positivity under trace with symmetric product states is not equivalent to positivity of the matrix η. Nevertheless, it is interesting to note that the relevant inequalities in the case of three qubits have a particular form in this representation. The matrix η that yields the symmetric strong subadditivity is obtained by choosing c −−− = −3,c −−+ =c −+− =c +−− = 1 and all other prefactors vanishing. The constraint A 2 ≤ 3, however, can be expressed by choosingc −−− = −3,c −++ =c +−+ =c ++− = 1. Usual state inversion is represented bỹ c −−− = 1. Therefore, it seems that the relevant inequalities correspond to some sort of extremal points in the set of coefficientsc that yield matrices that are positive under trace with positive product operators.

D Three-qubit states spanning the whole sector length space
In this section, we explicitly state families of states that cover the whole three-qubit sector-length space displayed in Fig. 2. First, we give families of states covering the yellow surface displayed in Fig. 3, corresponding to the state inversion bound: ρ B (p, α) = p |H(α) H(α)| + 1 − p 8 (1 + cos(α)Z11 + sin(α)XXX), with the abbreviations where 0 ≤ p ≤ 1, 0 ≤ q ≤ p and 0 ≤ α ≤ π.
All other states can be reached by mixing these states with their inverted states, defined byρ := Y ⊗n ρ T Y ⊗n .