Absolutely maximally entangled states of seven qubits do not exist

Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a seven-qubit state whose three-body marginals are all maximally mixed, or equivalently, a pure $((7,1,4))_2$ quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves the seven-particle problem as the last open case concerning maximally entangled states of qubits.

Pure multiparticle quantum states are called absolutely maximally entangled if all reduced states obtained by tracing out at least half of the particles are maximally mixed. We provide a method to characterize these states for a general multiparticle system. With that, we prove that a sevenqubit state whose three-body marginals are all maximally mixed, or equivalently, a pure ((7, 1, 4))2 quantum error correcting code, does not exist. Furthermore, we obtain an upper limit on the possible number of maximally mixed three-body marginals and identify the state saturating the bound. This solves the seven-particle problem as the last open case concerning maximally entangled states of qubits.
Introduction.-Multiparticle entanglement is central for the understanding of the possible quantum advantages in metrology or information processing. When investigating multiparticle entanglement as a resource, the question arises which quantum states are most entangled. For a pure multiparticle quantum state maximal entanglement is present across a bipartition if the smaller of the two corresponding reduced systems is maximally mixed. It is then a natural question to ask whether or not there exist quantum states for any number of parties n, such that all of its reductions to n/2 parties are maximally mixed [1,2]. If this is the case, maximal entanglement is present across all bipartitions and, accordingly, these states are also known as absolutely maximally entangled (AME) states [3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. These states have been shown to be a resource for open-destination and parallel teleportation [10], for threshold quantum secret sharing schemes [12], and are a type of quantum error correcting codes [3].
If the local dimension is chosen large enough, AME states always exist [12]. For qubits, however, the situation is only partially resolved. The three-qubit Greenberger-Horne-Zeilinger (GHZ) state is an AME state since all the single-qubit reduced states are maximally mixed. For four qubits it was shown that AME states do not exist [2] and best approximations of AME states (where not all reduced states are maximally mixed) have been presented [8]. Five-and six-qubit AME states are known [3,5,7]. These can be represented as graph states and correspond to additive or stabilizer codes used in quantum error correction [3,19]. For more than eight qubits, AME states do not exist [3,[20][21][22].
Despite many attempts, the case of seven qubits remained unresolved. Numerical results give some evidence for the absence of an AME state [5][6][7]. By exhaustive search, it was shown that such a state could not have the form of a stabilizer state [19]. Nevertheless, some approximation has been presented by making many but not all three-body marginals maximally mixed [9,23].
As shortly mentioned, AME states are a type of pure quantum error correcting codes (QECC), having the max-imal distance allowed by the Singleton bound [24]. In particular, AME states of n parties having local dimension D each correspond to a pure QECC in (C D ) ⊗n of distance n/2 + 1, denoted by ((n, 1, n/2 + 1)) D . Often, but not always, bounds on so-called non-additive (i.e. non-stabilizer) codes coincide with those for additive (stabilizer) codes. The seven qubit AME state would -if it existed -be one of the few examples where a nonadditive code outperformed an additive one. This possibility was noted already in a seminal article by Calderbank et al. [25]. Up to n = 30, there are only three other instances known where this could be the case for one-dimensional codes on qubits [26].
In this paper, we provide a method to characterize AME states and their approximations, making use of the Bloch representation [27]. The usefulness of this tool may be surprising at first sight, as the Bloch representation is designed to be a tool for mixed states. We were motivated to choose this approach by the fact that monogamy equalities [8,28] directly signal the non-existence of a four-qubit AME state, and the natural framework for deriving the monogamy equalities appears to be the Bloch representation [28].
The Bloch representation.-Any n-qubit state can be written as where the {α 1 , . . . , α n } ∈ {0, x, y, z} label combinations of the four Pauli matrices. For simplicity, we group the terms according to their weight, that is, their number of non-trivial (Pauli) operators. Let P j be the sum over terms of weight j, then the state can be written as We denote by P (V ) j a subset of P j , where V further specifies its support, i.e. its non-trivial terms are located on arXiv:1608.06228v3 [quant-ph] 18 May 2017 the subsystems in V . To give an example, a state of three qubits reads where, e.g., P (12) 2 = r α1,α2,0 σ α1 ⊗σ α2 ⊗1 and α 1 , α 2 = 0. When tracing out the third qubit, one drops the terms P 3 , P , and P 1 , as they do not contain an identity in the third subsystem. Also, the normalization prefactor is multiplied by the dimension of the parties over which the partial trace was performed, resulting in Accordingly, a three-qubit state having maximally mixed one-body reduced density matrices does not have terms of weight one, the terms P are absent. Similarly, in n-qubit AME states all operators P j with 1 ≤ j ≤ n/2 vanish.
Our further discussion rests on recognizing what terms may appear in the squared state 2 . For this, consider two terms A and B, both appearing in the Bloch expansion of the state. For computing 2 , the anticommutator {A, B} is required, and we state the following observation regarding its weight. Lemma 1 (parity rule). Let M, N be Hermitian operators proportional to n-fold tensor products of singlequbit Pauli operators, M = c M σ µ1 ⊗ · · · ⊗ σ µn , N = c N σ ν1 ⊗ · · · ⊗ σ νn , where c M , c N ∈ R. Let us denote their weights, that is, their number of nontrivial Paulioperators in their tensor expansion, by |M | and |N |. Then, if the anticommutator {M, N } does not vanish, its weight |{M, N }| fulfills Proof. The product M N , and thus also {M, N }, has at most weight |M | + |N |. This is attained, if the supports of M and N are disjoint. Each pair of equal, but nonzero indices µ j = ν j corresponds to some overlap of the supports and reduces the maximal weight |M | + |N | by two. In contrast, if a pair of non-zero indices are not equal (e.g., µ j = ν j ), the product M N contains the term σ µj σ νj = i µj νj χ σ χ . Consequently for each such pair |M | + |N | is reduced by only one. If an odd number of such pairs exists, the anticommutator has to vanish, as it is Hermitian. So, such pairs have to occur an even number of times, which proves the claim.
We can summarize the behavior of the weights of M and N and their anticommutator as follows: It follows that an analogous behavior holds for the P j . If j and k are either both even or both odd, the anticommutator {P j , P k } can only contribute to the P l where l is even. Similarly, if j is even and k is odd, it only contributes to the P l having odd l. Properties of AME state reductions.-First, recall that for a pure n-party state |ψ AB consisting of D-level systems, the complementary reduced states of any bipartition share the same spectrum. This follows from its Schmidt decomposition. Hence, if a (n − k)-body reduction B is maximally mixed, its complementary reduced state A of size k ≥ n/2 has all D (n−k) nonzero eigenvalues equal to λ = D −(n−k) . Thus the reduced state is proportional to a projector, This projector property alone is already enough to derive the following bounds on the existence of AME states, These bounds originate in work by Rains and were applied to AME states by Scott [3,20]. A proof using the projector property can be found in Appendix A [30]. By Schmidt decomposition, one further sees that the full state |ψ AB is an eigenvector of the reduced state A , Accordingly, for an AME state having all n/2 -body reduced states maximally mixed, any k-body reduced state (k) with n/2 ≤ k ≤ n fulfills relations (7) and (9). Let us now consider AME states of n qubits. We decompose Eq. (9) in the Bloch representation, using the reduced state (k) on the first k = n/2 + 1 parties of a qubit AME state, Because all n/2 -body marginals are maximally mixed, P j≤ n/2 = 0. We obtain the eigenvector relations By accounting for combinatorial factors, similar relations can be obtained in an iterative way for all P (1···j) j≥ n/2 +1 . With these building blocks in place, we are in the position to solve the last open qubit case -the existence of a seven qubit AME state. In the following, we will combine the projector property of a five qubit reduced state (5) with the eigenvector relations for P to obtain a contradiction from the parity rule stated in Lemma 1.
Observation 2. Consider a pure state of seven qubits. Then not all of its three-body reduced density matrices can be maximally mixed.
Note that while the proof requires the projector property only to hold on the first five qubits, Eq. (12) actually holds for all possible five-qubit reductions.
We will use these three equations to obtain a contradiction: Let us expand (4) and (5) in the Bloch basis There are five different terms P Inserting Eqs. (15,16) into Eqs. (13,14) results in the eigenvector relations We similarly insert Eq. (16) in Eq. (12) to obtain The key observation is now the parity rule stated in Lemma 1: Only certain products occurring on the lefthand side of Eq. (18) can contribute to P 5 on the right-hand side. Indeed, P 2 5 on the left-hand side cannot contribute to P 5 on the right-hand side. Similarly, 4 ⊗ 1 (j) ) 2 on the left-hand side cannot contribute to P 5 on the right-hand side.
Thus we can collect terms of weight five on both sides of the equation, Tensoring with the identity and multiplying by |φ from the right leads to However, using the eigenvector relations Eqs. (13,14), one arrives at a contradiction This ends the proof.
Upper bound for the number of maximally mixed reductions.-Note that in the derivation above not all constraints imposed by the reduced states have been taken into account. In fact, we only needed a single fivequbit reduced state (say, for definiteness, on the qubits {1, 2, 3, 4, 5}) fulfilling the Eqs. (12,14), whose threebody reduced density matrices are all maximally mixed [this was needed for Eq. (16)]. In addition, the five fourqubit reduced density matrices corresponding to the possible subsets of {1, 2, 3, 4, 5} have to obey Eq. (13).
We note that the existence of states where 32 of the three-body density matrices are maximally mixed was LC ≡ FIG. 1. The graph of the Fano (or seven-point) plane on the left, which can be transformed by local complementation (corresponding to local Clifford gates) to the wheel graph displayed on the right. The Fano plane plays a role in classical error correction, describing both a balanced block design as well as an error correcting code [31]. The corresponding graph state saturates the bound of Observation 3. The states are locally equivalent to the graph state depicted in Figs. 1 in [29], to No. 44 in Table V from Ref. [19], and to the states of Eq. (11) in Ref. [9] and of Eq. (26) in Ref. [23].
shown before: Refs. [9,23] presented such states, which are, up to local unitary transformation, a graph state occurring in Refs. [19,29]. As a graph state, the state can be described by the graphs in Fig. 1. It can be constructed from the graph as follows: Each vertex in a graph corresponds to a qubit. One prepares all the qubits in the state |+ = (|0 + |1 )/ √ 2. Then, for any edge connecting the qubits j and k one applies a twoqubit phase gate to the initial state. The fact that the marginals of this state have the right properties can also directly be checked in the stabilizer formalism, as explained in Ref. [19]. Finally, we add that there exists an AME state for seven three-dimensional systems, which is also a graph state [13]. AME states of n qubits.-The method presented for seven qubits can also be applied to the general n-qubit case. There, it can exclude that an AME state for a given number of qubits exists. It turns out that the qubit numbers n for which no contradiction is found (n = 2, 3, 5, 6) are exactly the ones for which AME states are known [15]. The proof is presented in the Appendix B [30].
Conclusion.-In summary, we have developed a method based on the Bloch representation for characterizing AME states. This allowed to rederive most of the known results for qubits in a very simple manner, but more importantly, it solved the long-standing question whether AME states of seven qubits exist or not. Also, the best approximation to such a state could be determined. For future work, it is very interesting to apply our methods to the question whether n-qubit states exist where all k-body reduced density are maximally mixed for k < n/2 . These are not AME states, but they are central for quantum error correction and many efforts have been devoted to finding them in the last years [32,33]. We hope that our method can also contribute to this problem.

APPENDIX A
From the projector property, we obtain bounds on AME states. These originate in work of Rains in the context of quantum codes and were applied to AME states by Scott [3,20]. Let {Λ α } form an orthonormal basis of Hermitian operators for a qudit system of local dimension D. Because of orthonormality, tr[Λ α Λ β ] = Dδ αβ .
A k-body reduced state on parties in V can then be written as r α1,...,αn Λ α1 ⊗ · · · ⊗ Λ αn ) . (23) Here, the sum runs over appropriate α, specifically, over those whose corresponding basis terms have nontrivial support only strictly within the reduced state under discussion, supp(α) ∈ V , cf. also Eq. (4). We recall that any subsystem of an AME state, having size k ≥ n/2 + 1, fulfills the projector property Expanding in the Bloch representation and taking the trace gives Thus the coefficients r α are constrained by For k > n/2 , the sum is strictly positive, because reductions of pure states to size n/2 + 1 can not be proportional to the identity, as one can see from its Schmidt decomposition.
Let us look at a specific reduced state of size n/2 +2, containing n/2 + 2 reduced systems of size n/2 + 1. Clearly, all coefficients appearing in the smaller subsystems also appear in the larger subsystem.
To obtain the bound, we require the coefficients corresponding to weight n/2 + 2 alone to be non-negative, This leads to the conditions which can be recast to the bounds of Refs. [3,20], This ends the proof.

APPENDIX B
The general case of determining which n-qubit AME states can possibly exist is detailed here. It follows the method which was used in the case of seven qubits: We combine the projector property of the reduced state of the first n/2 + 2 parties with the eigenvector relations for the terms P n/2 +1 and P n/2 +2 appearing in its expansion. Collecting terms with either even or odd weight, depending on the case, and applying the parity rule will lead to contradictions except in the cases of n = 2, 3, 5, 6 qubits. In the following, we will distinguish four cases, depending on n and n/2 being even or odd.
Thus qubit AME states do not exist when n is a multiple of 4.
We collect terms of odd weight, n/2 +1 ⊗ 1 (j) , P n/2 +2 }|φ Thus one requires whose only solution is n = 5. The corresponding AME state is the five-qubit ring-cluster state, depicted in Fig. 2.
Therefore, no AME qubit state with both n and n/2 being odd exists. We summarize: the only qubit AME states which are not excluded to exist by this method are the cases of two, three, five, and six parties, all of which are known. Their graph state representations are shown in Fig. 2.