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-

Contrail cirrus account for the major share of aviation’s climate impact. Yet, the links between jet fuel composition, contrail microphysics and climate impact remain unresolved. Here we present unique observations from two DLR-NASA aircraft campaigns that measured exhaust and contrail characteristics of an Airbus A320 burning either standard jet fuels or low aromatic sustainable aviation fuel blends. Our results show that soot particles can regulate the number of contrail cirrus ice crystals for current emission levels. We provide experimental evidence that burning low aromatic sustainable aviation fuel can result in a 50 to 70% reduction in soot and ice number concentrations and an increase in ice crystal size. Reduced contrail ice numbers cause less energy deposition in the atmosphere and less warming. Meaningful reductions in aviation’s climate impact could therefore be obtained from the widespread adoptation of low aromatic fuels, and from regulations to lower the maximum aromatic fuel content.

A pure multipartite quantum state is called absolutely maximally entangled (AME), if all reductions obtained by tracing out at least half of its parties are maximally mixed. Maximal entanglement is then present across every bipartition. The existence of such states is in many cases unclear. With the help of the weight enumerator machinery known from quantum error correction and the generalized shadow inequalities, we obtain new bounds on the existence of AME states in dimensions larger than two. To complete the treatment on the weight enumerator machinery, the quantum MacWilliams identity is derived in the Bloch representation. Finally, we consider AME states whose subsystems have different local dimensions, and present an example for a 2 × 3 × 3 × 3 system that shows maximal entanglement across every bipartition.