An absolutely maximally entangled (AME) state is one in which every possible division of a many-body system into two groups is as entangled as quantum mechanics allows. This makes AME states uniquely valuable as benchmarks for quantum theory and as resources for quantum technologies. Yet basic questions about their existence, structure and classification have remained unresolved, even after two decades of study.

In a new work, dedicated to Ryszard Horodecki, this field has been advanced in several important ways. First, the authors provided a comprehensive and up to date overview of known methods for constructing AME states, going beyond traditional approaches based on stabilizer and graph states. The authors showed how recent ideas from combinatorics, matrix and group theory generate entirely new families of highly entangled states that were previously unknown.

They also went on to study how entanglement behaves when particles are removed from an AME system. This reveals how robust these extreme states are to loss and noise, an essential consideration for real quantum technologies.

One highlight is a solution to the quantum version of Euler’s famous “36 officers” problem.  This puzzle asks whether 36 officers from six ranks and six regiments can be arranged in a 6 x 6 grid so that no row or column repeats a rank or regiment. Classical mathematics proves this is impossible.

The paper shows however, that quantum mechanics can bypass this restriction altogether. By using an absolutely maximally entangled quantum state, the researchers constructed a quantum version of the puzzle in which all constraints are satisfied simultaneously. The solution relies on superposition and quantum entanglement rather than fixed arrangements, illustrating how quantum theory enables outcomes forbidden in classical mathematics.

By mapping the limits of multipartite entanglement, this work connects abstract theory with practical goals such as quantum error correction, secure communication, and benchmarking future quantum computers.