Most headline-grabbing advances in quantum mechanics today are experimental in nature: more qubits, entangled particles, fewer errors. Often overlooked are the advances in the mathematics that underpins the behaviour of these quantum systems. The walled Brauer algebra is an abstract but increasingly important mathematical structure that appears in quantum information theory whenever physicists study particles, symmetries and transformations involving permutations and partial transposition. Work in this area inevitably leads to the question of how a system transforms when particles are permuted or when one part of a composite object is flipped (transposed) while the rest is left untouched. Collect all such operations together and you get the walled Brauer algebra. It plays an important role in the mathematical description of problems ranging from entanglement detection to advanced teleportation schemes. "The walled Brauer algebra (Credit: M. Horodecki, M. Studzi\u0144ski and M.…