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Genuinely entangled subspaces (GESs) are the class of completely entangledsubspaces that contain only genuinely multiparty entangled states. Theyconstitute a particularly useful notion in the theory of entanglement but alsohave found an application, for instance, in quantum error correction andcryptography. In a recent study (Demianowicz and Augusiak in Phys Rev A98:012313, 2018), we have shown how GESs can be efficiently constructed in anymultiparty scenario from the so-called unextendible product bases. The providedsubspaces, however, are not of maximal allowable dimensions, and our aim hereis to put forward an approach to building such. The method is illustrated withfew examples in small systems. Connections with other mathematical problems,such as spaces of matrices of equal rank and the numerical range, arediscussed.

An approach to constructing genuinely entangled subspaces of maximal dimension