Maximally entangled states, holographic codes and toy models of the bulk-boundary correspondence

Bell states are known to be maximally entangled among all two-qubit states, as the partial trace is maximally mixed. For four-qubit system there is no state, such that any of its two-party reduction, with respect to any possible splitting, is maximally mixed. We show that such states exist for four qutrits and discuss existence of such absolutely maximally entangled states in multipartite systems. Such states correspond to ’perfect tensors’ or multi-unitary matrices, i. e. unitary matrices, which remain unitary after certain reordering of their elements.

Absolutely maximally entangled states are useful to construct quantum error correction codes. Furthermore, we mention recent results of other groups which show that these very states are used to build holographic codes and toy models of the bulk-boundary correspondence, and they allow one to construct a partial isometry between the bulk and the boundary Hilbert spaces.

All interested persons are invited
M. Kowal, W. Piechocki, L. Roszkowski, J. Skalski