We introduce several classes of quantum combinatorial designs, namely quantum Latin squares, cubes, hypercubes and a notion of orthogonality between them. A further introduced notion, quantum orthogonal arrays, generalizes all previous classes of designs. We show that mutually orthogonal quantum Latin arrangements can be entangled in the same way than quantum states are entangled. Furthermore, we show that such designs naturally define a remarkable class of genuinely multipartite highly entangled states called k-uniform, i.e. multipartite pure states such that every reduction to k parties is maximally mixed. We derive infinitely many classes of mutually orthogonal quantum Latin arrangements and quantum orthogonal arrays having an arbitrary large number of columns. The corresponding multipartite k-uniform states exhibit a high persistency of entanglement, which makes them ideal candidates to develop multipartite quantum information protocols.

Absolutely maximally entangled (AME) states are pure multi-partite generalizations of the bipartite maximally entangled states with the property that all reduced states of at most half the system size are in the maximally mixed state. AME states are of interest for multipartite teleportation and quantum secret sharing and have recently found new applications in the context of high-energy physics in toy models realizing the AdS/CFT-correspondence. We work out in detail the connection between AME states of minimal support and classical maximum distance separable (MDS) error correcting codes and, in particular, provide explicit closed form expressions for AME states of n parties with local dimension q a power of a prime for all q ≥ n − 1. Building on this, we construct a generalization of the Bell-basis consisting of AME states and develop a stabilizer formalism for AME states. For every q ≥ n − 1 prime we show how to construct QECCs that encode a logical qudit into a subspace spanned by AME states. Under a conjecture for which we provide numerical evidence, this construction produces a family of quantum error correcting codes [[n, 1, n/2]]q for n even, saturating the quantum Singleton bound. We show that our conjecture is equivalent to the existence of an operator whose support cannot be decreased by multiplying it with stabilizer products and explicitly construct the codes up to n = 8.

Pure multipartite quantum states of n parties and local dimension q are called k-uniform if all reductions to k parties are maximally mixed. These states are relevant for our understanding of multipartite entanglement, quantum information protocols, and the construction of quantum error correction codes. To our knowledge, the only known systematic construction of these quantum states is based on classical error correction codes. We present a systematic method to construct other examples of k-uniform states and show that the states derived through our construction are not equivalent to any k-uniform state constructed from the so-called maximum distance separable error correction codes. Furthermore, we use our method to construct several examples of absolutely maximally entangled states whose existence was open so far.

We consider a boson gas on the stretched horizon of the Schwartzschild and Kerr black holes. It is shown that the gas is in a Bose-Einstein condensed state with the Hawking temperature Tc = TH if the particle number of the system be equal to the number of quantum bits of space-time N ≃ A/lp 2 . Entropy of the gas is proportional to the area of the horizon (A) by construction. For a more realistic model of quantum degrees of freedom on the horizon, we should presumably consider interacting bosons (gravitons). An ideal gas with intermediate statistics could be considered as an effective theory for interacting bosons. This analysis shows that we may obtain a correct entropy just by a suitable choice of parameter in the intermediate statistics.