We present new bounds on the existence of general quantum maximum distance separable codes (QMDS): the length $n$ of all QMDS codes with local dimension $D$ and distance $d \geq 3$ is bounded by $n \leq D^2 + d - 2$. We obtain their weight distribution and present additional bounds that arise from Rains' shadow inequalities. Our main result can be seen as a generalization of bounds that are known for the two special cases of stabilizer QMDS codes and absolutely maximally entangled states, and confirms the quantum MDS conjecture in the special case of distance-three codes. As the existence of QMDS codes is linked to that of highly entangled subspaces (in which every vector has uniform $r$-body marginals) of maximal dimension, our methods directly carry over to address questions in multipartite entanglement.
We investigate bounds on the parameters of the code that relate to the entanglement in the code, as manifested by maximally mixed marginals of the logical states. The first bound is the quantum Singleton bound, which has already been known very early in the theory of quantum error-correction. It is independent of the local dimension and can always be reached when the local dimension is sufficiently large. The corresponding codes are known as quantum maximum distance separable (QMDS) codes.
In this paper, we derive additional bounds on the existence of QMDS codes. Crucially, they are valid for all QMDS codes, including codes beyond the stabilizer formalism. We show that another characteristic property, the weight enumerator, is also independent of whether the QMDS code is of the stabilizer type or not.
In many cases the known stabilizer constructions match our upper bounds. It it surprising that these combinatorial, inherently classical constructions yield optimal codes also in the quantum case, dealing with arbitrary subspaces of complex vector spaces. We conclude with the open question whether or not there are QMDS codes which do not arise from classical MDS codes.
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