Quantum Codes of Maximal Distance and Highly Entangled Subspaces

Felix Huber1,2,3 and Markus Grassl4,5

1ICFO - The Institute of Photonic Sciences, 08860 Castelldefels (Barcelona), Spain
2Institut für Theoretische Physik, Universität zu Köln, 50937 Köln, Germany
3Naturwissenschaftlich-Technische Fakultät, Universität Siegen, 57068 Siegen, Germany
4International Centre for Theory of Quantum Technologies, University of Gdansk, 80-308 Gdańsk, Poland
5Max Planck Institute for the Science of Light, 91058 Erlangen, Germany

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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.

Quantum error-correcting codes are essential to protect quantum information against decoherence and interaction with the environment. While this interaction creates undesired entanglement between the system and its environment, it is again entanglement among the individual subsystems composing the code that allows to fight decoherence.

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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