Abstract
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≥3 is bounded by n≤D2+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.
Highlights
The processing of information with quantum particles is inevitably affected by disturbance from the environment
From the definition of a quantum error correcting codes (QECC) in Eq (3), it is not hard to see that a pure code with parameters ((n, K, r + 1))D implies the existence of an r-uniform subspace of (CD)⊗n with dimension K
From the bound in Theorem 10 it is seen that quantum maximum distance separable (QMDS) codes with distance d ≥ 3 can only exist if n + k ≤ 6 for qubits, n + k ≤ 16 for qutrits, n + k ≤ 30 for ququarts, and n + k ≤ 48 in the case of local dimension D = 5
Summary
The processing of information with quantum particles is inevitably affected by disturbance from the environment. The quantum Singleton bound can be seen as having its origins in the no-cloning theorem [15, 16] It states that the parameters of any quantum error correction code of distance d, encoding states from CK into a. Codes achieving this bound are called quantum maximum distance separable (QMDS) [3, 17]. The appendices contain proofs of the quantum Singleton bound and an overview on previous bounds for stabilizer QMDS codes and AME states This is followed by detailed tables on known QMDS constructions and bounds on their existence for small local dimensions
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
