2020
DOI: 10.22331/q-2020-06-18-284
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Quantum Codes of Maximal Distance and Highly Entangled Subspaces

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 t… Show more

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Cited by 52 publications
(49 citation statements)
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“…[20] or Theorem 2 in Ref. [21]. In next section, we give an analytical technique that can be applied to prove this and a much broader set of claims within the stabilizer formalism.…”
Section: Observationmentioning
confidence: 93%
See 2 more Smart Citations
“…[20] or Theorem 2 in Ref. [21]. In next section, we give an analytical technique that can be applied to prove this and a much broader set of claims within the stabilizer formalism.…”
Section: Observationmentioning
confidence: 93%
“…[32], does exist. It is however worth mentioning that a very recent paper shows that the existence of a [[n − 1, 1, n/2]] q code implies the existence of the [[n, 0, n/2 + 1]] q for n even [21].…”
Section: Corollary 23mentioning
confidence: 99%
See 1 more Smart Citation
“…For k = 3 there are [18,3,16] 16 linear MDS codes over F 16 which can be obtained by extending the Reed-Solomon codes or by taking the code generated by the matrix whose columns are the points of a Lunelli-Sce hyperoval [21]. These codes shorten to [17,3,15] 16 linear MDS codes over F 16 and dualise to [18,15,4] 16 linear MDS codes over F 16 which truncate to [17,15,3] 16 MDS codes and project to [17,14,4] 16 MDS codes.…”
Section: Additive Mds Codes Over Small Finite Fieldsmentioning
confidence: 99%
“…With this technique, one starts from a k-UNI state of n parties and constructs the QECC by taking partial trace over one particle, i.e., generates a code with n − 1 parties. Repeating this technique produces a family of QECCs with a different set of code parameters [11], [12]. This method can be called Shortening which refers to the connection it has with the classical codes and the subspace of constructed quantum codes which are spanned by the highly entangled states [5], [13]- [15].…”
Section: Introductionmentioning
confidence: 99%