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.
 C. H. Bennett, D. P. DiVincenzo, J. A. Smolin, and W. K. Wootters, ``Mixed-state entanglement and quantum error correction,'' Phys. Rev. A 54, 3824 (1996).
 J. Tillich and G. Zémor, ``Quantum LDPC Codes With Positive Rate and Minimum Distance Proportional to the Square Root of the Blocklength,'' IEEE Trans. Inf. Theory 60, 1193–1202 (2014).
 D. Chandra, Z. Babar, H. V. Nguyen, D. Alanis, P. Botsinis, S. X. Ng, and L. Hanzo, ``Quantum Coding Bounds and a Closed-Form Approximation of the Minimum Distance Versus Quantum Coding Rate,'' IEEE Access 5, 11557–11581 (2017).
 D. Gottesman, ``Lecture Notes CO639,'' available online at www.perimeterinstitute.ca/personal/dgottesman/CO639-2004/ (2004).
 J. Walgate and A. J. Scott, ``Generic local distinguishability and completely entangled subspaces,'' J. Phys. A: Math. Theor. 41, 375305 (2008).
 R. Sengupta, Arvind, and A. I. Singh, ``Entanglement properties of positive operators with ranges in completely entangled subspaces,'' Phys. Rev. A 90, 062323 (2014).
 M. Demianowicz and R. Augusiak, ``Entanglement of genuinely entangled subspaces and states: Exact, approximate, and numerical results,'' Phys. Rev. A 100, 062318 (2019).
 M. Demianowicz and R. Augusiak, ``An approach to constructing genuinely entangled subspaces of maximal dimension,'' Quantum Inf. Process. 19, 199 (2020).
 C. Eltschka, F. Huber, O. Gühne, and J. Siewert, ``Exponentially many entanglement and correlation constraints for multipartite quantum states,'' Phys. Rev. A 98, 052317 (2018).
 D. Gottesman, ``An Introduction to Quantum Error Correction,'' in Quantum computation: A grand mathematical challenge for the twenty-first century and the millennium, ed. S. J. Lomonaco, Jr., pp. 221–235, (American Mathematical Society, 2002) arXiv:quant-ph/0004072.
 F. Huber, C. Eltschka, J. Siewert, and O. Gühne, ``Bounds on absolutely maximally entangled states from shadow inequalities, and the quantum MacWilliams identity,'' J. Phys. A: Math. Theor. 51, 175301 (2018).
 A. Ashikhmin and A. Barg, ``Binomial moments of the distance distribution: bounds and applications,'' IEEE Transactions on Information Theory 45, 438–452 (1999).
 A. Winter, private communication (2019).
 F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes (North Holland, 1981).
 A. J. Scott, ``Multipartite entanglement, quantum-error-correcting codes, and entangling power of quantum evolutions,'' Phys. Rev. A 69, 052330 (2004).
 A. Ketkar, A. Klappenecker, S. Kumar, and P. K. Sarvepalli, ``Nonbinary Stabilizer Codes Over Finite Fields,'' IEEE Trans. Inf. Theory 52, 4892–4914 (2006).
 W. Helwig, W. Cui, J. I. Latorre, A. Riera, and H.-K. Lo, ``Absolute maximal entanglement and quantum secret sharing,'' Phys. Rev. A 86, 052335 (2012).
 F. Huber, O. Gühne, and J. Siewert, ``Absolutely Maximally Entangled States of Seven Qubits Do Not Exist,'' Phys. Rev. Lett. 118, 200502 (2017).
 F. Huber and N. Wyderka, ``Table of AME states,'' available online at http://www.tp.nt.uni-siegen.de/+fhuber/ame.html (2020).
 P. K. Sarvepalli and A. Klappenecker, ``Nonbinary quantum Reed-Muller codes,'' in Proceedings. Int. Symp. Inf. Theory, (ISIT 2005) (2005) pp. 1023–1027.
 L. Jin, S. Ling, J. Luo, and C. Xing, ``Application of classical Hermitian self-orthogonal MDS codes to quantum MDS codes,'' IEEE Trans. Inf. Theory 56, 4735–4740 (2010).
 Felix Huber, "Positive maps and trace polynomials from the symmetric group", Journal of Mathematical Physics 62 2, 022203 (2021).
 Simeon Ball, Guillermo Gamboa, and Michel Lavrauw, "On additive MDS codes over small fields", Advances in Mathematics of Communications (2021).
 Markus Grassl, "Algebraic quantum codes: linking quantum mechanics and discrete mathematics", International Journal of Computer Mathematics: Computer Systems Theory 1 (2020).
 Daniel Alsina and Mohsen Razavi, "Absolutely maximally entangled states, quantum-maximum-distance-separable codes, and quantum repeaters", arXiv:1907.11253, Physical Review A 103 2, 022402 (2021).
 Simeon Ball, "Some constructions of quantum MDS codes", Designs, Codes and Cryptography 89 5, 811 (2021).
 Paweł Mazurek, Máté Farkas, Andrzej Grudka, Michał Horodecki, and Michał Studziński, "Quantum error-correction codes and absolutely maximally entangled states", Physical Review A 101 4, 042305 (2020).
 Maciej Demianowicz and Remigiusz Augusiak, "Entanglement of genuinely entangled subspaces and states: Exact, approximate, and numerical results", Physical Review A 100 6, 062318 (2019).
 Maciej Demianowicz and Remigiusz Augusiak, "An approach to constructing genuinely entangled subspaces of maximal dimension", Quantum Information Processing 19 7, 199 (2020).
 Sathwik Chadaga, Mridul Agarwal, and Vaneet Aggarwal, "Encoders and Decoders for Quantum Expander Codes Using Machine Learning", arXiv:1909.02945.
 Zahra Raissi, "Modifying method of constructing quantum codes from highly entangled states", arXiv:2005.01426.
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