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Polynomial invariants of quantum codes

Rains, Eric M. (2000) Polynomial invariants of quantum codes. IEEE Transactions on Information Theory, 46 (1). pp. 54-59. ISSN 0018-9448. http://resolver.caltech.edu/CaltechAUTHORS:20170926-153330743

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Abstract

The weight enumerators (Shor and Laflamme 1997) of a quantum code are quite powerful tools for exploring its structure. As the weight enumerators are quadratic invariants of the code, this suggests the consideration of higher degree polynomial invariants. We show that the space of degree k invariants of a code of length n is spanned by a set of basic invariants in one-to-one correspondence with S^n_k. We then present a number of equations and inequalities in these invariants; in particular, we give a higher order generalization of the shadow enumerator of a code, and prove that its coefficients are nonnegative. We also prove that the quartic invariants of a ((4, 4, 2))_2 code are uniquely determined, an important step in a proof that any ((4, 4, 2))_2 code is additive (Rains 1999).


Item Type:Article
Related URLs:
URLURL TypeDescription
https://doi.org/10.1109/18.817508 DOIArticle
http://ieeexplore.ieee.org/document/817508PublisherArticle
https://arxiv.org/abs/quant-ph/9704042arXivDiscussion Paper
Additional Information:© 2000 IEEE. Manuscript received May 26, 1997; revised February 24, 1999.
Subject Keywords:Invariant, quantum code, shadow, weight enumerator
Record Number:CaltechAUTHORS:20170926-153330743
Persistent URL:http://resolver.caltech.edu/CaltechAUTHORS:20170926-153330743
Official Citation:E. M. Rains, "Polynomial invariants of quantum codes," in IEEE Transactions on Information Theory, vol. 46, no. 1, pp. 54-59, Jan 2000. doi: 10.1109/18.817508 URL: http://ieeexplore.ieee.org/stamp/stamp.jsp?tp=&arnumber=817508&isnumber=17721
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:81856
Collection:CaltechAUTHORS
Deposited By: Tony Diaz
Deposited On:26 Sep 2017 22:41
Last Modified:26 Sep 2017 22:41

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