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arXiv:1008.5337

Entanglement of stabilizer codewords

Xiaoyu Chen, Li-zhen Jiang

Published Aug 31, 2010• 1 citation
arXivHTMLPDF

Abstract

The geometric measure, the logarithmic robustness and the relative entropy of entanglement are proved to be equal for a stabilizer quantum codeword. The entanglement upper and lower bounds are determined with the generators of code. The entanglement of dual-containing CSS codes, Gottesman codes and the related codes are given. An iterative algorithm is developed to determine the exact value of the entanglement when the two bounds are not equal.

DOI
10.48550/arxiv.1008.5337

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Quantum communication and quantum computation are two important parts of quantum information science. Quantum algorithm usually uses real equally weighted states, among them are graph states, whose entanglement is well studied. Quantum communication is inevitably connected with quantum error correcting codes (QECC). The most important and frequently used QECCs are quantum stabilizer codes, which can be seen as the combination of graph states with classical error correcting codes. We argue that the complexity of quantum coding is closely related to the entanglement of the code. We prove that entanglement measured by the geometric measure, the robustness and the relative entropy of entanglement are equal for a stabilizer quantum codeword. The entanglement upper and lower bounds are determined from the generators of a quantum code. CSS codes are quantum codes derived from classical codes. We prove that the entanglement of CSS code equals its number of classical generators when the CSS code is dual-containing. We give the entanglement of codewords for Gottesman codes and related codes. An iterative algorithm is developed to calculate the entanglement numerically.
In this paper, we estimate the fidelity of stabilizer and CSS codes. First, we derive a lower bound on the fidelity of a stabilizer code via its quantum enumerator. Next, we find the average quantum enumerators of the ensembles of finite length stabilizer and CSS codes. We use the average quantum enumerators for obtaining lower bounds on the average fidelity of these ensembles. We further improve the fidelity bounds by estimating the quantum enumerators of expurgated ensembles of stabilizer and CSS codes. Finally, we derive fidelity bounds in the asymptotic regime when the code length tends to infinity. These results tell us which code rate we can afford for achieving a target fidelity with codes of a given length. The results also show that in symmetric depolarizing channel a typical stabilizer code has better performance, in terms of fidelity and code rate, compared with a typical CSS codes, and that balanced CSS codes significantly outperform other CSS codes. Asymptotic results demonstrate that CSS codes have a fundamental performance loss compared with stabilizer codes.
We consider design of the quantum stabilizer codes via a two-step, low-complexity approach based on the framework of codeword-stabilized (CWS) codes. In this framework, each quantum CWS code can be specified by a graph and a binary code. For codes that can be obtained from a given graph, we give several upper bounds on the distance of a generic (additive or nonadditive) CWS code, and the lower Gilbert-Varshamov bound for the existence of additive CWS codes. We also consider additive cyclic CWS codes and show that these codes correspond to a previously unexplored class of single-generator cyclic stabilizer codes. We present several families of simple stabilizer codes with relatively good parameters.
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