Non-negative Wigner functions in prime dimensions

Abstract

According to a classical result due to Hudson, the Wigner function of a pure, continuous variable quantum state is non-negative if and only if the state is Gaussian. We have proven an analogous statement for finite-dimensional quantum systems. In this context, the role of Gaussian states is taken on by stabilizer states. The general results have been published in [D. Gross, J. Math. Phys. 47, 122107 (2006)]. For the case of systems of odd prime dimension, a greatly simplified proof can be employed which still exhibits the main ideas. The present paper gives a self-contained account of these methods.

5 pages. Special case of a result proved in quant-ph/0602001. The proof is greatly simplified, making the general case more accessible. To appear in Appl. Phys. B as part of the proceedings of the 2006 DPG Spring Meeting (Quantum Optics and Photonics section)

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• Discrete phase space based on finite fields
• Hudson's Theorem for finite-dimensional quantum systems
• Quantum computers in phase space
• Classicality in discrete Wigner functions
• Discrete Wigner functions and quantum computational speedup
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