Wigner entropy conjecture and the interference formula in quantum phase space
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866911394341847040 |
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| author | Van Herstraeten, Zacharie Cerf, Nicolas J. |
| author_facet | Van Herstraeten, Zacharie Cerf, Nicolas J. |
| contents | Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. The Shannon differential entropy of the Wigner function of such states -- called Wigner entropy for brevity -- emerges as a fundamental information-theoretic measure in phase space and is subject to a conjectured lower bound, reflecting the uncertainty principle. In this work, we prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states, which are obtained by evolving a separable state through a balanced beam splitter and then discarding one mode. Our proof relies on known bounds on the $p$-norms of cross Wigner functions and on the interference formula, which relates the convolution of Wigner functions to the squared modulus of a cross Wigner function. Originally discussed in the context of signal analysis, the interference formula is not commonly used in quantum optics although it unveils a strong symmetry under convolution exhibited by Wigner functions of pure states. We provide here a simple proof of the formula and highlight some of its implications. Finally, we prove an extended conjecture on the Wigner-Rényi entropy of beam-splitter states, albeit in a restricted range for the Rényi parameter $α\geq 1/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_05562 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Wigner entropy conjecture and the interference formula in quantum phase space Van Herstraeten, Zacharie Cerf, Nicolas J. Quantum Physics Wigner-positive quantum states have the peculiarity to admit a Wigner function that is a genuine probability distribution over phase space. The Shannon differential entropy of the Wigner function of such states -- called Wigner entropy for brevity -- emerges as a fundamental information-theoretic measure in phase space and is subject to a conjectured lower bound, reflecting the uncertainty principle. In this work, we prove that this Wigner entropy conjecture holds true for a broad class of Wigner-positive states known as beam-splitter states, which are obtained by evolving a separable state through a balanced beam splitter and then discarding one mode. Our proof relies on known bounds on the $p$-norms of cross Wigner functions and on the interference formula, which relates the convolution of Wigner functions to the squared modulus of a cross Wigner function. Originally discussed in the context of signal analysis, the interference formula is not commonly used in quantum optics although it unveils a strong symmetry under convolution exhibited by Wigner functions of pure states. We provide here a simple proof of the formula and highlight some of its implications. Finally, we prove an extended conjecture on the Wigner-Rényi entropy of beam-splitter states, albeit in a restricted range for the Rényi parameter $α\geq 1/2$. |
| title | Wigner entropy conjecture and the interference formula in quantum phase space |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2411.05562 |