Baek Kyunghyun, Nha Hyunchul, Son Wonmin
Department of Physics, Texas A&M University at Qatar, Education City, P.O. Box 23874 Doha, Qatar.
Asia Pacific Center for Theoretical Physics, Pohang 37673, Korea.
Entropy (Basel). 2019 Mar 11;21(3):270. doi: 10.3390/e21030270.
We derive an entropic uncertainty relation for generalized positive-operator-valued measure (POVM) measurements via a direct-sum majorization relation using Schur concavity of entropic quantities in a finite-dimensional Hilbert space. Our approach provides a significant improvement of the uncertainty bound compared with previous majorization-based approaches (Friendland, S.; Gheorghiu, V.; Gour, G. , , 230401; Rastegin, A.E.; Życzkowski, K. , , , 355301), particularly by extending the direct-sum majorization relation first introduced in (Rudnicki, Ł.; Puchała, Z.; Życzkowski, K. , , 052115). We illustrate the usefulness of our uncertainty relations by considering a pair of qubit observables in a two-dimensional system and randomly chosen unsharp observables in a three-dimensional system. We also demonstrate that our bound tends to be stronger than the generalized Maassen-Uffink bound with an increase in the unsharpness effect. Furthermore, we extend our approach to the case of multiple POVM measurements, thus making it possible to establish entropic uncertainty relations involving more than two observables.
我们通过使用有限维希尔伯特空间中熵量的舒尔凹性的直和优超关系，推导出广义正算子值测量（POVM）测量的熵不确定关系。与之前基于优超的方法（Friendland, S.; Gheorghiu, V.; Gour, G.,,, 230401; Rastegin, A.E.; Życzkowski, K.,,,, 355301）相比，我们的方法显著改进了不确定度界限，特别是通过扩展首次在（Rudnicki, Ł.; Puchała, Z.; Życzkowski, K.,,, 052115）中引入的直和优超关系。我们通过考虑二维系统中的一对量子比特可观测量和三维系统中随机选择的非锐可观测量，来说明我们的不确定关系的有用性。我们还证明，随着非锐度效应的增加，我们的界限往往比广义马森 - 乌芬克界限更强。此外，我们将我们的方法扩展到多个POVM测量的情况，从而使得建立涉及两个以上可观测量的熵不确定关系成为可能。
