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首页> 外文期刊>Journal of physics, A. Mathematical and theoretical >General entropy-like uncertainty relations in finite dimensions
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General entropy-like uncertainty relations in finite dimensions

机译：有限维中一般熵的不确定性关系

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We revisit entropic formulations of the uncertainty principle (UP) for an arbitrary pair of positive operator-valued measures (POVM) A and B, acting on finite dimensional Hilbert space. Salicru generalized (h, phi)-entropies, including Renyi and Tsallis ones among others, are used as uncertainty measures associated with the distribution probabilities corresponding to the outcomes of the observables. We obtain a nontrivial lower bound for the sum of generalized entropies for any pair of entropic functionals, which is valid for both pure and mixed states. The bound depends on the overlap triplet (c(A), c(B), c(A),(B)) with c(A) (respectively c(B)) being the overlap between the elements of the POVM A (respectively B) and c(A),(B) the overlap between the pair of POVM. Our approach is inspired by that of de Vicente and Sanchez-Ruiz (2008 Phys. Rev. A 77 042110) and consists in a minimization of the entropy sum subject to the Landau-Pollak inequality that links the maximum probabilities of both observables. We solve the constrained optimization problem in a geometrical way and furthermore, when dealing with Renyi or Tsallis entropic formulations of the UP, we overcome the Holder conjugacy constraint imposed on the entropic indices by the Riesz-Thorin theorem. In the case of nondegenerate observables, we show that for given c(A,B) > 1/root 2, the bound obtained is optimal; and that, for Renyi entropies, our bound improves Deutsch one, but Maassen-Uffink bound prevails when c(A,B) <= 1/2. Finally, we illustrate by comparing our bound with known previous results in particular cases of Renyi and Tsallis entropies.
机译：我们对作用在有限维希尔伯特空间上的任意一对正算子值度量（POVM）A和B重新审视不确定性原理（UP）的熵公式。 Salicru广义（h，phi）熵（包括Renyi和Tsallis熵）被用作不确定性度量，其与对应于可观测结果的分布概率相关。我们获得了任意一对熵泛函的广义​​熵之和的非平凡下界，这对纯态和混合态均有效。边界取决于重叠三元组（c（A），c（B），c（A），（B）），其中c（A）（分别为c（B））是POVM A元素之间的重叠（分别是B）和c（A），（B）这对POVM之间的重叠。我们的方法受de Vicente和Sanchez-Ruiz（2008 Phys。Rev. A 77 042110）的启发，其方法是使受Landau-Pollak不等式影响的熵和最小，该不等式将两个可观测量的最大概率联系在一起。我们以几何方式解决约束优化问题，此外，在处理UP的Renyi或Tsallis熵公式时，我们克服了Riesz-Thorin定理对熵指数施加的Holder共轭约束。对于非简并可观测量，我们表明对于给定的c（A，B）> 1 /根2，获得的界是最优的;并且，对于Renyi熵，我们的界线提高了Deutsch一，但是当c（A，B）<= 1/2时，Maassen-Uffink界线占上风。最后，我们通过将我们的边界与已知的先前结果进行比较来说明在特定情况下的Renyi和Tsallis熵。

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