本文从矩阵理论视角统一分析 Kennard、Robertson、Schrödinger、Maccone 和 Pati 不确定关系，给出简短证明并刻画其饱和条件。
摘要：
我们从矩阵理论的角度，以统一方式分析了由 Kennard、Robertson、Schrödinger、Maccone 和 Pati 提出的不确定关系。文中给出了这些不确定关系的简短证明，并刻画了其饱和条件。
英文摘要：
We analyze uncertainty relations due to Kennard, Robertson, Schrödinger, Maccone and Pati in a unified way from matrix theory point of view. Short proofs are given to these uncertainty relations and characterizations of the saturation conditions are given.
Heisenberg's uncertainty principle was originally posed for the limit of the accuracy of simultaneous measurement of non-commuting observables as stating that canonically conjugate observables can be measured simultaneously only with the constraint that the product of their mean errors should be no less than a limit set by Planck's constant. However, Heisenberg with the subsequent completion by Kennard has long been credited only with a constraint for state preparation represented by the product of the standard deviations. Here, we show that Heisenberg actually proved the constraint for the ac…
Eigenvalues are defined for any element of an algebra of observables and do not require a representation in terms of wave functions or density matrices. A systematic algebraic derivation based on moments is presented here for the harmonic oscillator, together with a perturbative treatment of anharmonic systems. In this process, a collection of inequalities is uncovered which amount to uncertainty relations for higher-order moments saturated by the harmonic-oscillator excited states. Similar saturation properties hold for anharmonic systems order by order in perturbation theory. The new method,…
The measurement range problem, where one cannot determine the data outside the range of the detector, limits the characterization of entanglement in high-dimensional quantum systems when employing, among other tools from information theory, the entropic uncertainty relations. Practically, the measurement range problem weakens the security of entanglement-based large-alphabet quantum key distribution (QKD) employing degrees of freedom including time-frequency or electric field quadrature. We present a modified entropic uncertainty relation that circumvents the measurement range problem under ce…
We address entropic uncertainty relations between time and energy or, more precisely, between measurements of an observable $G$ and the displacement $r$ of the $G$-generated evolution $e^{-ir G}$. We derive lower bounds on the entropic uncertainty in two frequently considered scenarios, which can be illustrated as two different guessing games in which the role of the guessers are fixed or not. In particular, our bound for the first game improves the previous result by Coles et al.. Our derivation uses as a subroutine a recently proposed novel algebraic method, which can in general be used to d…
Implications and applications of the variance-based uncertainty equalities
在量子力学中，基于方差的海森堡型不确定性关系是一系列数学不等式，它们对状态准备的可实现精度构成了基本限制。与此相反，我们构造并公式化了两个量子测不准等式，它们对所有的不相容观测量对都成立，并表明了L. Maccone和A. K. Pati [Phys. Rev. Lett. 113，260401（2014）]。此外，我们提出了一个明确的解释背后的推导，并将这些关系到所谓的智能状态。作为例子，我们研究了量子比特系统中这些不确定性不等式的性质，得到了方差和的一个与状态无关的界。最后，我们将这些不等式应用于自旋压缩情形，并讨论了它在干涉灵敏度中的意义。
From the perspective of Markovian piecewise deterministic processes (PDPs), we investigate the derivation of a kinetic uncertainty relation (KUR), which was originally proposed in Markovian open quantum systems. First, stationary distributions of classical PDPs are explicitly constructed. Then, a tilting method is used to derive a rate functional of large deviations. Finally, based on an improved approximation scheme, we recover the KUR. These classical results are directly extended to the open quantum systems. We use a driven two-level quantum system to exemplify the quantum results.
We investigate quantum Rényi entropic quantities, specifically those derived from 'sandwiched' divergence. This divergence is one of several proposed Rényi generalisations of the quantum relative entropy. We may define Rényi generalisations of the quantum conditional entropy and mutual information in terms of this divergence, from which they inherit many desirable properties. However, these quantities lack some of the convenient structure of their Shannon and von Neumann counterparts. We attempt to bridge this gap by establishing divergence inequalities for valid combinations of Rényi order wh…
