Yu Watanabe, Takahiro Sagawa, Masahito Ueda
By invoking quantum estimation theory we formulate bounds of errors in quantum measurement for arbitrary quantum states and observables in a finite-dimensional Hilbert space. We prove that the measurement errors of two observables satisfy Heisenberg's uncertainty relation, find the attainable bound, and provide a strategy to achieve it.
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It is shown that all the known uncertainty relations are the secondary consequences of Robertson's relation. The basic idea is to use the Heisenberg picture so that the time development of quantum mechanical operators incorporate the effects of the measurement interaction. A suitable use of triangle inequalities then gives rise to various forms of uncertainty relations. The assumptions of unbiased measurement and unbiased disturbance are important to simplify the resulting uncertainty relations and to give the familiar uncertainty relations such as a naive Heisenberg error-disturbance relation…
A universally valid uncertainty relation proposed by Ozawa is re-investigated under for the generalized equation of motion with some boundary condition. Necessary conditions for violation (lessening) of the Heisenberg-type uncertainty relation is discussed in detail. An experimental test of lessening of the generalized uncertainty relation due to the boundary condition is shown by re-analyzing the data measuring electron densities of hydrogen molecule encapsulated in a cage of fullerene C60.
Uncertainty relations revisited
本研究提出一种适用于本科教学的统一方法，可推导各类标准不确定性关系，同时回顾相关早期历史，探讨方差量化不确定性的普及原因，并指出Robertson不等式取等态并非真正的最小不确定态。
We study the quantum-mechanical uncertainty relation originating from the successive measurement of two observables $\hat{A}$ and $\hat{B}$, with eigenvalues $a_n$ and $b_m$, respectively, performed on the same system. We use an extension of the von Neumann model of measurement, in which two probes interact with the same system proper at two successive times, so we can exhibit how the disturbing effect of the first interaction affects the second measurement. Detecting the statistical properties of the second {\em probe} variable $Q_2$ conditioned on the first {\em probe} measurement yielding $…
Early Gedanken Experiments of Quantum Mechanics Revisited
著名的量子力学的格丹肯实验在发展哥本哈根解释中发挥了至关重要的作用。它们在这里是从标准量子力学的角度来研究的，不涉及本体论的解释。玻尔对这些实验的研究，基于与他的解释的不确定性关系，是哥本哈根解释的起源，至今仍被广泛采用，但被证明与量子力学观点不一致。我们指出，在大多数这些gedanken实验中，纠缠起着至关重要的作用，而它的积累并没有改变的方式玻尔认为有关量的不确定性。特别是在γ射线显微镜和反冲双缝量子纠缠实验中，我们揭示了基于动量交换的纠缠。结果表明，即使在这种情况下，干涉的损失也仅仅是由于与其他自由度的纠缠，而迄今为止尚未受到质疑的测不准关系论点是不正确的。
Simultaneous optical phase and loss estimation revisited: measurement and probe incompatibility
本文研究了单个光模式中同时估计相位偏移和损耗的多参数计量问题，发现探测不兼容性在特定条件下可消失，但测量不兼容性在所有情况下均存在且无法克服。
Quantum Algorithms Revisited
量子计算机使用不同计算路径的量子干涉来增强正确的结果并抑制计算的错误结果。当量子计算被视为多粒子干涉时，可以识别支撑量子算法的共同模式。我们使用这种方法来审查（和改进）一些现有的量子算法，并显示它们是如何与不同的情况下的量子相位估计。我们提供了一个显式算法，用于生成任意精度的任意指定的干涉图案。
Minimum uncertainty measurements of angle and angular momentum
角和角动量的不确定性关系重新审视。我们使用角度的指数而不是角度本身，并采用色散作为分辨率的自然度量。我们找到的状态，最小化的不确定性产品的约束下，一个给定的角度或角动量的不确定性。这些状态用马蒂厄波函数描述，并且可以用冯·米塞斯分布近似，这是单位圆上高斯分布的最接近的类似物。我们使用束流光学证实了我们的预测报告的实验结果。
Quantum Parameter Estimation Uncertainty Relation
本文针对量子多参数估计问题，推导了双参数情形下由测量不相容性和参数相关性共同限制估计精度的紧致不确定性关系，并开发误差椭圆方法直观展示其应用，揭示了参数空间几何视角的威力。
We formulate some properties of a set of several mutually unbiased measurements. These properties are used for deriving entropic uncertainty relations. Applications of mutually unbiased measurements in entanglement detection are also revisited. First, we estimate from above the sum of the indices of coincidence for several mutually unbiased measurements. Further, we derive entropic uncertainty relations in terms of the Rényi and Tsallis entropies. Both the state-dependent and state-independent formulations are obtained. Using the two sets of local mutually unbiased measurements, a method of en…
Uncertainty relations in quantum mechanics express bounds on our ability to simultaneously obtain knowledge about expectation values of non-commuting observables of a quantum system. They quantify trade-offs in accuracy between complementary pieces of information about the system. In Quantum multiparameter estimation, such trade-offs occur for the precision achievable for different parameters characterizing a density matrix: an uncertainty relation emerges between the achievable variances of the different estimators. This is in contrast to classical multiparameter estimation, where simultaneou…
In 1930 Einstein argued against consistency of the time-energy uncertainty relation by discussing a thought experiment involving a measurement of mass of the box which emitted a photon. Bohr seemingly triumphed over Einstein by arguing that the Einstein's own general theory of relativity saves the consistency of quantum mechanics. We revisit this thought experiment from a modern point of view at a level suitable for undergraduate readership and find that neither Einstein nor Bohr was right. Instead, this thought experiment should be thought of as an early example of a system demonstrating nonl…
We address the question whether the super-Heisenberg scaling for quantum estimation is realizable. We unify the results of two approaches. In the first one, the original system is compared with its copy rotated by the parameter dependent dynamics. If the parameter is coupled to the one-body part of the Hamiltonian the precision of its estimation is known to scale at most as $N^{-1}$ (Heisenberg scaling) in terms of the number of elementary subsystems used, $N$. The second approach considers fidelity at criticality often leading to an apparent super-Heisenberg scaling. However, scaling of time …
In 2004 Ambainis and Regev formulated a certain form of quantum adiabatic theorem and provided an elementary proof which is especially accessible to computer scientists. Their result is achieved by discretizing the total adiabatic evolution into a sequence of unitary transformations acting on the quantum system. Here we continue this line of study by providing another elementary and shorter proof with improved bounds. Our key finding is a succinct integral representation of the difference between the target and the actual states, which yields an accurate estimation of the approximation error. …
Quantum reference frames, revisited
本文指出量子参考系主流处理方法的内在不一致，提出拒绝以酉算子表示参考系变换的替代定义，并给出可实验判别的方案及其基础意义。
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