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理论物理通讯：英文版

Uncertainty and Certainty Relations for Successive Projective Measurements of a Qubit in Terms of Tsallis＇ Entropies

Alexey, Rastegin

Published Jan 1, 2015• 0 citations
OpenAlex

Abstract

我们为二维的 observables 的二连续大小学习无常和必然关系。在连续测量的无常在下列二种情形以内被考虑。在第一种情形，第二大小在与完全擦掉的信息在第一大小以后产生的量状态上被执行。在第二种情形，第二大小在在实际测量结果上调节的 post-first-measurement 状态上被执行。导致的量无常借助于 Tsallis 熵被描绘。为 qubit 的二连续射影的大小，我们获得导致的无常的相关 entropic 措施的最小、最大的价值。在第二种情形发现的一些结论被扩大到任意的有限维数。特别地，有相互的不偏的一个连接被强调。

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我们为二维的 observables 的二连续大小学习无常和必然关系。在连续测量的无常在下列二种情形以内被考虑。在第一种情形，第二大小在与完全擦掉的信息在第一大小以后产生的量状态上被执行。在第二种情形，第二大小在在实际测量结果上调节的 post-first-measurement 状态上被执行。导致的量无常借助于 Tsallis 熵被描绘。为 qubit 的二连续射影的大小，我们获得导致的无常的相关 entropic 措施的最小、最大的价值。在第二种情形发现的一些结论被扩大到任意的有限维数。特别地，有相互的不偏的一个连接被强调。
Abstract We study uncertainty and certainty relations for two successive measurements of two-dimensional observables. Uncertainties in successive measurement are considered within the following two scenarios. In the first scenario, the second measurement is performed on the quantum state generated after the first measurement with completely erased information. In the second scenario, the second measurement is performed on the post-first-measurement state conditioned on the actual measurement outcome. Induced quantum uncertainties are characterized by means of the Tsallis entropies. For two successive projective measurement of a qubit, we obtain minimal and maximal values of related entropic measures of induced uncertainties. Some conclusions found in the second scenario are extended to arbitrary finite dimensionality. In particular, a connection with mutual unbiasedness is emphasized.
We address the generalized uncertainty principle in scenarios of successive measurements. Uncertainties are characterized by means of generalized entropies of both the Rényi and Tsallis types. Here, specific features of measurements of observables with continuous spectra should be taken into account. First, we formulated uncertainty relations in terms of Shannon entropies. Since such relations involve a state-dependent correction term, they generally differ from preparation uncertainty relations. This difference is revealed when the position is measured by the first. In contrast, state-independent uncertainty relations in terms of Rényi and Tsallis entropies are obtained with the same lower bounds as in the preparation scenario. These bounds are explicitly dependent on the acceptance function of apparatuses in momentum measurements. Entropic uncertainty relations with binning are discussed as well.
We obtain uncertainty and certainty relations of state-independent form for the three Pauli observables with use of the Rényi entropies of order α ∈ (0; 1]. It is shown that these entropic bounds are tight in the sense that they are always reached with certain pure states. A new result is the condition for equality in Rényi-entropy uncertainty relations for the Pauli observables. Upper entropic bounds in the pure-state case are also novel. Combining the presented bounds leads to a band, in which the rescaled average Rényi α-entropy ranges for a pure measured state. A width of this band is compared with the Tsallis formulation derived previously.