量子計量學中適界的可達性

Abstract

在量子系統中高精度地估計參數是許多領域的重要工作，從工程、物理到生物學皆 是如此。Holevo-Cramer-Rao界限(HCRB)是量子計量學中的一個重要的適界(tight bound)。一般來說，它在漸近(asymptotical)情況下是可達到的(achievable)， 但在有限樣本(finite copy)—特別是單樣本(single shot)—的情況下並不總是可 達到的。在這本論文中，我們刻畫了估計量(estimator)和最佳測量在單樣本下飽 和HCRB的幾個必要/充分條件。特別是，我們提供了兩個關於HCRB可達性的刻畫。 其中之一(定理4.1)作為物理圖像和達到HCRB的充分條件。我們提供了一個條件 (定理4.1中的條件5)，它作為該定理的刻畫。在某些條件下，這種刻畫可以簡化為 其他量子Cramer-Rao界限的可達性必要條件(見頁碼30的討論)。然而，總的來說， 該定理並不是HCRB可達性的必要條件。另一個刻畫(定理5.1)提供了HCRB可達性 的數學刻畫，這是必要且充分的條件。通過這種刻畫(定理5.1)，其他量子Cramer- Rao界限的可達性可以很容易地確立(推論5.2和推論5.3)。

Estimating parameter in quantum system with high precision is an important work in many field, ranging from engineering, physics to biology. The Holevo Cramer-Rao bound (HCRB) is an important tight bound in quantum metrology. In general, it is asymptotic achievable, while it is not always achievable with finite copy, in particular, single shot level. In this work, we characterize several necessary/sufficient conditions for the estimator and the optimal measurement to saturate the HCRB in single copy level. In particular, we develop 2 characterizations for the achievability of the HCRB. First one of them (Theorem 4.1) is served as a physical picture and sufficient condition for saturating the HCRB. We provide a condition (condition 5 in Theorem 4.1) which is served as a characterization of the theorem. With some condition, this characterization can be reduced to a necessary condition for the achievabilty of the other quantum Cramer-Rao bound (see discussion on page 30). In general, however, the the- orem is not a necessary condition for the achievability of the HCRB. Second one of them (Theorem 5.1) gives a mathematical characterization for the saturation of the HCRB, it is necessary and sufficient condition. With this characterization (Theorem 5.1), the achievability of other quantum Cramer-Rao bound can be easily established (Corollary 5.2 and Corollary 5.3).