To investigate the role of initial quantum coherence in work-probability distribution, it is necessary to consider an incomplete or partial measurement, in which the energy cannot be fully discriminated by the detector. In this paper, we use a harmonic oscillator with a coherent or squeezing state to realize this incomplete or partial measurement, and propose a unified framework of quantum work statistics for a closed system with an arbitrary initial state. We find that work is proportional to the change of the real part of the coherent state parameter, i.e., quantum work can be estimated by the coherent state parameter. The resulting work-probability distribution includes the initial quantum coherence, and can be reduced to the result of the traditional two projective energy measurement scheme (TPM) by squeezing the state of the harmonic oscillator. Our measurement scheme reveals the fundamental connections between measurement error and coherent work. By introducing a ‘coherent work-to-noise ratio’, we find the optimal measurement error, which is determined by the energy difference between the superposed energy levels. As an application, we consider a driven two-level system and investigate the effects of driving velocity on work statistics. We find that only when the driving velocity matches the transition frequency of the system can initial quantum coherence play an important role.

为了研究初始量子相干性在工作概率分布中的作用，有必要考虑不完整或部分测量，其中检测器无法完全区分能量。在本文中，我们使用具有相干或挤压状态的谐振子来实现这种不完全或部分测量，并针对具有任意初始状态的封闭系统提出了量子功统计的统一框架。我们发现功与相干态参数实部的变化成正比，即量子功可以通过相干态参数来估计。由此产生的工作概率分布包括初始量子相干性，并且可以通过压缩谐振子的状态将其简化为传统的两个投影能量测量方案（TPM）的结果。我们的测量方案揭示了测量误差和相干工作之间的基本联系。通过引入“相干功噪比”，我们找到了最佳测量误差，该误差由叠加能级之间的能量差决定。作为一个应用，我们考虑一个驱动的两级系统并研究驱动速度对工作统计的影响。我们发现只有当驱动速度与系统的跃迁频率匹配时，初始量子相干性才能发挥重要作用。我们找到了最佳测量误差，这是由叠加能级之间的能量差决定的。作为一个应用，我们考虑一个驱动的两级系统并研究驱动速度对工作统计的影响。我们发现只有当驱动速度与系统的跃迁频率匹配时，初始量子相干性才能发挥重要作用。我们找到了最佳测量误差，这是由叠加能级之间的能量差决定的。作为一个应用，我们考虑一个驱动的两级系统并研究驱动速度对工作统计的影响。我们发现只有当驱动速度与系统的跃迁频率匹配时，初始量子相干性才能发挥重要作用。