标题： 当弱变强：通过时间对称量子选择的有效可观测量 显示英文标题

标题： When the Weak Becomes Strong: Effective Observables via Time-Symmetric Quantum Selection

摘要： 我们研究在时间对称量子力学框架下弱值的顺序组合。 具体而言，我们考虑从预选态$\ket{\psi}$到后选态$\ket{\phi}$的正向“弱测量”，接着进行反向“弱测量”。 我们证明这两个弱值的乘积对应于一个强的、状态条件可观测量$B = A P_\psi A$的归一化期望值，其中$P_\psi = \ket{\psi}\bra{\psi}$是投影到预选态上的投影算符。 分析$B$的结构，我们展示了它是如何编码干涉信息的，特别是在$\ket{\psi}$是叠加态而非$A$的本征态时。 这种表述通过将$P_\psi$替换为一般的密度矩阵$\rho$自然地扩展到混合态，将该构造与广义量子测量的形式主义联系起来。 我们在量子信息中展示了实际应用，包括量子计算中的特定状态误差见证，并展示了如何在纯态情况下通过强测量推断弱值的相位。 显示英文摘要

摘要： We investigate the sequential composition of weak values in the framework of time-symmetric quantum mechanics. Specifically, we consider a forward'' weak measurement from a preselected state $\ket{\psi}$ to a post-selected state $\ket{\phi}$, followed by a reverse'' weak measurement. We show that the product of these two weak values corresponds to the normalized expectation value of a strong, state-conditioned observable $B = A P_\psi A$, where $P_\psi = \ket{\psi}\bra{\psi}$ is the projector onto the preselected state. Analyzing the structure of $B$, we demonstrate how it encodes interference information, particularly when $\ket{\psi}$ is a superposition rather than an eigenstate of $A$. This formulation extends naturally to mixed states by replacing $P_\psi$ with a generic density matrix $\rho$, linking the construction to the formalism of generalized quantum measurements. We illustrate practical applications in quantum information, including state-specific error witnessing in quantum computing, and show how the phase of a weak value can be inferred via strong measurements in the pure-state case.