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

Abstract

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 $|\psi ⟩$ to a post-selected state $|\varphi ⟩$, 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 $P_{\psi }=|\psi ⟩⟨\psi |$ is the projector onto the preselected state. Analyzing the structure of $B$, we demonstrate how it encodes interference information, particularly when $|\psi ⟩$ 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.