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

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 ψ to a post-selected state ϕ, 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_ψA, where P_ψ= ψψ is the projector onto the preselected state. Analyzing the structure of B, we demonstrate how it encodes interference information, particularly when ψ is a superposition rather than an eigenstate of A. This formulation extends naturally to mixed states by replacing P_ψ with a generic density matrix $ρ$, 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.