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

TL;DR

This paper reveals that the product of sequential weak measurements in time-symmetric quantum mechanics corresponds to a strong, state-conditioned observable that encodes interference, with applications in quantum information processing.

Contribution

It introduces a novel connection between sequential weak values and strong observables, extending the formalism to mixed states and practical quantum information tasks.

Findings

Product of weak values equals a strong, state-conditioned observable.

The observable encodes interference, especially for superpositions.

Application to error witnessing and phase inference in quantum computing.

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 $\ket{\psi}$ 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.