Experimental Observation of Anomalous Complementary Weak Values from Correlated Pairwise Two-State Vectors

In standard quantum mechanics, the Two-State Vector Formalism (TSVF) describes a system using both a pre-selected initial state and a post-selected final state. Weak measurements within this framework can produce anomalous weak values (WVs)—values that lie outside the eigenvalue spectrum of an observable—which are highly useful for precision metrology. However, these anomalous values typically only appear in one specific post-selection branch, while the complementary branch remains bounded. This creates an operational asymmetry that limits the efficiency and utility of weak value amplification.

To address this, the researchers developed a post-selection-controlled architecture. Instead of using a fixed pre-selected state, they utilize a maximally entangled state between the system and an ancilla. By applying a controlled-unitary operation that depends on the post-selection outcome, they create a one-to-one correlation between the post-selected state of the system and its corresponding pre-selected state. This effectively turns post-selection from a passive filtering process into an active engineering tool, allowing both complementary post-selection branches to exhibit anomalous weak values simultaneously.

Findings

Using a photonic platform, the team experimentally verified that their protocol allows both complementary post-selection branches to produce anomalous weak values. In their tests, they observed that while the standard protocol restricted the complementary branch to values within the eigenvalue spectrum, their new method enabled both branches to reach anomalous magnitudes (e.g., reaching values of ~5.02 compared to ~0.1 in the standard approach). This confirms that the protocol successfully removes the structural asymmetry of the TSVF.

This work provides a more efficient way to utilize quantum events in metrology. By doubling the success probability of observing anomalous weak values and providing a more symmetric operational framework, the approach improves the resource efficiency of weak-value-based sensing. Furthermore, it offers a versatile platform for exploring generalized time-symmetric quantum structures and potential applications in quantum information processing.