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Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE

Weak measurements, weak values, and entanglement

Jeff Tollaksen, Debabrata Ghoshal

Published Apr 27, 2007• 4 citations
Publisher

Abstract

As a resource, quantum entanglement provides enormous power to quantum information processing and quantum communication. We focus on new properties of entanglement as revealed by quantum weak measurements. Weak measurements are performed between a pre-selected and post-selected states, one, or both or which are entangled.
DOI
10.1117/12.719534

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Violations of Bell inequalities are a hallmark of entanglement, with only entangled states capable of exceeding classical bounds in standard Bell tests. Here we analyze anomalous weak values of the CHSH-Bell operator in pre- and post-selected (PPS) quantum ensembles, using them to define separability-constrained bounds on Bell-type nonlocal correlations in the presence of post-selection. Fixing the overlap between the pre- and post-selected states, we compare three scenarios: unrestricted boundary states, one separable boundary state, and both boundary states separable. For each case, we derive both the maximal weak value for a fixed Bell operator and the maximal bound obtained by further optimizing over all CHSH operators. Our results show that post-selection and entanglement are distinct operational resources: post-selection alone can enhance correlations, but entanglement is necessary to exceed the corresponding separable PPS bounds, and their combination yields the strongest attainable correlations. Thus the separable PPS bound plays the role of a post-selected separability benchmark, distinct from the standard Bell bound. We further show that the enhancement beyond the separable bound closely tracks the concurrence of the states that optimize the bounds, identifying entanglement as the source of the additional correlation strength. Finally, we show that nonlocal weak values provide post-selected entanglement witnesses, and we give a constructive protocol that detects every pure two-qubit source state with nonzero concurrence in the ideal state-adapted setting, even in regimes where the corresponding standard CHSH entanglement test is inconclusive. More broadly, our results motivate hybrid protocols that combine post-selection and entanglement, with possible applications to improved quantum sensing, weak-value amplification, and quantum information processing.
Weak values (WVs) arise from weak measurements performed within a time-symmetric formulation of quantum mechanics, where a system is both pre- and post-selected. Anomalous WVs that lie far outside the eigenvalue spectrum of the observable hold both fundamental and practical significance. However, their generation typically relies on near-orthogonal pre- and post-selection, which confines them to a single post-selection outcome with extremely low success probability. This constraint limits experimental accessibility and hinders the full exploitation of time symmetry. To overcome these limitations, we utilize quantum entanglement and post-selection-controlled operations to generate correlated pairwise two-state vectors. By changing the role of post-selection from passive filtering to active engineering, this approach enables the observation of anomalous complementary WVs associated with mutually exclusive post-selection branches. Our results extend the operational accessibility of time-symmetric quantum structures associated with the two-state vector formalism, and open new avenues for exploring the applications of time symmetry in quantum information processing.