The equation of motion for a time-dependent weak value of a quantum-mechanical observable is known to contain a complex valued energy factor (the weak energy of evolution) that is defined by the dynamics of the pre-selected and post-selected states which specify the observable's weak value. In this paper, the mechanism responsible for the creation of this energy is identified and it is shown that the cumulative effect over time of this energy is manifested as dynamical phases and pure geometric phases (the intrinsic phases of evolution) which govern the evolution of the weak value during its measurement process. These phases are simply related to a Pancharatnam phase and Fubini–Study metric distance defined by the Hilbert space evolution of the associated pre-selected and post-selected states. A characterization of time-dependent weak value evolution as Pancharatnam phase angle rotations and Fubini–Study distance scalings of a vector in the Argand plane is discussed as an application of this relationship. The theory of weak values is also reviewed and simple ‘gedanken experiments’ are used to illustrate both the time-independent and the time-dependent versions of the theory. It is noted that the direct experimental observation of the weak energy of evolution would strongly support the time-symmetric paradigm of quantum mechanics and it is suggested that weak value equations of motion represent a new category of nonlocal equations of motion.
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.
We prove uncertainty relations that quantitatively express the impossibility of jointly sharp preparation of pre- and post-selected quantum states for measuring incompatible observables during the weak measurement. By defining a suitable operator whose average in the pre-selected quantum state gives the weak value, we show that one can have new uncertainty relations for variances of two such operators corresponding to two non-commuting observables. These generalize the recent stronger uncertainty relations that give non-trivial lower bounds for the sum of variances of two observables which fully capture the concept of incompatible observables. Furthermore, we show that weak values for two non-commuting projection operators obey a complementarity relation. Specifically, we show that for a pre-selected state if we measure a projector corresponding to an observable
A weakly followed by the strong measurement of another observable
B (for the post-selection) and, for the same pre-selected state we measure a projector corresponding to an observable
B weakly followed by the strong measurement of the observable
A (for the post-selection), then the product of these two weak values is always less than one. This shows that even though individually they are complex and can be large, their product is always bounded.