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Amplification limit of weak measurements: a variational approach

Post-selected weak measurement has been widely used in experiments to observe weak effects in various physical systems. However, it is still unclear how large the amplification ability of a weak measurement can be and what determines the limit of this ability, which is fundamental to understanding and applying weak measurements. The limitation of the conventional weak value formalism for this problem is the divergence of weak values when the pre- and post-selections are nearly orthogonal. In this paper, we study this problem by a variational approach for a general Hamiltonian $H_{\mathrm{int}}=gA\otimesΩδ(t-t_{0}),\, g\ll1$. We derive a general asymptotic solution, and show that the amplification limit is essentially independent of $g$, and determined only by the initial state of the detector and the number of distinct eigenvalues of $A$ or $Ω$. An example of spin-$\frac{1}{2}$ particles with a pair of Stern-Gerlach devices is given to illustrate the results. The limiting case of continuous variable systems is also investigated to demonstrate the influence of system dimension on the amplification limit.