Postselected 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 postselections are nearly orthogonal. In this paper, we study this problem by a variational approach for a general Hamiltonian $H_{\mathrm{int}}=gA\otimes \Omega \delta (t-t_0),g\ll 1$. We derive a general asymptotic solution and show that the amplification limit is essentially independent of $g$ and is determined by only the initial state of the detector and the number of distinct eigenvalues of $A$ or $\Omega$. 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.

• Received 9 July 2013
• Revised 6 May 2014

DOI:https://doi.org/10.1103/PhysRevA.90.012108