Modified Kubo formula with a complex force term for weak measurement

In a seminal work, Aharonov, Albert, and Vaidman showed that by having a weak interaction between a system and a detecting apparatus, the average output of the latter could be much larger than the maximum eigenvalue of the observed quantity (times the amplification factor). This does not always happen, however: the observed system must subsequently undergo a second measurement, on the output of which the result of the first one is conditioned. This procedure is known as postselection. On the other hand, linear response theory describes how the observables of a quan- tum system change upon perturbation by a weak classical external force. In a measurement, the measured system applies a generalized force to the measuring apparatus, leading to an observable change in the latter. It appears natural, then, to unify the treatment of weak measurements with an extended version of linear response theory that accounts for a force introduced by an external quantum system. Here, we show how the postselection introduces a complex force term and we provide a modified Kubo formula working in the non-linear regime.