Sequential weak measurement

The notion of weak measurement provides a formalism for extracting information from a quantum system in the limit of vanishing disturbance to its state. Here we extend this formalism to the measurement of sequences of observables. When these observables do not commute, we may obtain information about joint properties of a quantum system that would be forbidden in the usual strong measurement scenario. As an application, we provide a physically compelling characterization of the notion of counterfactual quantum computation.

Figure 1

The double interferometer: an optical circuit in which a photon, injected along path $A$, passes through two interferometers, represented by paths $B$ and $C$ and paths $E$ and $F$. Finally, the photon is postselected at the detector $D$. The beamsplitters are shown with their reflecting surface marked in black.

Figure 2

Paths through the double interferometer, and the number of photons that follow the indicated path. Thus for instance $N_{BE}=N∕2$. Note however the curious prediction $N_{BF}=-N∕2$.

Figure 3

The double interferometer restricted to its second interferometer. According to Eq. (22), the ratio of the weak values $E_w∕F_w$ in the second interferometer, with photons injected along $C$, is the same as the ratio of the sequential weak values $(E,C{)}_w∕(F,C{)}_w$ in the double interferometer with photons injected along $A$.

Figure 4

The double interferometer of Fig. 1 treated as a protocol with computer insertions (black rectangles) in paths $B$ and $F$. If a photon passes down either of these paths, the computer runs.