Weak value beyond conditional expectation value of the pointer readings

It is argued that a weak value of an observable is a robust property of a single pre- and postselected quantum system rather than a statistical property. During an infinitesimal time a system with a given weak value affected other systems as if it had been in an eigenstate with eigenvalue equal to the weak value. This differs significantly from the action of a system preselected only and possessing a numerically equal expectation value. The weak value has a physical meaning beyond a conditional average of a pointer in the weak measurement procedure. The difference between the weak value and the expectation value has been demonstrated on the example of photon polarization. In addition, the weak values for systems pre- and postselected in mixed states are considered.

Figure 1

Experimental setup for comparing pointer wave functions. By varying the phase $\alpha$, we can change between constructive and destructive interference in order to measure the maximal visibility and thus to find the Bures angle. The first polarizer in the reference beam (ref) is used to vary the intensity of the reference beam. The postselection polarizer in the test beam (test) used when analyzing the weak value (WV) case is removed and put in front of the ${\mathrm{YVO}}_4$ when observing the expectation value in the test arm (ExpV) to keep the number of optical components equal. Note that for practical reasons a folded Mach-Zehnder interferometer, i.e., a nondegenerate version of a Michelson interferometer, is used, where the phase can be scanned by means of a retroreflecting prism coupled to a piezo. The shift of the center of the test beam caused by the ${\mathrm{YVO}}_4$ is compensated by a lateral displacement of the beam.

Figure 2

Maximally achievable visibility for different couplings $g\epsilon$ in the interference experiment between the undisturbed beam and the test beam (weak value case, blue boxes with blue solid line; expectation value case, red circles with red dashed line) as a function of the coupling strength $g\epsilon /2\mathrm{\Delta }$. The lines represent the theoretical curves with fitted offsets. Each setting was measured ten times to reduce the errors introduced by the optimization procedure.

Figure 3

Bures angles between the spatial state of the undisturbed beam (3) and the spatial states of the beam coupled to the weak value $A_w=0$ (blue boxes with blue solid fitted line) and the expectation value $\langle A\rangle =0$ (red circles with red dashed fitted line) as a function of the coupling strength $g\epsilon /2\mathrm{\Delta }$. The fitting according to Eqs. (24) and (23) includes offset parameters ${\xi }_1$ and ${\xi }_2$ appearing due to imperfections of the optical setup.

Figure 4

Preparation of a mixed two-state vector $({\rho }_{\text{post}},{\rho }_{\text{pre}})$. System $S$ and ancilla $A_1$ are prepared at $t_1$ in an entangled state ${|\mathrm{\Psi }\rangle }_{S,A_1}$ such that the reduced density matrix of $S$ is ${\rho }_{\text{pre}}$. The system is then found at $t_2$ in another entangled state ${|\mathrm{\Phi }\rangle }_{S,A_2}$ with ancilla $A_2$ such that the reduced density matrix of $S$ is ${\rho }_{\text{post}}$. Just before $t_2$, ancilla $A_2$ was prepared in a maximally entangled state with yet another ancilla $A_3$. At intermediate times all ancilla systems are protected to avoid any interaction.

Figure 5

The pointer reading and the weak value as a function of the strength of the measurement interaction. The pointer reading $\frac{\theta }{g\tau }$ (calibrated to correctly show eigenvalues of $A$) is given by the continuous red line. The weak value $A_w$ is given by the blue dashed line.

Figure 6

In order to calculate the weak value, a (hypothetical) verification measurement is introduced at $t_3$ which provides a pure two-state vector for the system and ancilla $A_1$ and thus a generalized two-state vector for the system.