Significance of the imaginary part of the weak value

Unlike the real part of the generalized weak value of an observable, which can in a restricted sense be operationally interpreted as an idealized conditioned average of that observable in the limit of zero measurement disturbance, the imaginary part of the generalized weak value does not provide information pertaining to the observable being measured. What it does provide is direct information about how the initial state would be unitarily disturbed by the observable operator. Specifically, we provide an operational interpretation for the imaginary part of the generalized weak value as the logarithmic directional derivative of the postselection probability along the unitary flow generated by the action of the observable operator. To obtain this interpretation, we revisit the standard von Neumann measurement protocol for obtaining the real and imaginary parts of the weak value and solve it exactly for arbitrary initial states and postselections using the quantum operations formalism, which allows us to understand in detail how each part of the generalized weak value arises in the linear response regime. We also provide exact treatments of qubit measurements and Gaussian detectors as illustrative special cases, and show that the measurement disturbance from a Gaussian detector is purely decohering in the Lindblad sense, which allows the shifts for a Gaussian detector to be completely understood for any coupling strength in terms of a single complex weak value that involves the decohered initial state.

Figure 1

Schematic for a von Neumann measurement. An initially prepared system state ${\widehat{\mathit{\rho }}}_i$ and detector state $|\psi \rangle \langle \psi |$ become entangled with the von Neumann unitary interaction ${\widehat{\mathit{U}}}_T$ [Eq. (6)] over a time interval $T$. Measuring a particular detector position $x$ after the interaction updates the detector state to $|x\rangle \langle x|$ and also updates the system state to ${\mathcal{M}}_x\left({\widehat{\mathit{\rho }}}_i\right)$, where ${\mathcal{M}}_x$ [Eq. (11)] is an effective measurement operation that encodes the entanglement with, and subsequent measurement of, the detector.

Figure 2

Schematic for a sequence of two indirect measurements. After the von Neumann interaction and measurement of $x$ illustrated in Fig. 1 that produces the effective measurement operation ${\mathcal{M}}_x$ upon the initial system state, a second detector interacts impulsively with the system for a time interval $T_2-T$. The second detector is then measured to have a particular outcome $f$, which updates the system state to ${\mathcal{P}}_f({\mathcal{M}}_x\left({\widehat{\mathit{\rho }}}_i\right))$, where ${\mathcal{P}}_f$ is another measurement operation. Taking the trace of the final system state will then produce the joint probability densities (24a).

Figure 3

The projection onto the plane $r_3=0$ of the qubit Bloch sphere, showing the vector field ${\delta }_{{\sigma }_3}\left({\widehat{\mathit{\rho }}}_i\right)=-r_2{\widehat{\mathit{\sigma }}}_1+r_1{\widehat{\mathit{\sigma }}}_2$ for arbitrary initial states ${\widehat{\mathit{\rho }}}_i=(\widehat{\mathbf{1}}+r_1{\widehat{\mathit{\sigma }}}_1+r_2{\widehat{\mathit{\sigma }}}_2+r_3{\widehat{\mathit{\sigma }}}_3)/2$. The curves of the flow through this vector field are the Rabi oscillations around the $r_3$ axis that are generated by the unitary action of ${\widehat{\mathit{\sigma }}}_3$. The quantity $2\mathrm{Im}{A}_w$ [Eq. (63a)] is the logarithmic rate of change of the postselection probability (41) along this vector field.

Figure 4

Two projections of the Bloch sphere showing the pure decoherence of the specific state ${\widehat{\mathit{\rho }}}_i\left(\epsilon \right)=\exp \left(\epsilon \mathcal{L}\left[{\widehat{\mathit{\sigma }}}_3\right]\right)\left({\widehat{\mathit{\rho }}}_i\right)=[\widehat{\mathbf{1}}+\exp (-2\epsilon )\sqrt{3}{\widehat{\mathit{\sigma }}}_2/2+{\widehat{\mathit{\sigma }}}_3/2]/2$ due to the Gaussian detector (74). (Left) The projection onto the plane $r_1=0$ showing the progressive collapse of ${\widehat{\mathit{\rho }}}_i\left(\epsilon \right)$ onto the $r_3$ axis with increasing $\epsilon$. (Right) The projection onto the plane $r_3=0$ showing the vector field ${\delta }_{{\sigma }_3}({\widehat{\mathit{\rho }}}_i\left(\epsilon \right))$ during the progressive collapse. Notably, the quantity $2\mathrm{Im}{A}_w\left(\epsilon \right)$ [Eq. (73a)] is the rate of change of the postselection probability (41) along this vector field for all $\epsilon$, but not along the purely decohering trajectory that ${\widehat{\mathit{\rho }}}_i\left(\epsilon \right)$ actually follows.