Contextual-value approach to the generalized measurement of observables

We present a detailed motivation for and definition of the contextual values of an observable, which were introduced by Dressel et al. [Phys. Rev. Lett. 104, 240401 (2010)]. The theory of contextual values is a principled approach to the generalized measurement of observables. It extends the well-established theory of generalized state measurements by bridging the gap between partial state collapse and the observables that represent physically relevant information about the system. To emphasize the general utility of the concept, we first construct the full theory of contextual values within an operational formulation of classical probability theory, paying special attention to observable construction, detector coupling, generalized measurement, and measurement disturbance. We then extend the results to quantum probability theory built as a superstructure on the classical theory, pointing out both the classical correspondences to and the full quantum generalizations of both Lüder's rule and the Aharonov-Bergmann-Lebowitz rule in the process. As such, our treatment doubles as a self-contained pedagogical introduction to the essential components of the operational formulations for both classical and quantum probability theory. We find in both cases that the contextual values of a system observable form a generalized spectrum that is associated with the independent outcomes of a partially correlated and generally ambiguous detector; the eigenvalues are a special case when the detector is perfectly correlated and unambiguous. To illustrate the approach, we apply the technique to both a classical example of marble color detection and a quantum example of polarization detection. For the quantum example we detail two devices: Fresnel reflection from a glass coverslip, and continuous beam displacement from a calcite crystal. We also analyze the three-box paradox to demonstrate that no negative probabilities are necessary in its analysis. Finally, we provide a derivation of the quantum weak value as a limit point of a pre- and postselected conditioned average and provide sufficient conditions for the derivation to hold.

Figure 1

Diagram of the relationship between the sample space of atomic propositions $X$, the Boolean algebra of propositions ${\Sigma }_X$, and the algebra of observables ${\Sigma }_X^{\mathbb{R}}$. The probability state $P$ is a measure from ${\Sigma }_X$ to the interval $[0,1]$. The expectation functional $\langle ·\rangle$ is a linear extension of $P$ that maps ${\Sigma }_X^{\mathbb{R}}$ to the reals $\mathbb{R}$; by construction $\langle ·\rangle =P(·)$ whenever both are defined.

Figure 2

Coverslip polarization measurement. A laser beam passes through a preselection $x$ polarizer, a glass microscope coverslip, and a postselection $z$ polarizer. The transmission probabilities for each segment of the apparatus are shown. By assigning appropriate contextual values $f_Y\left(t\right)$ and $f_Y\left(r\right)$ (12) to the output ports of the coverslip, the polarization observable $F_X=f_X\left(h\right)h+f_X\left(v\right)v$ can be measured using the equivalent expansion in terms of the appropriate measurement context $F_X=f_Y\left(t\right){\mathcal{E}}_t\left(1_X\right)+f_Y\left(r\right){\mathcal{E}}_r\left(1_X\right)$. Averaging the same contextual values with pre- and postselected conditional probabilities ${}_z\langle \tilde{t}\rangle {}_x={\langle x{\mathcal{E}}_t\left(z\right)x\rangle }_X/[{\langle x{\mathcal{E}}_t\left(z\right)x\rangle }_X+{\langle x{\mathcal{E}}_r\left(z\right)x\rangle }_X]$ and ${}_z\langle \tilde{r}\rangle {}_x={\langle x{\mathcal{E}}_r\left(z\right)x\rangle }_X/[{\langle x{\mathcal{E}}_t\left(z\right)x\rangle }_X+{\langle x{\mathcal{E}}_r\left(z\right)x\rangle }_X]$ produces the conditioned average (127) ${}_z\langle \widetilde{F_X}\rangle {}_x=f_Y{\left(t\right)}_z\langle \tilde{t}\rangle {}_x+f_Y{\left(r\right)}_z\langle \tilde{r}\rangle {}_x$.

Figure 3

Preferred CVs $f_Y\left(y\right)$ given in Eq. (132c) for a calcite position measurement that targets the polarization observable $F_X=h-v$, shown for strong separation ($\epsilon =1$), wimpy separation ($\epsilon =0.1$), and weak separation ($\epsilon =0.02$) of the polarizations. (top row) Initial Gaussian beam profile. (middle row) Initial Laplace beam profile. (bottom row) Initial top-hat beam profile. Note that the top-hat CVs are the eigenvalues of $\pm 1$ under strong separation, but become amplified as the distributions start to overlap; moreover, the top-hat CVs cancel out in the perfectly ambiguous overlapping region. The amplification and cancellation behavior of the CVs is more complicated for less-definite detector profiles.

Figure 4

Pre- and postselected detector probability densities ${}_z\tilde{p}{}_x\left(y\right)$ for the calcite position measurement (131), shown for strong separation ($\epsilon =1$), wimpy separation ($\epsilon =0.1$), and weak separation ($\epsilon =0.02$) of the polarizations. The preselection is $x=|x\rangle \langle x|$ with associated vector $|x\rangle =\cos (4\pi /6\left)\right|h\rangle +\sin (4\pi /6)|v\rangle$. The postselection is $z=|z\rangle \langle z|$ with associated vector $|z\rangle =\left(\right|h\rangle +|v\rangle )/\sqrt{2}$. (top row) Initial Gaussian beam profile. (middle row) Initial Laplace beam profile. (bottom row) Initial top-hat beam profile. Note that the Gaussian profile tilts to approximate a single-shifted Gaussian under weak separation, as leveraged in the weak-measurement protocol introduced in Ref. [17].

Figure 5

Pre- and postselected conditioned average densities $f_Y\left(y\right){}_z\tilde{p}{}_x\left(y\right)$ for a calcite position measurement targeting the observable $F_X=h-v$ with CVs as in Fig. 3, shown for strong separation ($\epsilon =1$), wimpy separation ($\epsilon =0.1$), and weak separation ($\epsilon =0.02$) of the polarizations. The conditioned averages ${}_z\langle \widetilde{F_X}\rangle {}_x={\int }_Y{f}_Y\left(y\right){}_z\tilde{p}{}_x\left(y\right)dy$ are the areas under the curves and are shown inset. As in Fig. 4, the preselection is $x=|x\rangle \langle x|$, where $|x\rangle =\cos (4\pi /6\left)\right|h\rangle +\sin (4\pi /6)|v\rangle$. The postselection is $z=|z\rangle \langle z|$, where $|z\rangle =\left(\right|h\rangle +|v\rangle )/\sqrt{2}$. (top row) Initial Gaussian beam profile. (middle row) Initial Laplace beam profile. (bottom row) Initial top-hat beam profile. For sufficiently strong separation, all three detector profiles will produce the strong-conditioned average ${}_z\langle \widetilde{F_X}\rangle {}_x=-1/2$. For weak separation, all three profiles approximate the weak value ${}_z\langle \widetilde{F_X}\rangle {}_x^w=-2-\sqrt{3}\approx -3.73$. However, the different detector profiles converge to the weak value at different rates with decreasing $\epsilon$.