Generalized modular-value-based scheme and its generalized modular value

We consider a generalized modular-value-based scheme based on the standard von Neumann measurement. We model the scheme as an interaction between a quantum system and a discrete quantum pointer where the pointer operator is a projection operator onto one of the states of the basis of the pointer Hilbert space. The interaction strength is made arbitrarily large. After post-selection onto the system, the results of the pointer measurement are the so-called conditional probabilities. We first explicitly derive the analytical expressions of the conditional probabilities, the expectation value, and the average displacement in the measured value of a pointer observable that we name as the pointer quantities. We also provide an expression for a generalized modular value and discuss the relationship between the generalized modular value and generalized weak values. The study then shows that the generalized modular value can characterize these pointer quantities. Then we give applications of our proposal to the cases of a spin-$s$ particle pointer and a semiclassical pointer state. One of the key results is that the amplification effect, similar to the weak-value case, is also observed in the case of the generalized modular value. Our study can also apply to the cases of nonclassical pointer states.

Figure 1

The probability displacements $\mathrm{\Delta }p(\mu =s|{\widehat{\rho }}_f)$ and $\mathrm{\Delta }p(\mu \ne s|{\widehat{\rho }}_f)$ in Eq. (20) as functions of ${\left(A\right)}_{\mathrm{m}}^s$ for $\gamma =0.4$, 0.6, and 0.8, and $s=1$, i.e., spin-1 pointer. Inset: The modular value of the system observable ${\widehat{\sigma }}_z/2$ varies as a function of $\epsilon$, where the quantum system states are chosen as in Eqs. (21) and (22), also we fix the value of $g=\pi$.

Figure 2

Contour plot of the average displacement in the measured value of the pointer observable ${\widehat{S}}_{\mathrm{p}}^z$ of spin-2 ($s=2$) particle. The vertical line at ${\left(A\right)}_{\mathrm{m}}^s=1$ indicates the value zero, which means no “displacement”. The quantum system is chosen the same as in Fig. 1.

Figure 3

The SNRs are shown as functions of ${\left(A\right)}_{\mathrm{m}}^s$ and $\gamma$ for $s=\frac{1}{2},2,\frac{7}{2}$, and 5 as can be seen in each panel. The quantum system is chosen the same as in Fig. 1.

Figure 4

The conditional probability versus $n$ curves with different modular values ${\left(A\right)}_{\mathrm{m}}^n$ and ${\left|\alpha \right|}^2$. All curves show the increasing with ${\left(A\right)}_{\mathrm{m}}^n$ for each ${\left|\alpha \right|}^2$. This phenomenon can be regarded as the amplification effect of the modular value. The quantum system can be chosen the same as in Fig. 1, i.e., interactions between spin and photon.