Uncertainty Relations for Coarse-Grained Measurements: An Overview

Abstract

Uncertainty relations involving incompatible observables are one of the cornerstones of quantum mechanics. Aside from their fundamental significance, they play an important role in practical applications, such as detection of quantum correlations and security requirements in quantum cryptography. In continuous variable systems, the spectra of the relevant observables form a continuum and this necessitates the coarse graining of measurements. However, these coarse-grained observables do not necessarily obey the same uncertainty relations as the original ones, a fact that can lead to false results when considering applications. That is, one cannot naively replace the original observables in the uncertainty relation for the coarse-grained observables and expect consistent results. As such, several uncertainty relations that are specifically designed for coarse-grained observables have been developed. In recognition of the 90th anniversary of the seminal Heisenberg uncertainty relation, celebrated last year, and all the subsequent work since then, here we give a review of the state of the art of coarse-grained uncertainty relations in continuous variable quantum systems, as well as their applications to fundamental quantum physics and quantum information tasks. Our review is meant to be balanced in its content, since both theoretical considerations and experimental perspectives are put on an equal footing.

Keywords: continuous variables; quantum foundations; quantum information; quantum uncertainty.

Figure 1

Multi-element detector array illustrating the standard coarse-graining geometry.

Figure 2

Coarse-grained distributions (blue bars) according to the standard model. The red solid line indicates the underlying continuous distribution used to generate the discretised versions. The used resolution $\Delta$ and positioning degree of freedom $u_{\mathrm{cen}}$ is indicated beside each distribution. For each resolution, two distinct distributions are shown, each of which associated with a different positioning of the coarse-graining bins.

Figure 3

Periodic coarse-graining design with $d=T_u/s_u=5$ detection outcomes. The parameter $T_u$ is the periodicity in which bins of size $s_u$ are arranged.

Figure 4

In panel (a) the full line is the graph of the function $f(\alpha )={\alpha }^{\frac{1}{2-2\alpha }}{\beta }^{\frac{1}{2-2\beta }}$, with $0<\alpha \le 1$, and where $\beta (\alpha )=\alpha /(2\alpha -1)$ that stems from the condition $1/\alpha +1/\beta =2$. The horizontal dashed line is drawn to indicate the limit ${lim}_{\alpha \to 1}f(\alpha )=1/e$. In panel (b) we plot the behaviour of $g(y)=(1/2)R_{00}(y,1)$ as a function of $y:=\Delta \delta /(4\hslash |\gamma |)$. Although $g(y)$ is shown in the range $0\le y\le 50$, it is important to note that $g(y)$ is continuous monotonically decreasing function in the positive real axis such ${lim}_{y\to \infty }g(y)=0$.

Figure 5

Numerical results testing entanglement criteria for the two-mode vacuum state, a separable pure state. The entanglement criteria are based on URs following the PPT argument outlined in Section 3. The criteria are evaluated as a function of the bin widths $\Delta =\delta$, which are given in units of the standard deviations ${\sigma }_{P_u}$ and ${\sigma }_{P_v}$. We note that ${\sigma }_{P_u}={\sigma }_{P_v}$ for the two-mode vacuum state. The red circles show the variance product UR Equation (8), where we apply the naive approach in which the variances of the continous variables are calculated from the discretized data using Equation (42). One can see that in this case we obtain a false-positive for entanglement when the coarse-graining widths are large. The blue squares show the coarse-grained variance product UR Equation (69), both applied to the global operators Equations (29) and (30). Here the lower bounds for both inequalities have been subtracted, so that a negative value indicates entanglement. The lines are merely guides for the eye.