Information-Theoretic Analysis of Weak Measurements and Their Reversal

Abstract

We study trade-off relations in information extraction from quantum systems subject to null-result weak measurements, where the absence of a detected photon continuously updates the system state. We present a detailed analysis of qubit and qutrit systems and investigate a general framework for a multilevel quantum system. We develop a dynamical characterization of null-result weak measurements that quantifies the information extracted over time, revealing the amount of the obtained information and also the rate of the information accumulation. The characterizations are obtained by examining the time-dependent evolution of the information theoretic quantities. More specifically, we consider Shannon entropy, mutual information, fidelity, and relative entropy to characterize the weak measurement dynamics. Our results provide an information theoretic analysis of the weak measurement process and highlight the dynamical nature of information extraction and reversibility in the weak measurement processes.

Figures (4)

• Figure 1: Information-theoretic quantities as functions of $2\gamma t$ for four different prior distributions for the state $\ket{\psi}=c_0\ket{0}+c_1\ket{1}$. Each panels correspond to different priors: a) $p(x_n) = [1/2,\,1/2]$, b) $p(x_n) = [0.6,\,0.4]$, c) $p(x_n) = [0.2,\,0.8]$, and d) $p(x_n) = [0.3,\,0.7]$.
• Figure 2: Information-theoretic quantities as functions of $2\gamma t$ for four different prior distributions for the state $\ket{\psi}=c_0\ket{0}+c_1\ket{1}+c_2\ket{2}$. Each panel corresponds to different priors: a) $p(x_n) = [1/3,\,1/3,\,1/3]$, b) $p(x_n) = [0.2,\,0.4,\,0.4]$, c) $p(x_n) = [0.5,\,0.3,\,0.2]$, and d) $p(x_n) = [0.2,\,0.2,\,0.6]$.
• Figure 3: Instantaneous rates of change of information gain, fidelity, and reversal probability for a qubit under null-result monitoring as functions of $2\gamma t$. Each panel corresponds to a different prior distribution: a) $p(x_n) = [1/2,\,1/2]$, b) $p(x_n) = [0.6,\,0.4]$, c) $p(x_n) = [0.2,\,0.8]$, and d) $p(x_n) = [0.3,\,0.7]$.
• Figure 4: Instantaneous rates of change of information gain, fidelity, and reversal probability for a qutrit under null-result monitoring as functions of $2\gamma t$. Each panel corresponds to a different prior distribution: a) $p(x_n) = [1/3,\,1/3,\,1/3]$, b) $p(x_n) = [0.2,\,0.4,\,0.4]$, c) $p(x_n) = [0.5,\,0.3,\,0.2]$, and d) $p(x_n) = [0.2,\,0.2,\,0.6]$.