Information conservation relations for weak measurement and its reversal

TL;DR

The paper addresses how information is allocated among measurement outcomes, the quantum system, and the environment in weak measurements of decaying multilevel systems under continuous monitoring. It develops exact conservation-type information relations that hold for each record outcome and extends these to weak-measurement reversal and to finite-count (k-click) photon-number detection. The approach hinges on conditional probabilities, posterior distributions, and relative entropy, yielding explicit formulas linking decay and reversal information and exposing a unified information-flow framework for monitored open quantum dynamics. These results have potential implications for quantum control, error mitigation, and the design of measurement-based strategies in noisy quantum technologies.

Abstract

We investigate the information distribution among different entities in the weak measurements protocol. Focusing on multilevel, decaying systems under continuous (no-click) monitoring, we derive exact, conservation-type information relations that hold for each outcome in the record. Analogous relations hold when an explicit reversal is applied, with the reversal success probability entering the relation. We extend the framework to finite-count outcomes (arbitrary photon numbers) obtaining quantitative trade-offs that link information change in the weak measurement process to the entities to which the information is distributed. These results provide a unified, outcome-resolved account of information flow in monitored open quantum dynamics and provide insight into a deeper understanding of open-system dynamics and its control.

Figures (2)

• Figure 1: Information-theoretic quantities as functions of $2\gamma t$ for four different prior distributions for the state $\ket{\psi}=c_0\ket{0}+c_1\ket{1}+c_2\ket{2}$. 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 2: Informatics elements in Eq. \ref{['k-click']} as functions of the rescaled time $2\gamma t$ for the state $\ket{\psi}=c_0\ket{0}+c_1\ket{1}+c_2\ket{2}+c_3\ket{3}$. Each panel corresponds to the uniform prior distribution $p=[1/4,\,1/4,\,1/4,\,1/4]$. The four panels correspond to detection outcomes (a) $k=0$, (b) $k=1$, (c) $k=2$, and (d) $k=3$.