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.
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.
We derive a tight bound between the quality of estimating a quantum state by measurement and the success probability of undoing the measurement in arbitrary dimensional systems, which completely describes the tradeoff relation between the information gain and reversibility. In this formulation, it is clearly shown that the information extracted from a weak measurement is erased through the reversing process. Our result broadens the information-theoretic perspective on quantum measurement as well as provides a standard tool to characterize weak measurements and reversals.
I compare the role of the information in classical and quantum dynamics by examining the relation between information flows in measurements and the ability of observers to reverse evolutions. I show that in the Newtonian dynamics reversibility is unaffected by the observer's retention of the information about the measurement outcome. By contrast-even though quantum dynamics is unitary, hence, reversible-reversing quantum evolution that led to a measurement becomes, in principle, impossible for an observer who keeps the record of its outcome. Thus, quantum irreversibility can result from the information gain rather than just its loss-rather than just an increase of the (von Neumann) entropy. Recording of the outcome of the measurement resets, in effect, initial conditions within the observer's (branch of) the Universe. Nevertheless, I also show that the observer's friend-an agent who knows what measurement was successfully carried out and can confirm that the observer knows the outcome but resists his curiosity and does not find out the result-can, in principle, undo the measurement. This relativity of quantum reversibility sheds new light on the origin of the arrow of time and elucidates the role of information in classical and quantum physics. Quantum discord appears as a natural measure of the extent to which dissemination of information about the outcome affects the ability to reverse the measurement.This article is part of a discussion meeting issue 'Foundations of quantum mechanics and their impact on contemporary society'.
We consider the information flow on a system observable
X corresponding to a positive-operator-valued measure under a quantum measurement process
Y described by a completely positive instrument from the viewpoint of the relative entropy. We establish a sufficient condition for the relative-entropy conservation law which states that the average decrease in the relative entropy of the system observable
X equals the relative entropy of the measurement outcome of
Y, i.e., the information gain due to measurement. This sufficient condition is interpreted as an assumption of classicality in the sense that there exists a sufficient statistic in a joint successive measurement of
Y followed by
X such that the probability distribution of the statistic coincides with that of a single measurement of
X for the premeasurement state. We show that in the case when
X is a discrete projection-valued measure and
Y is discrete, the classicality condition is equivalent to the relative-entropy conservation for arbitrary states. The general theory on the relative-entropy conservation is applied to typical quantum measurement models, namely, quantum nondemolition measurement, destructive sharp measurements on two-level systems, a photon counting, a quantum counting, homodyne and heterodyne measurements. These examples except for the nondemolition and photon-counting measurements do not satisfy the known Shannon-entropy conservation law proposed by Ban [M. Ban, J. Phys. A: Math. Gen. 32, 1643 (1999)], implying that our approach based on the relative entropy is applicable to a wider class of quantum measurements.
論文題目 Relative-entropy conservation law in quantum measurement and its applications to continuous measurements (量子測定における相対エントロピーの保存則と その連続測定への応用) 氏名 倉持 結In a quantum measurement process, there is a back-action on the system and the information about the measured observable is in general disturbed by the measurement back-action.Still in some quantum measurements such as a quantum non-demolition measurement or a photon-counting measurement, the information about the measured observable is conserved.Ban discussed these types of measurements quantitatively based on the Shannon entropy and the mutual information [1, 2, 3].In his formulation, the obtained information is quantified in terms of the mutual information between the measured observable X and the measurement outcome Y .He established a condition for the Shannon entropy conservation which states that the amount of the obtained information is equal to the decrease in the system's Shannon entropy of X due to the measurement.However, the Shannon entropy for a continuous variable cannot be interpreted as the proper information content in general since its value changes by a one-to-one transformation, which is just a relabelling and should not increase or decrease the information.Furthermore, the physical meaning of the established condition for the Shannon entropy conservation is not clear.The aim of this thesis is to study these information-conserving measurement processes based on the relative entropy [4] and establish the conservation relation for this kind of information content.This thesis consists of eight chapters and two appendices.Chapter 1 is the introduction.Chapters 2, 3 and 4 are review parts.Chapters 5, 6 and 7 contain the new results obtained by the present author.Chapter 8 summarizes this thesis.Chapter 2 reviews the quantum measurement theory and in this chapter we introduce the positive operator-valued measure (POVM) and the completely positive (CP) instrument.The POVM is a generalization of the projection-valued measure (PVM) associated with a measurement on a self-conjugate operator and it determines the statistics of the measurement outcome for a non-ideal quantum measurement.The CP instrument describes both the measurement outcome and the conditional state change due to the measurement back-action.In the introduction of these concepts,
We present the amount of information, fidelity, and reversibility obtained by arbitrary quantum measurements on completely unknown states. These quantities are expressed as functions of the singular values of a measurement operator corresponding to the obtained outcome. As an example, we consider a class of quantum measurements with highly degenerate singular values to discuss trade-offs among information, fidelity, and reversibility. The trade-offs are at the level of a single outcome, in the sense that the quantities pertain to each single outcome rather than the average over all possible outcomes.