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arXiv:1406.2130 · Physical Review A

Classicality condition on a system observable in a quantum measurement and a relative-entropy conservation law

Yui Kuramochi, Masahito Ueda

Published Mar 16, 2015• 5 citations
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Abstract

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.

DOI
10.1103/physreva.91.032110

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We consider the information flow on a system observable XXX corresponding to a positive-operator-valued measure under a quantum measurement process YYY 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 XXX equals the relative entropy of the measurement outcome of YYY, 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 YYY followed by XXX such that the probability distribution of the statistic coincides with that of a single measurement of XXX for the premeasurement state. We show that in the case when XXX is a discrete projection-valued measure and YYY 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,
For a quantum measurement process described by a quantum instrument I and a system observable corresponding to a positive-operator valued measure (POVM) E, I is said to conserve the information of E if the joint successive measurement of I followed by E is equivalent to a single measurement of E. We show that for any quantum instrument I we can construct a POVM conserved by I. Intuitively, the construction gives the infinite joint successive measurement of I. We also show that the constructed POVM is the least informative observable among POVMs conserved by I, i.e., the constructed POVM can be realized by a classical post-processing of any POVM conserved by I. As typical examples of quantum instruments, we explicitly evaluate POVMs of infinite successive measurements for photon counting and quantum counter instruments.
For a quantum measurement process described by a quantum instrument I\mathcal{I}I and a system observable corresponding to a positive-operator valued measure (POVM) E,E ,E, I\mathcal{I}I is said to conserve the information of EEE if the joint successive measurement of I\mathcal{I}I followed by EEE is equivalent to a single measurement of E.E .E. We show that for any quantum instrument I\mathcal{I}I we can construct a POVM conserved by I\mathcal{I}I. Intuitively the construction gives the infinite joint successive measurement of I.\mathcal{I}.I. We also show that the constructed POVM is the least informative observable among POVMs conserved by I\mathcal{I}I, i.e. the constructed POVM can be realized by a classical post-processing of any POVM conserved by I.\mathcal{I} .I. As typical examples of quantum instruments, we explicitly evaluate POVMs of infinite successive measurements for photon counting and quantum counter instruments.
The concept of entropy rst arose in the study of thermodynamics and statistical me- chanics. In information theory the so-called Shannon entropy plays a signi cant role, which measures how much information we gain, on average, when we learn the value of a random variable. An alternative interpretation is that the entropy of the random variable quanti es the amount of uncertainty about the random variable before we learn its value. The concept was introduced by Claude E. Shannon in his 1948 paper "A Mathematical Theory of Communication". In the rst part of the present work we consider this fundamental measure of information and show some of its properties, such as non-negativity. The relative entropy or the Kullback-Leibler divergence is a very useful measure of the distance between two prob- ability distributions. After introducing its de nition and other notions (joint entropy, conditional entropy or mutual information) we can prove additional assertions about the Shannon entropy and present statements about the relations between the above men- tioned quantities. In quantum information theory, there is a quantity, namely von Neumann entropy, which corresponds to Shannon entropy and originates from it. In the section Von Neu- mann entropy we will introduce this quantity which is a continuous functional on the states, or on their mathematical representatives, the density operators, and it ful ls in- equalities such as subadditivity and strong subaddititvity. Just as with the Shannon entropy, it is extremely useful to de ne a quantum version of the relative entropy, which is one of the most important numerical quantities appearing in quantum information theory. It is used as a measure of distinguishability between quantum states. Among others its properties are jointly convexity, monotonicity which are speci cally convenient. Transformations on structures which preserve operations, quantities or relations among elements appear in many areas of mathematics. These are the so-called preservers which show up also in physics and even in chemistry where they are usually called symmetries. Therefore, the study of preserver transformations is an important area of research. The third section of this work is based on a paper, in which Lajos Molnár described the structure every bijective map on the space of all density operators on a given Hilbert space, which preserves the relative entropy. This result is closely related to Wigner's famous theorem on the form of quantum mechanical symmetry transformations. The above theorem of Molnár can be extended into two directions. On the one hand, the result holds also in the case when the transformation under consideration acts on the space of all positive semide nite operators not only on density operators (i.e., we omit the condition of unit trace). On the other hand the statement remains valid even if we omit the strongly restrictive condition that the transformation is bijective. The second result is a joint work with Prof. Lajos Molnár which appeared in the journal Linear Algebra and Its Applications.
This study proposes a time-symmetric framework for quantum measurement that restores microscopic reversibility at the level of the dynamical description while remaining compatible with causality and thermodynamic consistency. Instead of invoking a stochastic wavefunction collapse, the measurement process is modeled as a bidirectional informational update between a forward-evolving state and a backward-propagating effect, governed by a completely positive generator and its adjoint. Within this operator-based formalism, pre- and post-selected statistics are treated on an equal footing, yielding a unified description of both. The proposed scheme rigorously preserves complete positivity, normalization, and the no-signalling principle, and it is shown to satisfy Spohn's inequality for the associated quantum Markov semigroup, thereby ensuring non-negative entropy production within this setting. The framework admits a direct experimental interpretation across a range of scenarios, including weak measurements, EPR-Bell tests, homodyne detection, and photon counting. Furthermore, in the classical limit, the bidirectional update is demonstrated to reduce to the well-established Kalman filter and Rauch-Tung-Striebel (RTS) smoother used in classical estimation theory. These results support the view that the apparent temporal asymmetry of quantum measurement arises not from fundamental dynamical irreversibility, but from informational conditioning, specifically, the one-sided way in which measurement outcomes are incorporated into our description. In this sense, the arrow of time in measurement theory may be understood as an arrow of information.
Abstract We investigate four partial orderings on the space of quantum measurements (i.e. on POVMs or positive operator valued measures), describing four notions of coarse/fine-ness of measurement. These are the partial orderings induced by: (1) classical post-processing, (2) measured relative entropy, (3) observational entropy, and (4) linear relation of POVMs. The orderings form a hierarchy of implication, where e.g. post-processing relation implies all the others. We show that this hierarchy is strict for general POVMs, with examples showing that all four orderings are strictly inequivalent. Restricted to projective measurements, all are equivalent. Finally we show that observational entropy equality S M = S N (for all ρ) holds if and only if POVMs M ≡ N are post-processing equivalent, which shows that the first three orderings induce identical equivalence classes.