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,
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 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.
Information obtained by a quantum measurement process performed on a physical system and the entropy change of the measured physical system are considered in detail. It is shown that the condition for the amount of information obtained by the quantum measurement process to be represented by the Shannon mutual entropy is that the intrinsic observable of the measured physical system commutes with the operational observable defined by the quantum measurement process. When some measurement outcome is obtained, the decrease of the Shannon entropy of the measured system is compared with that of the von Neumann entropy. Furthermore, a condition is established under which the amount of information that can be established by the quantum measurement process becomes equal to the decrease of the Shannon entropy of the measured physical system.
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.
In this chapter, we discuss several information measures that are important for quantifying the amount of information and correlations that are present in quantum systems. The first fundamental measure that we introduce is the von Neumann entropy (or simply quantum entropy ). It is the quantum generalization of the Shannon entropy, but it captures both classical and quantum uncertainty in a quantum state.1 The quantum entropy gives meaning to the notion of an information qubit . This notion is different from that of the physical qubit, which is the description of a quantum state of an electron or a photon. The information qubit is the fundamental quantum informational unit of measure, determining how much quantum information is present in a quantum system. The initial definitions here are analogous to the classical definitions of entropy, but we soon discover a radical departure from the intuitive classical notions from the previous chapter: the conditional quantum entropy can be negative for certain quantum states. In the classical world, this negativity simply does not occur, but it takes on a special meaning in quantum information theory. Pure quantum states that are entangled have stronger-than-classical correlations and are examples of states that have negative conditional entropy. The negative of the conditional quantum entropy is so important in quantum information theory that we even have a special name for it: the coherent information. We discover that the coherent information obeys a quantum data-processing inequality, placing it on a firm footing as a particular informational measure of quantum correlations. We then define several other quantum information measures, such as quantum mutual information, that bear similar definitions as in the classical world, but with Shannon entropies replaced with quantum entropies. This replacement may seem to make quantum entropy somewhat trivial on the surface, but a simple calculation reveals that a maximally entangled state on two qubits registers two bits of quantum mutual information (recall that the largest the mutual information can be in the classical world is one bit for the case of two maximally correlated bits). We then discuss several entropy inequalities that play an important role in quantum information processing: the monotonicity of quantum relative entropy, strong subadditivity, the quantum data-processing inequalities, and continuity of quantum entropy.
Abstract Information is often considered classical in a definite state rather than in a superposition of states. It seems rather strange to consider information in superpositions. Some people would, on the basis of this argument, conclude that quantum information can never exist and we can only have access to classical information. It turns out, however, that quantum information can be quantified in the same way as classical information using Shannon's prescription. There is a unique measure (up to a constant additive or multiplicative term) of quantum information such that S (the von Neumann entropy) is purely a function of the probabilities of outcomes of measurements made on a quantum system (that is, a function of a density operator); S is a continuous function of probability; S is additive. This chapter discusses the fidelity of pure quantum states, Helstrom's discrimination, quantum data compression, entropy of observation, conditional entropy and mutual information, relative entropy, and statistical interpretation of relative entropy.
In this work, we use the theory of quantum states over time to define joint entropy for timelike-separated quantum systems. For timelike-separated systems that admit a dual description as being spacelike-separated, our notion of entropy recovers the usual von Neumann entropy for bipartite quantum states and thus may be viewed as a spacetime generalization of von Neumann entropy. Such an entropy is then used to define dynamical extensions of quantum joint entropy, quantum conditional entropy, and quantum mutual information for systems separated by the action of a quantum channel. We provide an in-depth mathematical analysis of such information measures and the properties they satisfy. We also use such a dynamical formulation of entropy to quantify the information loss/gain associated with the dynamical evolution of quantum systems, which enables us to formulate a precise notion of information conservation for quantum processes. Finally, we show how our dynamical entropy admits an operational interpretation in terms of quantifying the amount of state disturbance associated with a positive operator- valued measurement.