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Journal of Physics A Mathematical and General

State reduction, information and entropy in quantum measurement processes

Masashi Ban

Published Jan 1, 1999• 10 citations
Publisher

Abstract

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.

DOI
10.1088/0305-4470/32/9/012

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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.
It is shown that the entropy reduction (the information gain in the initial terminology) of an efficient (ideal or pure) quantum measurement coincides with the generalized quantum mutual information of a quantum-classical channel mapping an a priori state to the corresponding posteriori probability distribution of the outcomes of the measurement. As a result the entropy reduction is defined for arbitrary a priori states (not only for states with finite von Neumann entropy) and its analytical properties is studied in detail by using general properties of the quantum mutual information. By using this approach it is shown that the entropy reduction of an efficient quantum measurement is a nonnegative lower semicontinuous concave function on the set of all a priori states having continuous restrictions to subsets on which the von Neumann entropy is continuous. Monotonicity and subadditivity of the entropy reduction are also easily proved by this method. A simple continuity condition for the entropy reduction and for the mean posteriori entropy considered as functions of a pair (a priori state, measurement) is obtained. A characterization of an irreducible measurement (in the Ozawa sense) which is not efficient is considered in the Appendix.
Quantum and classical informations are considered in quantum measurement processes described by pure operations. The quantum information is given by coherent information in the state change of the measured physical system, and the classical information represented by the Shannon mutual information is obtained from the measurement outcomes. It is shown that if the maximum classical information is obtained, the quantum information becomes zero and if any classical information is not obtained, all the quantum information that the physical system has can be transmitted.
A partially reversible operation is introduced in quantum measurement processes and the information-theoretical property is investigated. The dual map of a partially reversible operation becomes reversible with respect to the intrinsic observable of a physical system. It is shown that the amount of information about the intrinsic observable of the physical system, obtained in a quantum measurement process described by a partially reversible operation, is equal to the decrease of the entropy of the measured physical system.
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.
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,
A generalization of Shannon’s amount of information into quantum measurements of continuous observables is introduced. A necessary and sufficient condition for measuring processes to have a non-negative amount of information is obtained. This resolves Groenewold’s conjecture completely including the case of measurements of continuous observables. As an application the approximate position measuring process considered by von Neumann and later by Davies is shown to have a non-negative amount of information.