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Construction of the least informative observable conserved by a given quantum instrument

Yui Kuramochi

Published Sep 28, 2015• 3 citations
OpenAlex

Abstract

For a quantum measurement process described by a quantum instrument $\mathcal{I}$ and a system observable corresponding to a positive-operator valued measure (POVM) $E ,$ $\mathcal{I}$ is said to conserve the information of $E$ if the joint successive measurement of $\mathcal{I}$ followed by $E$ is equivalent to a single measurement of $E .$ We show that for any quantum instrument $\mathcal{I}$ we can construct a POVM conserved by $\mathcal{I}$. Intuitively the construction gives the infinite joint successive measurement of $\mathcal{I}.$ We also show that the constructed POVM is the least informative observable among POVMs conserved by $\mathcal{I}$, i.e. the constructed POVM can be realized by a classical post-processing of any POVM conserved by $\mathcal{I} .$ As typical examples of quantum instruments, we explicitly evaluate POVMs of infinite successive measurements for photon counting and quantum counter instruments.

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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.
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.
Studying sequential measurements is of the utmost importance to both the foundational aspects of quantum theory and the practical implementations of quantum technologies, with both of these applications being abstractly described by the concatenation of quantum instruments into a sequence of a certain length. In general, the choice of instrument at any given step in the sequence can be conditionally chosen based on the classical results of all preceding instruments. For two instruments in a sequence we consider the conditional second instrument as an effective way of postprocessing the first instrument into a new one. This is similar to how a measurement described by a positive operator-valued measure (POVM) can be postprocessed into another by way of classical randomization of its outcomes using a stochastic matrix. In this work we study the postprocessing relation of instruments and the partial order it induces on their equivalence classes. We characterize the greatest and the least element of this order, give examples of postprocessings between different types of instruments, and draw connections between postprocessings of some of these instruments and their induced POVMs.
For any quantum observable (positive operator valued measure (POVM)), we show that its compatible quantum instruments can be identified with certain channels and can be seen as combinations of Lüders operations and channels. We prove that a POVM is rank-1 exactly when its compatible instruments are nuclear and their associate channels are entanglement-breaking. Any instrument can be maximally refined into a rank-1 instrument. We present a characterization for (minimal) pure realizations of instruments and characterize completely the minimal pure measurement models of POVMs. The standard model of quantum measurement theory is generalized for arbitrary observables. Finally, we study post measurement states.
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,
Various forms of optimality for quantum observables described as normalized positive-operator-valued measures (POVMs) are studied in this paper. We give characterizations for observables that determine the values of the measured quantity with probabilistic certainty or a state of the system before or after the measurement. We investigate observables that are free from noise caused by classical post-processing, mixing, or pre-processing of quantum nature. Especially, a complete characterization of pre-processing and post-processing clean observables is given, and necessary and sufficient conditions are imposed on informationally complete POVMs within the set of pure states. We also discuss joint and sequential measurements of optimal quantum observables.