In this paper, we aim to establish foundations of measurement theory in local quantum physics. For this purpose, we discuss a representation theory of completely positive (CP) instruments on arbitrary von Neumann algebras. We introduce a condition called the normal extension property (NEP) and establish a one-to-one correspondence between CP instruments with the NEP and statistical equivalence classes of measuring processes. We show that every CP instrument on an atomic von Neumann algebra has the NEP, extending the well-known result for type I factors. Moreover, we show that every CP instrument on an injective von Neumann algebra is approximated by CP instruments with the NEP. The concept of posterior states is also discussed to show that the NEP is equivalent to the existence of a strongly measurable family of posterior states for every normal state. Two examples of CP instruments without the NEP are obtained from this result. It is thus concluded that in local quantum physics not every CP instrument represents a measuring process, but in most of physically relevant cases every CP instrument can be realized by a measuring process within arbitrary error limits, as every approximately finite dimensional von Neumann algebra on a separable Hilbert space is injective. To conclude the paper, the concept of local measurement in algebraic quantum field theory is examined in our framework. In the setting of the Doplicher-Haag-Roberts and Doplicher-Roberts theory describing local excitations, we show that an instrument on a local algebra can be extended to a local instrument on the global algebra if and only if it is a CP instrument with the NEP, provided that the split property holds for the net of local algebras.

To improve the spectral efficiency and increase network capacity, wavelength signals must be multiplexed densely, i.e. ultra-dense wavelength-division-multiplexing (WDM) networks. However, the spectrum of each WDM signal is narrowed by the wavelength-selective switches placed in optical nodes, so the transmissible distance and node hop count of the signal are strictly limited. To counter this spectrum narrowing problem caused by optical node traversal in ultra-dense WDM networks, pre-filtering techniques have been proposed. This paper comprehensively investigates the performance of four pre-filtering techniques: conventional Nyquist filtering, pre-emphasis filtering, partial-response filtering, and the combination of pre-emphasis and partial-response filtering. We conduct extensive computer simulations and discuss the optimality of pre-filtering techniques. The simulation results show that the adaptive use of pre-filtering can substantially extend the maximum attainable transmission distance and hop counts of 4/16/64-QAM signals.

In this study, we develop quantum measurement theory for quantum systems described by C${}^{\ast }$-algebras. This is the first step to establish measurement theory for interacting quantum fields with off-shell momenta. Unlike quantum mechanics (i.e., quantum systems with finite degrees of freedom), measurement theory for quantum fields is still in development because of the difficulty of quantum fields that are typical quantum systems with infinite degrees of freedom. Furthermore, the mathematical theory of quantum measurement is formulated in the von Neumann algebraic setting in previous studies. In the paper, we aim to extend the applicable area of quantum measurement theory to quantum systems described by C${}^{\ast }$-algebras from a mathematical viewpoint, referring to the sector theory that is related to symmetry and based on the theory of integral decomposition of states. In particular, we define central subspaces of the dual space of a C${}^{\ast }$-algebra and use them to define instruments. This attempt makes the connection between measurement theory and sector theory explicit and enables us to understand the macroscopic nature and the physical meaning of measurement.

In this paper, we attempt to establish quantum measurement theory in the Heisenberg picture. First, we review foundations of quantum measurement theory, that is usually based on the Schr\"{o}dinger picture. The concept of instrument is introduced there. Next, we define the concept of system of measurement correlations and that of measuring process. The former is the exact counterpart of instrument in the (generalized) Heisenberg picture. In quantum mechanical systems, we then show a one-to-one correspondence between systems of measurement correlations and measuring processes up to complete equivalence. This is nothing but a unitary dilation theorem of systems of measurement correlations. Furthermore, from the viewpoint of the statistical approach to quantum measurement theory, we focus on the extendability of instruments to systems of measurement correlations. It is shown that all completely positive (CP) instruments are extended into systems of measurement correlations. Lastly, we study the approximate realizability of CP instruments by measuring processes within arbitrarily given error limits.