^{1,a)}and Masanao Ozawa

^{1,b)}

### Abstract

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.

K.O. would like to thank Professor Izumi Ojima and Dr. Hayato Saigo for their warm encouragement and useful comments. The authors thank the anonymous referee for useful comments. This work was supported by the John Templeton Foundations, No. 35771 and the JSPS KAKENHI, No. 26247016 and No. 15K13456.

I. INTRODUCTION II. PRELIMINARIES III. COMPLETELY POSITIVE INSTRUMENTS AND QUANTUM MEASURING PROCESSES IV. APPROXIMATIONS BY CP INSTRUMENTS WITH THE NEP V. EXISTENCE OF A FAMILY OF POSTERIOR STATES AND ITS CONSEQUENCES VI. DHR-DR THEORY AND LOCAL MEASUREMENT

^{∗}-inclusions,” Commun. Math. Phys. 110, 325–348 (1987).

^{56}A local state T on with an inclusion pair of regions is defined as a unital CP map T on satisfying the following conditions, where such that : (i) T(AB) = T(A) B for all and ; (ii) there exists such that T(A) = φ(A)1 for all . On the other hand, we need states on to describe the given physical system. By using local states, states on can be regarded as local states with in the limit of tending to the whole space ℝ

^{4}. The authors believe that the use of local states would be helpful for developing measurement theory in local quantum physics in the future.

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### Abstract

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.

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