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Measurement theory in local quantum physics
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80. One may consider that we can prepare states only in bounded local regions, i.e., (normal) states on for some . This intuition is realized as the concept of local state.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 ℝ4. 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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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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