# Asymptotic measurement schemes for every observable of a quantum field theory

March 17, 2022

In quantum measurement theory, a measurement scheme describes how an
observable of a given system can be measured indirectly using a probe. The
measurement scheme involves the specification of a probe theory, an initial
probe state, a probe observable and a coupling between the system and the
probe, so that a measurement of the probe observable after the coupling has
ceased reproduces (in expectation) the result of measuring the system
observable in the system state. Recent work has shown how local and causal
measurement schemes may be described in the context of model-independent
quantum field theory (QFT), but has not addressed the question of whether such
measurement schemes exist for all system observables. Here, we present two
treatments of this question. The first is a proof of principle which provides a
measurement scheme for every local observable of the quantized real linear
scalar field if one relaxes one of the conditions on a QFT measurement scheme
by allowing a non-compact coupling region. Secondly, restricting to compact
coupling regions, we explicitly construct asymptotic measurement schemes for
every local observable of the quantized theory. More precisely, we show that
for every local system observable $A$ there is an associated collection of
measurement schemes for system observables that converge to $A$. The
convergence holds in particular in GNS representations of suitable states on
the field algebra as well as the Weyl algebra. In this way, we show that every
observable can be asymptotically measured using locally coupled probe theories.

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