No measurement is possible without any theory. With respect to a theory under consideration one has to distinguish between direct and indirect measurements. Direct measurements are defined by pre-theories, i.e. without the theory under consideration. The direct measurements for quantum theory are given by measurable processes on macroscopic devices. It is demonstrated how indirect measurements in quantum mechanics can be reduced to direct measurements. A theory of the measuring device is in this context not possible by quantum mechanics alone, but by a more comprehensive theory of macrosystems.No measurement is possible without any theory. With respect to a theory under consideration one has to distinguish between direct and indirect measurements. Direct measurements are defined by pre-theories, i.e. without the theory under consideration. The direct measurements for quantum theory are given by measurable processes on macroscopic devices. It is demonstrated how indirect measurements in quantum mechanics can be reduced to direct measurements. A theory of the measuring device is in this context not possible by quantum mechanics alone, but by a more comprehensive theory of macrosystems.

MetrologiaJan 1, 19869 citations

Does a classical probability space for two-dimensional quantum measurement theory exist?

Koji Nagata, Tadao Nakamura

Recently, a new measurement theory based on the truth values is proposed \cite{NN1}. The results of measurements are either 0 or 1. The measurement theory accepts a hidden variables model for a single Pauli observable. Therefore we can introduce a classical probability space for the measurement theory in this case. On the other hand, we discuss the fact that the projective measurement theory (the results of measurements are either $+1$ or $-1$ ) does not meet a hidden variables model for a single Pauli observable. Hence we cannot introduce a classical probability space for the projective measurement theory in this case. Our discussion provides new insight to formulate quantum measurement theory, by using the measurement theory based on the truth values.Recently, a new measurement theory based on the truth values is proposed \cite{NN1}. The results of measurements are either 0 or 1. The measurement theory accepts a hidden variables model for a single Pauli observable. Therefore we can introduce a classical probability space for the measurement theory in this case. On the other hand, we discuss the fact that the projective measurement theory (the results of measurements are either $+1$ or $-1$) does not meet a hidden variables model for a single Pauli observable. Hence we cannot introduce a classical probability space for the projective measurement theory in this case. Our discussion provides new insight to formulate quantum measurement theory, by using the measurement theory based on the truth values.

Jun 1, 20160 citations

Theory of Measurements

Wolfgang Ernst Friederich Pauli

Springer eBooksJan 1, 198022 citations

Measurement simulability and incompatibility in quantum theory and other operational theories

Leevi Leppäjärvi

In this thesis, we consider the properties of measurements in quantum theory and other operational theories. After having introduced the framework of operational theories, we consider a communication scheme based on an experimental prepare-and-measure scenario and demonstrate this with different communication tasks. This gives us context for how the different communication tasks can be implemented in different theories, in doing so establishing quantum theory intuitively as an operational theory among other theories. The main property of measurements we focus on in this work is the simulation of measurements, which consists of manipulating the inputs and outputs of the measurement devices. We study how using this process on existing measurement devices can be used to operationally imitate new devices, and what kind of structure the simulation process induces on measurements. We also consider applications of simulability. Firstly, we consider operational restrictions imposed upon measurements. We argue that the restricted set of physical measurements must be closed with respect to the simulation process since the simulation of physical devices must lead to other physically feasible devices. We demonstrate different types of restrictions by classifying them. As a second application we see how the simulation of measurements relates to compatibility of measurements and how it can be viewed as a generalisation of it. This allows us to present an operational principle previously known to quantum theory, the no-free-information principle, according to which any measurement that is compatible with all other measurement must not provide any useful, and therefore free, information about the system. Whilst this principle holds in quantum theory, there are non-classical theories for which it is violated, and so enforcing this principle may be considered a way to exclude some unphysical theories.In this thesis, we consider the properties of measurements in quantum theory and other operational theories. After having introduced the framework of operational theories, we consider a communication scheme based on an experimental prepare-and-measure scenario and demonstrate this with different communication tasks. This gives us context for how the different communication tasks can be implemented in different theories, in doing so establishing quantum theory intuitively as an operational theory among other theories. The main property of measurements we focus on in this work is the simulation of measurements, which consists of manipulating the inputs and outputs of the measurement devices. We study how using this process on existing measurement devices can be used to operationally imitate new devices, and what kind of structure the simulation process induces on measurements. We also consider applications of simulability. Firstly, we consider operational restrictions imposed upon measurements. We argue that the restricted set of physical measurements must be closed with respect to the simulation process since the simulation of physical devices must lead to other physically feasible devices. We demonstrate different types of restrictions by classifying them. As a second application we see how the simulation of measurements relates to compatibility of measurements and how it can be viewed as a generalisation of it. This allows us to present an operational principle previously known to quantum theory, the no-free-information principle, according to which any measurement that is compatible with all other measurement must not provide any useful, and therefore free, information about the system. Whilst this principle holds in quantum theory, there are non-classical theories for which it is violated, and so enforcing this principle may be considered a way to exclude some unphysical theories.

arXiv:2106.03588Jun 7, 20211 citation

Why measurements are made of effects

Tobias Fritz

Both in quantum theory and in general probabilistic theories, measurements with $n$ outcomes are modelled as $n$ -tuples of \emph{effects} summing up to the unit effect. Why is this the case, and can this assumption be meaningfully relaxed? Here we develop \emph{generalized measurement theories (GMTs)} as a mathematical framework for physical theories that is complementary to general probabilistic theories, and where this kind of question can be made precise and answered. We then give a definition of \emph{probabilistic state} on a GMT, prove that measurements are made of effects in every GMT in which the probabilistic states separate the measurements, and also argue that this separation condition is physically well-motivated. Finally, we also discuss when a GMT should be considered classical and characterize GMTs corresponding to Boolean algebras as those that are strongly classical and projective.Both in quantum theory and in general probabilistic theories, measurements with $n$ outcomes are modelled as $n$-tuples of \emph{effects} summing up to the unit effect. Why is this the case, and can this assumption be meaningfully relaxed? Here we develop \emph{generalized measurement theories (GMTs)} as a mathematical framework for physical theories that is complementary to general probabilistic theories, and where this kind of question can be made precise and answered. We then give a definition of \emph{probabilistic state} on a GMT, prove that measurements are made of effects in every GMT in which the probabilistic states separate the measurements, and also argue that this separation condition is physically well-motivated. Finally, we also discuss when a GMT should be considered classical and characterize GMTs corresponding to Boolean algebras as those that are strongly classical and projective.

quant-pharXiv:2602.18898Feb 24, 20260 citations

+1

-1

) does not meet a hidden variables model for a single Pauli observable. Hence we cannot introduce a classical probability space for the projective measurement theory in this case. Our discussion provides new insight to formulate quantum measurement theory, by using the measurement theory based on the truth values.

Both in quantum theory and in general probabilistic theories, measurements with

n

outcomes are modelled as

n

-tuples of \emph{effects} summing up to the unit effect. Why is this the case, and can this assumption be meaningfully relaxed? Here we develop \emph{generalized measurement theories (GMTs)} as a mathematical framework for physical theories that is complementary to general probabilistic theories, and where this kind of question can be made precise and answered. We then give a definition of \emph{probabilistic state} on a GMT, prove that measurements are made of effects in every GMT in which the probabilistic states separate the measurements, and also argue that this separation condition is physically well-motivated. Finally, we also discuss when a GMT should be considered classical and characterize GMTs corresponding to Boolean algebras as those that are strongly classical and projective.