No-free-information principle in general probabilistic theories

Teiko Heinosaari1, Leevi Leppäjärvi1, and Martin Plávala2

1QTF Centre of Excellence, Department of Physics and Astronomy, University of Turku, Turku 20014, Finland
2Mathematical Institute, Slovak Academy of Sciences, Štefánikova 49, Bratislava, Slovakia

In quantum theory, the no-information-without-disturbance and no-free-information theorems express that those observables that do not disturb the measurement of another observable and those that can be measured jointly with any other observable must be trivial, i.e., coin tossing observables. We show that in the framework of general probabilistic theories these statements do not hold in general and continue to completely specify these two classes of observables. In this way, we obtain characterizations of the probabilistic theories where these statements hold. As a particular class of state spaces we consider the polygon state spaces, in which we demonstrate our results and show that while the no-information-without-disturbance principle always holds, the validity of the no-free-information principle depends on the parity of the number of vertices of the polygons.

We have introduced a new information-theoretic principle, the no-free-information principle, and compared it to the previously known no-broadcasting and no-information-without-disturbance principles. The no-broadcasting principle states that information cannot be copied, whilst the no-information-without-disturbance principle states that if we want to obtain any information about a system, we must disturb it, thereby changing the state it is in. The newly introduced no-free-information principle states that there is no type of information that can be learned about a system every time we perform any other measurement on it.

By noting that in quantum theory these principles hold, we formalized this problem in a more general class of theories, namely within the framework of general probabilistic theories (GPTs). These theories form a wide class of operational theories where many of the key features of quantum theory can be formulated more generally. By including both quantum and classical theory, as well as countless toy theories, the study of GPTs allows us to compare these theories based on their respective features and quantify their properties.

In the presented manuscript, we have mathematically characterized the principles in question within the GPT framework, tying our work with previously known results. We show that all three principles are strictly different, which means that there are operationally valid theories (even non-classical theories) where one can obtain non-trivial information about a system without disturbing it and where one can always choose to get some fixed non-trivial information when performing any other measurement. We have presented several examples of theories to demonstrate our claims.

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