# Information and disturbance in operational probabilistic theories

QUIT group, Physics Dept., Pavia University, and INFN Sezione di Pavia, via Bassi 6, 27100 Pavia, Italy

### Abstract

Any measurement is intended to provide $information$ on a system, namely knowledge about its state. However, we learn from quantum theory that it is generally impossible to extract information without disturbing the state of the system or its correlations with other systems. In this paper we address the issue of the interplay between information and disturbance for a general operational probabilistic theory. The traditional notion of disturbance considers the fate of the system state after the measurement. However, the fact that the system state is left untouched ensures that also correlations are preserved only in the presence of local discriminability. Here we provide the definition of disturbance that is appropriate for a general theory. Moreover, since in a theory without causality information can be gathered also on the effect, we generalise the notion of no-information test. We then prove an equivalent condition for no-information without disturbance---$\textit{atomicity of the identity}$---namely the impossibility of achieving the trivial evolution---the $identity$---as the coarse-graining of a set of non trivial ones. We prove a general theorem showing that information that can be retrieved without disturbance corresponds to perfectly repeatable and discriminating tests. Based on this, we prove a structure theorem for operational probabilistic theories, showing that the set of states of any system decomposes as a direct sum of perfectly discriminable sets, and such decomposition is preserved under system composition. As a consequence, a theory is such that any information can be extracted without disturbance only if all its systems are classical. Finally, we show via concrete examples that no-information without disturbance is independent of both local discriminability and purification.

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### Cited by

[1] John van de Wetering, "From diagrams to quantum theory", Quantum Views 5, 54 (2021).

[2] Giacomo Mauro D’Ariano, Paolo Perinotti, and Alessandro Tosini, "Incompatibility of observables, channels and instruments in information theories", Journal of Physics A: Mathematical and Theoretical 55 39, 394006 (2022).

[3] Paolo Perinotti, "Causal influence in operational probabilistic theories", Quantum 5, 515 (2021).

[4] Paolo Perinotti, Alessandro Tosini, and Leonardo Vaglini, "Shannon theory beyond quantum: Information content of a source", Physical Review A 105 5, 052222 (2022).

[5] Giacomo Mauro D'Ariano, Marco Erba, and Paolo Perinotti, "Classicality without local discriminability: Decoupling entanglement and complementarity", Physical Review A 102 5, 052216 (2020).

[6] Giacomo Mauro D'Ariano, Marco Erba, and Paolo Perinotti, "Classical theories with entanglement", Physical Review A 101 4, 042118 (2020).

[7] Lorenzo Giannelli, "Bit Commitment in Operational Probabilistic Theories", arXiv:2101.09171.

[8] Kartik Patekar and Holger F. Hofmann, "The role of system-meter entanglement in controlling the resolution and decoherence of quantum measurements", arXiv:1905.09978.

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