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---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.
Highlights
The possibility that gathering information on a physical system may affect the state of the system itself was introduced by Heisenberg in his famous gedanken experiment [1], which became the first paradigm of quantum mechanics
For some probabilistic theories which can be reframed as operational probabilistic theories (OPTs), the definitions of information and disturbance have been investigated in the presence of local discriminability, purifica
In the absence of local discriminability, it is still possible to have other theories where all the information can be extracted without disturbance. This has been proved in Ref. [35] where the authors describe an OPT whose systems of any dimension are classical, but with a parallel composition that differs from the usual classical one, leading to a violation of local discriminability, more precisely to a 2-local theory according to Definition 10
Summary
We review the framework of operational probabilistic theories (OPT) (we refer to [10, 9, 11] for further details). The equivalence classes of preparation events and observation events of A will be denoted by the same symbols as their elements |ρ)A and An operational probabilistic theory is defined as a collection of systems and transformations with the above rules for parallel and sequential composition and with a probability associated to any closed circuit 2. Given a system A, in the dilations sets DSt(A) and DEff(A), there could be states and effects with the following property. Among the properties discussed in the following there is causality, which induces an asymmetry in the structure of states and effects of the theory As it happens in both classical and quantum theory, causality forces the existence of a unique deterministic effect, while the set of states typically presents several deterministic elements in the presence of causality
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