Abstract
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.
Highlights
Quantum theory implies three simple, yet significant and powerful theorems: the nobroadcasting theorem [1], the no-information-without-disturbance theorem [2], and the no-free-information theorem
The no-broadcasting theorem says that quantum states cannot be copied; the no-informationwithout-disturbance theorem states that a quantum observable that can be measured without any disturbance must be trivial, meaning that it does not give any information on the input state; and the no-free-information theorem states that a quantum observable that can be measured jointly with any other observable must be a trivial observable
In quantum theory these sets are seen to coincide, we show that in general only the inclusions T1 ⊆ T2 ⊆ T3 hold
Summary
Quantum theory implies three simple, yet significant and powerful theorems: the nobroadcasting theorem [1], the no-information-without-disturbance theorem [2], and the no-free-information theorem (which can be extracted e.g. from [3, Prop. 3.25]). GPTs constitute a wide class of theories that are based on operational notions such as states, measurements and transformations, where many of the key features of quantum theory, such as non-locality and incompatibility, can be formulated more generally. Including both quantum and classical theory as well as countless toy theories, GPTs allow us to compare these theories to each other based on their features and quantify their properties. The no-broadcasting principle is known to be valid in any non-classical general probabilistic theory [4, 5].
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