Post-measurement states in generalized probabilistic theories

In summary, the framework of "convex theories" is still a developing area of research and there may not be a definitive answer to the question of assigning post-measurement states. Some approaches that have been proposed include considering the state as a probability distribution over ontic states or as a convex combination of pure states. It may also be helpful to reach out to the authors of related papers for further insights and discussions.
  • #1
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Hello everybody

I am currently trying to understand attempts to create a framework of generalized probabilistic theories in which quantum theory and classical theory appear as special cases. More precisely, I try to understand the framework which is sometimes called the framework of "convex theories", in which the set of states is considered to be a convex subset an ordered vector space. For those not familiar with this framework, have a look at e.g. the following papers:
  • H. Barnum: Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory, Arxiv preprint arXiv:0908.2354
  • J. Barret et al.: Limits on nonlocal correlations from the structure of the local state space, New Journal of Physics 13 (2011) 063024 (24pp)
  • O. Dahlsten et al.: Unifying typical entanglement and coin tossing: on randomization in probabilistic theories, Arxiv preprint arXiv:1107.6029

There is one thing I don't understand. So far, I have not found a paper in which it is explained how performing a measurement changes the state of a system. I'm trying to establish a rule for updating the state from the pre- to the post-measurement state. Let me tell you what I have come up with so far:

I think it is clear that one cannot give a rule for updating the state for every effect of a theory (see papers above to learn what it means to be an effect). If this were possible, then one could assign a post-measurement state to every element of a POVM in quantum mechanics. But in the case of a POVM which is not a projective measurement, this is not possible, because if one regards this POVM as being induced by a measurement on a larger system containing the system in question as a subsystem, then the post-measurement state might depend on the state of the ancilla system which is needed to extend the system in question to the larger system.
But (still talking about the quantum case) one can assign a post-measurement state to the system if the the POVM is a projective measurement. Noting that a POVM element is a projector if and only if it is an extremal point of the set of proper effects, one might hope that one can generalize the state-update-rule to the case of a generalized probabilistic theory.

One property which, I would say, is save to demand from a post-measurement state is the following one:
If one measures a state [itex]\rho[/itex] and gets the result corresponding to effect [itex]f[/itex], then the post-measurement state [itex]\rho_{post}[/itex] should satisfy [itex]f(\rho_{post}) = 1[/itex].
The problem is that in general, this condition does not specify [itex]\rho_{post}[/itex] uniquely. In the quantum case, the condition specifies [itex]\rho_{post}[/itex] uniquely if [itex]f[/itex] is induced by a one-dimensional projector, but it does not in the case where it is a 2- or higher-dimensional projector.

Now one might say the following (I hope this claim is correct; i haven't proved it so far): In the quantum case, [itex]f(\rho_{post})=1[/itex] specifies [itex]\rho_{post}[/itex] uniquely (i.e. [itex]f[/itex] is a one-dimensional projector) if and only if [itex]f[/itex] (in addition to being an extreme point of the set of proper effects) is ray extremal (i.e. lies on an extremal ray of [itex]V^*_+[/itex]). In the classical and quantum case, restriction to the set of extremal and ray extremal points of the set of proper effects is sufficient to get unique post-measurement states by the condition [itex]f(\rho_{post})=1[/itex].
But again, this does not suffice. In boxworld (see the n=4 polygon theory of the second paper listed above), there are extremal and ray-extremal effects for which [itex]f(\rho_{post})=1[/itex] is satisfied for a whole 1-dimensional face of the set of states.



After all, I don't know how to establish a rule for assigning post-measurement states in generalized probabilistic theories. Can anyone help me? Any ideas or even solutions to this problem?

Thanks in advance.
 
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  • #2




It seems like you have already done a lot of research and thinking on this topic. I can offer some perspectives and ideas that may help in your understanding of this framework.

Firstly, it is important to note that the framework of "convex theories" is still a work in progress and there is ongoing research in this area. Therefore, it is possible that there may not be a definitive answer to your question yet. However, there are some interesting ideas and approaches that have been proposed in this framework.

One approach is to consider the state as a probability distribution over a set of "ontic states" which represent the underlying physical reality. This idea is similar to the ontological models proposed in the field of quantum foundations. In this approach, a measurement would be seen as updating the probability distribution over the ontic states based on the outcome of the measurement. This idea has been explored in papers such as "Generalized probabilistic theories: what determines the structure of quantum theory?" by O. Dahlsten et al. (2012) and "Ontological models and generalized probabilistic theories" by R. Spekkens (2007).

Another approach is to consider the state as a convex combination of "pure states", which represent the most specific information that can be obtained about a system. In this approach, a measurement would be seen as a transformation of the state, where the pure states that are consistent with the measurement outcome are given more weight in the convex combination. This idea has been explored in papers such as "The geometry of quantum states and generalized probabilistic theories" by G. Chiribella et al. (2010) and "Convex operational theories: a framework for quantum and non-locality" by M. Navascués et al. (2008).

I hope these ideas can provide some insight into the problem you are trying to solve. It may also be helpful to reach out to the authors of the papers you have mentioned for further clarification and discussions. Good luck with your research!
 

1. What are post-measurement states in generalized probabilistic theories?

Post-measurement states in generalized probabilistic theories refer to the state of a physical system after a measurement has been performed on it. In these theories, the outcome of a measurement is not determined solely by the initial state of the system, but also by the measurement apparatus and the measurement process itself.

2. How are post-measurement states different from pre-measurement states?

Pre-measurement states refer to the state of a physical system before any measurement has been performed on it. Post-measurement states, on the other hand, refer to the state of the system after a measurement has been performed. In generalized probabilistic theories, post-measurement states take into account the effects of the measurement process, while pre-measurement states do not.

3. Can post-measurement states be predicted beforehand?

In generalized probabilistic theories, post-measurement states cannot be predicted beforehand with certainty. This is because the outcome of a measurement is determined not only by the initial state of the system, but also by the measurement process itself. Therefore, the exact post-measurement state of a system cannot be known beforehand.

4. How do post-measurement states relate to the uncertainty principle?

The uncertainty principle states that the more precisely one property of a particle is measured, the less precisely another complementary property can be known. In generalized probabilistic theories, post-measurement states take into account the effects of the measurement process, which can lead to uncertainty in the post-measurement state of the system. Therefore, post-measurement states are directly related to the uncertainty principle.

5. What is the significance of post-measurement states in physics?

Post-measurement states in generalized probabilistic theories are significant because they provide a more complete understanding of the effects of measurement on physical systems. They also play a crucial role in the development of quantum mechanics, as they allow for a probabilistic description of measurement outcomes that is consistent with experimental results.

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