# Operations in operational probabilistic theories

Tags:
1. Aug 22, 2015

### msumm21

I'm trying to read this recent paper (http://arxiv.org/abs/1507.07745) but I don't understand the definition of an "operation" in the paper. It is defined as a set of events on the first page, but I don’t know what an event is in this context. Looks like it’s some kind of function but I can’t verify that or tell any details. E.g. is it really a function, if so is it a linear transformation, what kind of spaces does it map between, …?

An example would be great as well. Anyone know if there’s a link with a simple example?

2. Aug 22, 2015

### atyy

They give an example at the bottom paragraph starting "Quantum theory ...", left column, page 2.

3. Sep 2, 2015

### msumm21

Thanks. I can see the definition now in context of Hilbert spaces, but now I'm struggling to understand it. Let me try to make a simple example which should illustrate a lot of my problems.

I want to make an OPT circuit for an electron's spin that I think has 3 operations:
1. Preparation: measure the spin about "X"
2. Place in a uniform magnetic field for T seconds.
3. Measurement: measure the spin about "Y"
From the paper the 1st operation is a set of events, each event being a type of function that maps linear transformations on "I" (I don't see "I" defined anywhere and don't know what it means here) to linear transformation on the 2d hilbert space H. Is "I" supposed to be 1D hilbert space? Anyone know what operation and events would be associated with step 1 above? I know they later say there's a isomorphism identifying a preparation event with a linear transformation on H, but I'm trying to understand this in general first. I guess I'll wait till I understand #1 before even trying to continue...