In the last weeks, I came across expressions of the kind $$\int p(\rho) \rho^{\otimes N} d\nu,$$ where $\rho$ is a density operator and $\nu$ is "some appropriate" measure. It also often appeared as $\int p(\rho) \rho^{\otimes N} d\rho$, where it seems like they chose some particular measure.
For example, it appears often in the context of the Quantum de-Finetti theorem (see, for example, page 5).
Nevertheless, I haven't ever seen a proper definition of an integral over some set of density operators is and I also couldn't find anything on the internet that could clarify my confusion about this term. I try to imagine this as some weighted mixture of density matrices, where $p(\rho)$ is the weight we assign to every $\rho$, but in the lack of a clean and unambigious definition, I do not feel like I understood this mathematical term properly.
Can anyone give me a proper definition and/or notion how to think about this term in order to improve my understanding and to be able to work with this object?
Best Answer
In the formula : $$\rho^{(N)} = \int \rho^{\otimes N}P(\rho)\text d\rho $$ the measure $\text dP(\rho) = P(\rho)\text d\rho$ is any probability measure over density operators. (You can define a probability measure on any set (with some additional structure if you want to get into the detail), and the set of density matrices on a Hilbert space is no exception). There are a lot of such measure, and a lot of way to define some.
For example, it could be a combination of delta functions : let $p_1,\ldots, p_n \geq 0$ with $p_1 + \ldots + p_n = 1$ and $\rho_1,\ldots,\rho_n$ some density matrices. Then, we can define a probability measure $\text{d}P = \sum p_i \delta_{\rho_i}$ by : $$\int f(\rho)\text dP(\rho) = \sum p_if(\rho_i)$$ for any function $f$.
If the Hilbert space we are working on is finite dimensional, then so is the space of density matrices, so we can define a lot of probability densities using the Lebesgue measure.
If the Hilbert space is infinte dimensional, then there is no real equivalent of the Lebesgue measure, but we can still make up a lot of density matrices, for example by restricting to some finite dimensional subspace.
To recap, I would say : it is a probability measure just like any other probability measure. There is not much to worry about.