# The joint distribution of marginals of multipartite random quantum states

Together with Stephane Dartois and Luca Lionni, we have a new preprint on the arXiv, On the joint distribution of the marginals of multipartite random quantum states. The paper is about the joint distribution of the set of (twisted) marginals of random multipartite quantum states. We study mainly random Wishart tensors, which, after normalization, yield multipartite random quantum states, so our work can also be seen as a contribution to the theory of random tensors.

Start with  copies of a finite dimensional complex Hilbert space, say . Consider  a random complex Gaussian tensor, that is a vector of size  with i.i.d. standard complex Gaussian entries. for a subset  of cardinality , we define the marginal

obtained by partial tracing the tensor legs belonging to , the complementary set of . Moreover, we allow for twisted marginals: for a permutation , let

where  permutes the tensor legs as follows:

We prove the following result.

Theorem 1 The set of  twisted marginals  are asymptotically free as . In other words, they jointly converge in distribution to a family of free standard free Poisson elements.

Recall that a free Poisson (or a Marchenko-Pastur) random variable has distribution

where  is a positive parameter (in our result, ). We display below the density (red curve) and numerical simulations for 

and for 

The message of our main theorem is that although the (overlapping) marginals are correlated random variables, their distributions become asymptotically free as the size of the individual Hilbert spaces grows to infinity.

The freeness result proven in the balanced asymptotical regime (the tensor factors scale in the same way) fails in the unbalanced asymptotical regime (some of the tensor factors have fixed dimensions, whereas others grow to infinity). We emphasize this fact in a particular situation, for  (4-partite tensors). Write , where

and  denote, respectively, the four tensor factors above. It is natural to consider the joint distribution of the marginals , which are not asymptotically free anymore:

Theorem 2 As  and  are fixed, the marginals  have the following asymptotical free cumulants:

where  is an arbitrary word in the letters , and  is the number of different consecutive values of , counted cyclically:

where .

The difference with the previous situation is that the amount of ''fresh'' randomness in the marginals  is relatively small (fixed  vs. ), and not enough to ensure the asymptotical freeness. The natural framework to tackle such situations is that of freeness with amalgamation, and we shall study these cases in some future work.