Has the $N$-representability problem for bosons been solved? The $N$-representability problem in non-relativistic quantum mechanics is the question whether a 1- or 2-particle mixed state is a reduced density matrix of a pure $N$-particle wavefunction. Has this problem ever been solved in its general form for bosons?
 A: tl;dr: For the one-particle reduced density matrix yes and for the two-particle reduced density matrix no.
Long answer:
Preliminaries: If $\mathfrak h$ denotes the (complex) one-particle Hilbert space, then the Hilbert space of $N$ indistinguishable bosons follows as the $N$-fold symmetrized tensor product of $\mathfrak h$:  $H_N:=\vee^N \mathfrak h $. We define the Fock space as $F:=\bigoplus\limits_{N=0}^\infty H_N$ with the usual creation and annihilation operators $a^\dagger (f)$ and $a(f)$ for $f\in \mathfrak h$. We assume $\mathbb N\ni N\geq 2$.
Consider a normalized bosonic state $\psi \in F$ with with $\langle \psi,N^2\psi\rangle_F<\infty$. Following [1], we define the one- and two-particle reduced density matrices for $\psi$ by the following relations:
\begin{align}
\langle f,\gamma_\psi \,g\rangle_{\mathfrak h}&:=\langle \psi, a^\dagger(g)\, a(f)\,\psi\rangle_{F}\tag{1}\\
\langle f_1\otimes f_2, \Gamma_\psi\, g_1 \otimes g_2\rangle_{\mathfrak h\otimes \mathfrak h}&:= \langle \psi, a^\dagger(g_2)\,a^\dagger(g_1)\, a(f_1)\,a(f_2)\,\psi\rangle_F \tag{2} \quad .
\end{align}
We can show that $\gamma_\psi$ and $\Gamma_\psi$ are of trace-class, self-adjoint and positive semi-definite operators on $\mathfrak h$ and $\mathfrak h\otimes \mathfrak h$, obeying $\mathrm{Tr}_{\mathfrak h}\, \gamma_\psi =N$ and $\mathrm{Tr}_{\mathfrak h\otimes \mathfrak h}\, \Gamma_\psi =N(N-1)$ if $\psi \in H_N$, respectively. Equivalently, we could've defined these operators in first quantization. Moreover, the above definition can be generalized for mixed states, i.e. general density operators (respecting the indistinguishability).

$N$-representability: We now ask if a given trace-class, self-adjoint and positive semi-definite operator $\gamma$ on $\mathfrak h$ or $\Gamma$ on $\mathfrak h\otimes \mathfrak h$, obeying the above trace conditions (and in the case of $\Gamma$ obeying some symmetry condition), arises from some $\psi \in H_N \subset F$, i.e. whether $\gamma=\gamma_\psi$ or $\Gamma=\Gamma_\psi$ in the sense of equations $(1)$ and $(2)$, respectively. If this is the case, then we call the respective operator $N$-representable. The above described problem is known as the $N$-representability problem.
We are mostly interested in practicable necessary and sufficient conditions. See also the end of this answer for why this problem is indeed very relevant in certain areas of physics and chemistry.
Answer: The $N$-representability problem for the one-particle reduced density matrix (1-RDM) is solved and trivial. Indeed, as shown in [2], for every $\gamma$ on $L^2(\mathbb R)$ with the above discussed properties, there is at least one $\psi \in H_N$ such that $\gamma=\gamma_\psi$. The proof goes by construction: First, note that $\gamma$ admits a spectral representation of the form
$$ \gamma=\sum\limits_j \lambda_j \, |f_j\rangle\langle f_j| \quad ,\tag{3}$$
where $\lambda_j \geq 0$ are its eigenvalues and $f_j \in L^2(\mathbb R)$ its orthonormal eigenfunctions. Second, define
$$ \psi(z_1,z_2,\ldots,z_N):=N^{-1/2}\, \sum\limits_j\, \lambda_j^{1/2}\, \prod_{i=1}^N f_j(z_i)\tag{4}$$
and verify that this is indeed a normalized $N$-particle bosonic state, i.e. $\psi \in H_N$, yielding $\gamma=\gamma_\psi$ in the sense of equation $(1)$.
The $N$-representability problem for the two-particle reduced density matrix (2-RDM) is non-trivial and there are no "closed" solutions known. There are only formal conditions which are not practicable (not applicable in practice), however, cf. Ref. [3-4]. Interestingly, the problem of deciding whether a general $\Gamma$ is $N$-representable is QMA complete, as shown in Ref. [5].

Outlook: A few comments first: We can define similar quantities for the fermionic case. There, the $N$-representability problem for the 2-RDM is unsolved, too. For the 1-RDM, things are more complicated. It is known for a rather long time under which conditions a fermionic 1-RDM is ensemble-representable, i.e. under which conditions it arises from a (generally mixed) density operator, cf. Ref. [6]. The (pure) $N$-representability problem was solved only- more or less- recently, see Ref. [7-8] and these conditions are known as generalized Pauli constraints.
In general, all of this is important and relevant in certain (non-relativistic) condensed matter/ electronic structure or quantum chemistry computations. To see this, note that generic Hamiltonians in these applications are often of the form $$H=\sum\limits_{ij} t_{ij}\, a^\dagger(f_j)\, a(f_i)  + \sum\limits_{ijkl} W_{ijkl}\,a^\dagger(f_i)\,a^\dagger(f_j)\,a(f_l)\,a(f_k) \quad . \tag{5}$$
It is an easy exercise to verify that the energy in a normalized state $\psi \in H_N$ follows as
$$ E_{\psi}:= \langle \psi, H\psi\rangle_F= \mathrm{Tr}_{\mathfrak h} t\, \gamma_\psi +  \mathrm{Tr}_{\mathfrak h\otimes \mathfrak h}\, W\, \Gamma_\psi \quad .\tag{6}$$
Now we are often interested in the ground state, so we have to compute
$$E_0(N):=\min\limits_{\psi \in H_N\\ ||\psi||_{H_N}=1} E_\psi\quad \tag{7}  \quad .$$
This is a rather demanding task and the appeal of equation $(6)$ is the following: We could in principle avoid the minimization $(7)$ and instead minimize $(6)$ over the set of $N$-representable 2-RDMs. This is preferable, since the "dimensions" of the 2-RDM do not scale with the number of particles, and hence one does not encounter the so-called exponential wall.
However, as explained, this set is not characterized completely by any practical means.
An interesting final note is that, under certain conditions, we can prove that the ground state energy is a unique functional of the 1-RDM. This approach is called Reduced Density Matrix Functional Theory (a sort of generalization of Density Functional Theory). For more on this, see [9-11].

References:
[1] Solovej, J. P. Many body quantum mechanics. Lecture Notes 2007.
[2] Lieb, E.H. and Seiringer, R. The Stability of Matter in Quantum Mechanics. Cambridge University Press 2010. Section 3.1.5.
[3] Mazziotti, D. A. Structure of fermionic density matrices: Complete N-representability conditions. Physical Review Letters 108.26 (2012): 263002.
[4] Garrod,C  and Percus, J.K. Reduction of the N‐particle variational problem. Journal of Mathematical Physics 5.12 (1964): 1756-1776. See also Kummer, H. n‐Representability Problem for Reduced Density Matrices. Journal of Mathematical Physics 8.10 (1967): 2063-2081.
[5] Wei, Tzu-Chieh, Michele Mosca, and Ashwin Nayak. Interacting boson problems can be QMA hard. Physical review letters 104.4 (2010): 040501. arxiv.
[6] Coleman, A. J. (1963). Structure of fermion density matrices. Reviews of modern Physics, 35(3), 668.
[7] Klyachko, A. A. (2006, April). Quantum marginal problem and N-representability. In Journal of Physics: Conference Series (Vol. 36, No. 1, p. 014). IOP Publishing.
[8] Altunbulak, M., & Klyachko, A. (2008). The Pauli principle revisited. Communications in Mathematical Physics, 282(2), 287-322.
[9] Gilbert, T. L. (1975). Hohenberg-Kohn theorem for nonlocal external potentials. Physical Review B, 12(6), 2111.
[10] Levy, M. (1979). Universal variational functionals of electron densities, first-order density matrices, and natural spin-orbitals and solution of the v-representability problem. Proceedings of the National Academy of Sciences, 76(12), 6062-6065.
[11] A matter modeling answer discussing this approach a bit more.
