general nature; for instance, the subsystems can describe different degrees of freedom such as position and spin. Depending on whether the particles are in principle distinguishable or indistinguishable, we distinguish two basic classes of composite systems, which are of fundamentally different nature.

If the quantum system is composed of $N$ *distinguishable particles*, its global quantum state is described by a density matrix on the tensor product $\mathcal H = \mathcal H_1 \otimes \ldots \otimes \mathcal H_N$, where the $\mathcal H_j$ are the Hilbert spaces describing the individual particles. Quantum mechanics also tells us that observables $O_j$ acting on a single subsystem $\mathcal H_j$ correspond to tensor product observables ${\mathbf 1}_{\mathcal H_1 \otimes \ldots \otimes \mathcal H_{j-1}} \otimes O_j \otimes {\mathbf 1}_{\mathcal H_{j+1} \otimes \ldots \otimes \mathcal H_N}$, which act by the identity on all other subsystems. By non-degeneracy of the inner product, there exists a unique density matrix $\rho_j$ on $\mathcal H_j$ such that $$\label{definition reduced density matrix}
  {\mathrm{tr}\left(({\mathbf 1}_{\mathcal H_1 \otimes \ldots \otimes \mathcal H_{j-1}} \otimes O_j \otimes {\mathbf 1}_{\mathcal H_{j+1} \otimes \ldots \otimes \mathcal H_N}) \rho\right)} = {\mathrm{tr}\left(O_j \rho_j\right)}$$ for all observables $O_j$. It describes the quantum state of the $j$-th subsystem.

The density matrix $\rho_j$ is called 