  The eigenvectors of the particle number operator in second quantization are
characterized by the block sparsity of their matrix product state
representations. This is shown to generalize to other classes of operators.
Imposing block sparsity yields a scheme for conserving the particle number that
is commonly used in applications in physics. Operations on such block
structures, their rank truncation, and implications for numerical algorithms
are discussed. Explicit and rank-reduced matrix product operator
representations of one- and two-particle operators are constructed that operate
only on the non-zero blocks of matrix product states.
