In complete erasure any arbitrary pure quantum state is transformed to a fixed pure state by irreversible operation. Here we ask if the process of partial erasure of quantum information is possible by general quantum operations, where partial erasure refers to reducing the dimension of the parameter space that specifies the quantum state. We prove that quantum information stored in qubits and qudits cannot be partially erased, even by irreversible operations. The ‘no-partial erasure’ theorem applies to spin and bosonic coherent states, and continuous variable quantum information schemes as well. Our theorem suggests an integrity principle that quantum information is indivisible.