  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. Here we prove that quantum
information stored in qubits and qudits cannot be partially erased, even by
irreversible operations. The no-flipping theorem, which rules out the existence
of a universal NOT gate for an arbitrary qubit, emerges as a corollary to this
theorem. The `no partial erasure' theorem is shown to apply to spin and bosonic
coherent states, with the latter result showing that the `no partial erasure'
theorem applies to continuous variable quantum information schemes as well. The
no partial erasure theorem suggests an integrity principle that quantum
information is indivisible.
