Proof
The theorem holds for quantum states in a Hilbert space of any dimension. For simplicity,
consider the deleting transformation for two identical qubits. If two qubits are in orthogonal states, then deletion requires that
- ,
- .
Let be the state of an unknown qubit. If we have two copies of an unknown qubit, then by linearity of the deleting transformation we have
In the above expression, the following transformation has been used:
However, if we are able to delete a copy, then, at the output port of the deleting machine, the combined state should be
- .
In general, these states are not identical and hence we can say that the machine fails to delete a copy. If we require that the final output states are same, then we will see that there is only one option:
and
Since final state of the ancilla is normalized for all values of it must be true that and are orthogonal. This means that the quantum information is simply in the final state of the ancilla. One can always obtain the unknown state from the final state of the ancilla using local operation on the ancilla Hilbert space. Thus, linearity of quantum theory does not allow an unknown quantum state to be deleted perfectly.