No-deleting theorem


In physics, the no-deleting theorem of quantum information theory is a no-go theorem which states that, in general, given two copies of some arbitrary quantum state, it is impossible to delete one of the copies. It is a time-reversed dual to the no-cloning theorem, which states that arbitrary states cannot be copied. This theorem seems remarkable, because, in many senses, quantum states are fragile; the theorem asserts that, in a particular case, they are also robust. Physicist Arun K. Pati along with Samuel L. Braunstein proved this theorem.
The no-deleting theorem, together with the no-cloning theorem, underpin the interpretation of quantum mechanics in terms of category theory, and, in particular, as a dagger symmetric monoidal category. This formulation, known as categorical quantum mechanics, in turn allows a connection to be made from quantum mechanics to linear logic as the logic of quantum information theory.

Overview of quantum deletion

Suppose that there are two copies of an unknown quantum state. A pertinent question in this context is to ask if it is possible, given two identical copies, to delete one of them using quantum mechanical operations? It turns out that one cannot. The no-deleting theorem is a consequence of linearity of quantum mechanics. Like the no-cloning theorem this has important implications in quantum computing, quantum information theory and quantum mechanics in general.
The process of quantum deleting takes two copies of an arbitrary, unknown
quantum state at the input port and outputs a blank state along with the original. Mathematically,
this can be described by:
where is the deleting operation which is not necessarily unitary, is the unknown quantum
state, is the blank state, is the initial state of
the deleting machine and is the final state of the machine.
It may be noted that classical bits can be copied and deleted, as can qubits in orthogonal states. For example, if we have two identical qubits and then we can transform to and. In this case we have deleted the second copy. However, it follows from linearity of quantum theory that there is no that can perform the deleting operation for any arbitrary state.

Formal statement of the no-deleting theorem

Let be an unknown quantum state in some Hilbert space. Then,
there is no linear isometric transformation such that
, with the final state of the ancilla being independent of

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.

Consequence