Foundational theorem of quantum information processing
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. It was proved by Arun K. Pati and Samuel L. Braunstein. Intuitively, it is because information is conserved under unitary evolution.
This theorem seems remarkable, because, in many senses, quantum states are fragile; the theorem asserts that, in a particular case, they are also robust. 
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 (in exact analogy to classical logic being founded on Cartesian closed categories).
Overview.
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:
formula_1
where formula_2 is a unitary operator, formula_3 is the unknown quantum 
state, formula_4 is the blank state, formula_5 is the initial state of 
the deleting machine and formula_6 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 formula_7 and formula_8 then we can transform to formula_7 and formula_10. In this case we have deleted the second copy. However, it follows from linearity of quantum theory that there is no formula_2 that can perform the deleting operation for any arbitrary state formula_12.
Formal statement.
Given three Hilbert spaces for systems formula_13, such that the Hilbert spaces for systems formula_14 are identical.
If formula_2 is a unitary transformation, and formula_16 is an ancilla state, such that formula_17where the final state of the ancilla formula_18 may depend on formula_19, then formula_2 is a swapping operation, in the sense that the map formula_21 is an isometric embedding.
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 
formula_22,
formula_23.
Let formula_24 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 
formula_25
formula_26
In the above expression, the following transformation has been used: 
formula_27
However, if we are able to delete a copy, then, at the output port of the deleting machine, the combined state should be 
formula_28.
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:
formula_29
and
formula_30
Since final state formula_31 of the ancilla is normalized for all values of formula_32 it must be true that formula_33 and formula_34 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.
References.
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