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:
U | ψ ⟩ A | ψ ⟩ B | A ⟩ C = | ψ ⟩ A | 0 ⟩ B | A ′ ⟩ C {\displaystyle U|\psi \rangle _{A}|\psi \rangle _{B}|A\rangle _{C}=|\psi \rangle _{A}|0\rangle _{B}|A'\rangle _{C}}
where U {\displaystyle U} is a unitary operator, | ψ ⟩ A {\displaystyle |\psi \rangle _{A}} is the unknown quantum state, | 0 ⟩ B {\displaystyle |0\rangle _{B}} is the blank state, | A ⟩ C {\displaystyle |A\rangle _{C}} is the initial state of the deleting machine and | A ′ ⟩ C {\displaystyle |A'\rangle _{C}} 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 | 00 ⟩ {\displaystyle |00\rangle } and | 11 ⟩ {\displaystyle |11\rangle } , then we can transform to | 00 ⟩ {\displaystyle |00\rangle } and | 10 ⟩ {\displaystyle |10\rangle } . In this case we have deleted the second copy. However, it follows from linearity of quantum theory that there is no U {\displaystyle U} that can perform the deleting operation for any arbitrary state | ψ ⟩ {\displaystyle |\psi \rangle } .
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