No-deleting theorem

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 |\psi\rangle_A |\psi\rangle_B |A\rangle_C = |\psi\rangle_A |0\rangle_B |A'\rangle_C where U is a unitary operator, |\psi\rangle_A is the unknown quantum state, |0\rangle_B is the blank state, |A\rangle_C is the initial state of the deleting machine and |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 \rangle and |11 \rangle , then we can transform to |00 \rangle and |10 \rangle . In this case we have deleted the second copy. However, it follows from linearity of quantum theory that there is no U that can perform the deleting operation for any arbitrary state |\psi\rangle. == Formal statement ==

Given three Hilbert spaces for systems A, B, C, such that the Hilbert spaces for systems A, B are identical. If U is a unitary transformation, and |A \rangle_C is an ancilla state, such that U |\psi\rangle_A |\psi\rangle_B |A\rangle_C = |\psi\rangle_A |0\rangle_B |A_\psi\rangle_C, \quad \forall | \psi \rangle where the final state of the ancilla |A_\psi\rangle_C may depend on |\psi\rangle , then U is a swapping operation, in the sense that the map |\psi\rangle_A \mapsto |A_\psi\rangle_C 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 :|0 \rangle_A |0 \rangle_B |A\rangle_C \rightarrow |0\rangle_A |0\rangle_B |A_0\rangle_C, :|1 \rangle_A |1 \rangle_B |A\rangle_C \rightarrow |1 \rangle_A |0\rangle_B |A_1\rangle_C. Let |\psi\rangle = \alpha |0\rangle + \beta |1 \rangle 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 :|\psi\rangle_A |\psi\rangle_B |A\rangle_C = [\alpha^2 |0 \rangle_A |0\rangle_B + \beta^2 : \qquad \rightarrow \alpha^2 |0 \rangle_A |0\rangle_B |A_0\rangle_C + \beta^2 In the above expression, the following transformation has been used: :1/{\sqrt 2}(|0\rangle_A |1\rangle_B + |1 \rangle_A |0\rangle_B ) |A \rangle_C \rightarrow |\Phi \rangle_{ABC} . However, if we are able to delete a copy, then, at the output port of the deleting machine, the combined state should be : |\psi\rangle_A |0\rangle_B |A'\rangle_C = (\alpha |0 \rangle_A |0\rangle_B + \beta |1\rangle_A |0\rangle_B) |A'\rangle_C. 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: : |\Phi\rangle = 1/{\sqrt 2}(|0 \rangle_A |0\rangle_B |A_1\rangle_C + |1\rangle_A |0\rangle_B |A_0\rangle_C), and : |A'\rangle_C = \alpha |A_0\rangle_C + \beta |A_1\rangle_C . Since final state |A' \rangle of the ancilla is normalized for all values of \alpha, \beta it must be true that |A_0\rangle_C and |A_1\rangle_C 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==

• If it were possible to delete an unknown quantum state, then, using two pairs of EPR states, we could send signals faster than light. Thus, violation of the no-deleting theorem is inconsistent with the no-signalling condition. • The no-cloning and the no-deleting theorems point to the conservation of quantum information. • Stronger versions of the no-cloning theorem and the no-deleting theorem provide permanence to quantum information. To create a copy one must import the information from some part of the universe and to delete a state one needs to export it to another part of the universe where it will continue to exist. ==See also==