Deletion of imperfect cloned copies

In this work, we design a deleting machine and shown that for some given condition on machine parameters, it gives slightly better result than P-B deleting machine [5,6]. Also it is shown that for some particular values of the machine parameters it acts like Pati-Braunstein deleting machine. We also study the combined effect of cloning and deleting machine, where at first the cloning is done by some standard cloning machines such as Wootters-Zurek [1] and Buzek-Hillery [2] cloning machine and then the copy mode is deleted by Pati-Braunstein deleting machine or our prescribed deleting machine. After that we examine the distortion of the input state and the fidelity of deletion . PACS: 03.67.-a, 03.65.Bz, 89.70.+c In quantum information theory, it is well known that an unknown quantum state cannot be cloned or deleted [1,4,5]. But we cannot rule out the possibility of constructing the approximate cloning machine [1,2,3]. We can divide the approximate cloning machines into two categories: (i) State dependent cloning machine: A cloning machine that depends on input state such as Wootter-Zurek cloning machine [1]. It can clone better for some state while it gives worst clone for some other states. (ii) Universal quantum cloning machine : A cloning machine which does not depend on input state such as Buzek-Hillery cloning machine [2]. The fidelity of cloning is same for all input state while cloning with this cloning machine. Like cloning machine, deleting machines have not performed well. This was first observed by Pati and Braunstein and they showed that the linearity of quantum theory does not allow to delete a copy of an arbitrary quantum state perfectly. But ignoring the problem of perfect deletion we can construct approximate deleting machines which are input state dependent such as Pati-Braunstein deleting machine. These deleting machines are not perfect in the sense that they can neither delete the copy mode perfectly nor can retain the input state. It may happen that we first clone an unknown quantum state by using known cloning machine and then after using the copy mode, we want to delete it with known deleting machine. After completing the whole procedure, it is natural to ask about the fidelity of deletion and the distortion of the input state. We are trying to give the answer to the above question in this paper. In section I, We briefly discuss the Wootter-Zurek quantum copying machine and Buzek-Hillery Universal quantum cloning machine. In section II, we discuss shortly about the Pati-Braunstein deleting machine and D.Qiu’s non-optimal universal quantum deleting machine [8]. In section III, we construct a quantum deleting machine which is input state dependent. Then we have shown that the minimum average distortion of the input qubit and maximum fidelity of deletion approaches to