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arXiv:0902.1622

Quantum Cloning and Deletion in Quantum Information Theory

Satyabrata Adhikari

Published Feb 10, 2009• 2 citations
arXivHTMLPDF

Abstract

Quantum mechanics put restriction on performing some task which we can do classically. One such restriction is that we cannot copy an arbitrary quantum state. This is known as No-cloning theorem. Although quantum mechanics forbid us to construct a perfect cloner, there is no restriction to construct an imperfect cloner. Therefore, we have constructed and studied a special kind of quantum cloning machine named as Hybrid quantum cloning machine which is nothing but a different combination of pre-existing quantum cloning machines. We showed that this hybrid quantum cloning machine sometime gives better quality copy than the pre-existing quantum cloners. Quantum entanglement is a very astonishing property in a multi-partite system and it is a very useful ingredient of quantum information theory. Therefore broadcasting (cloning) of entanglement is a very important subject to study. In this regard we have studied the broadcasting of entanglement using state dependent quantum cloning machine. Further, we have investigated the problem of secretly broadcasting of three-qubit entangled state between two distant partners. Linearity of quantum mechanics put another restriction on deletion of an arbitrary quantum state i.e. an arbitrary quantum state against a copy cannot be perfectly deleted. Like cloning machine one can construct an approximate deletion machine also. Therefore, we have constructed an universal deletion machine in the sense that the fidelity of deletion does not depend on the input state and showed that the fidelity of deletion is 3/4 . Further we showed that the fidelity of deletion depends on the blank state at the output and it can be improved from the value 3/4 for certain blank state. Moreover, we have studied the case where two cloned copies are given at the input of the deletion machine.

DOI
10.48550/arxiv.0902.1622

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Quantum mechanics put restriction on performing some task which we can do classically. One such restriction is that we cannot copy an arbitrary quantum state. This is known as No-cloning theorem. Although quantum mechanics forbid us to construct a perfect cloner, there is no restriction to construct an imperfect cloner. Therefore, we have constructed and studied a special kind of quantum cloning machine named as Hybrid quantum cloning machine which is nothing but a different combination of pre-existing quantum cloning machines. We showed that this hybrid quantum cloning machine sometime gives better quality copy than the pre-existing quantum cloners. Quantum entanglement is a very astonishing property in a multi-partite system and it is a very useful ingredient of quantum information theory. Therefore broadcasting (cloning) of entanglement is a very important subject to study. In this regard we have studied the broadcasting of entanglement using state dependent quantum cloning machine. Further, we have investigated the problem of secretly broadcasting of three-qubit entangled state between two distant partners. Linearity of quantum mechanics put another restriction on deletion of an arbitrary quantum state i.e. an arbitrary quantum state against a copy cannot be perfectly deleted. Like cloning machine one can construct an approximate deletion machine also. Therefore, we have constructed an universal deletion machine in the sense that the fidelity of deletion does not depend on the input state and showed that the fidelity of deletion is 3/4 . Further we showed that the fidelity of deletion depends on the blank state at the output and it can be improved from the value 3/4 for certain blank state. Moreover, we have studied the case where two cloned copies are given at the input of the deletion machine.
Suppose we are given two identical copies of an unknown quantum state and we wish to delete one copy from among the given two copies. The quantum no-deletion principle restricts us from perfectly deleting a copy but it does not prohibit us from deleting a copy approximately. Here we construct two types of a ``universal quantum deletion machine'' which approximately deletes a copy such that the fidelity of deletion does not depend on the input state. The two types of universal quantum deletion machines are (1) a conventional deletion machine described by one unitary operator and (2) a modified deletion machine described by two unitary operators. Here it is shown that the modified deletion machine deletes a qubit with fidelity 3∕43∕43∕4, which is the maximum limit for deleting an unknown quantum state. In addition to this we also show that the modified deletion machine retains the qubit in the first mode with average fidelity 0.77 (approx.) which is slightly greater than the fidelity of measurement for two given identical states, showing how precisely one can determine its state [S. Massar and S. Popescu, Phys. Rev. Lett. 74, 1259 (1995)]. We also show that the deletion machine itself is input state independent, i.e., the information is not hidden in the deleting machine, and hence we can delete the information completely from the deletion machine.
Using the reduced density operator of quantum state to study the deletion and clone of quantum information.Testify that there no exists quantum deleting machine which makes the deleted states and the blank states be correlated;also we can not construct a machine to perfectly delete arbitrary mixed quantum states.At last,we consider the problem of cloning and deleting of mixed quantum states in the case of probabilistic form,provide a sufficient and necessary condition in the form of probability with a unification formula.By using the results obtained,the processing of quantum information can be further comprehended and has aspiring significance for quantum computer.
Due to the axioms of quantum mechanics, perfect cloning of an unknown quantum state is impossible. But since imperfect cloning is still possible, a question arises: ``Is there an optimal quantum cloning machine?'' Buzek and Hillery answered this question and constructed their famous B-H quantum cloning machine. The B-H machine clones the state of an arbitrary single qubit in an optimal manner and hence it is universal. Generalizing this machine for a two-qubit system is straightforward, but during this procedure, except for product states, this machine loses its universality and becomes a state-dependent cloning machine. In this paper, we propose some classes of optimal universal local quantum state cloners for a particular class of two-qubit systems, more precisely, for a class of states with known Schmidt basis. We then extend our machine to the case that the Schmidt basis of the input state is deviated from the local computational basis of the machine. We show that more local quantum coherence existing in the input state corresponds to less fidelity between the input and output states. Also we present two classes of a state-dependent local quantum copying machine. Furthermore, we investigate local broadcasting of two aspects of quantum correlations, i.e., quantum entanglement and quantum discord, defined, respectively, within the entanglement-separability paradigm and from an information-theoretic perspective. The results show that although quantum correlation is, in general, very fragile during the broadcasting procedure, quantum discord is broadcasted more robustly than quantum entanglement.
We further verify the impossibility of deleting an arbitrary unknown quantum state, and also show it is impossible to delete two nonorthogonal quantum states as a consequence of unitarity of quantum mechanics. A quantum approximate (deterministic) deleting machine and a probabilistic (exact) deleting machine are constructed. The estimation for the global fidelity characterizing the efficiency of the quantum approximate deleting is given. We then demonstrate that unknown nonorthogonal states chosen from a set with their multiple copies can evolve into a linear superposition of multiple deletions and failure branches by a unitary process if and only if the states are linearly independent. It is notable that the proof for necessity is somewhat different from Pati's [Phys. Rev. Lett. 83, 2849 (1999)]. Another deleting machine for the input states that are unnecessarily linearly independent is also presented. The bounds on the success probabilities of these deleting machines are derived. So we expound some preliminary analogies between quantum cloning and deleting.
We show that using a fixed cloning machine, whether deterministic inexact or probabilistic exact, one can take an arbitrary blank state. Quantum information cannot be cloned. There cannot exist a machine which can produce two (or more) exact copies of an arbitrary state in a deterministic manner [1]. Replication is not allowed even when the input state is taken at random from a given set of two non-orthogonal states [2]. It has been further shown that probabilistic cloning is not possible if the input state is from a given linearly dependent set [3]. Henceforth we deal explicitly with qubits although all the results can be extended to higher dimensions. Although exact cloning is not possible, one can approximately clone an arbitrary input state [4, 5, 6]. In this scheme a fixed state is taken as the blank state, depending on which and the initial machine state, the cloning operation is constructed. We extend this operation such that any arbitrary (pure or mixed) state taken as the blank copy can do the job. The optimal universal 1 → 2 inexact qubit cloner of Brußet al.[5] takes an arbitrary qubit |ψ 〉 〈ψ | = 1 2 (I + − → s. − → σ) along with a fixed blank qubit |b 〉 and a machine state |M 〉 as input. An entangled state of the three qubits is produced as the output such that the reduced density matrices of the first two qubits are two similar approximate copies ρ = 1 2 (I + η− → s. − → σ) of |ψ 〉 〈ψ | with η = 2 3. The unitary operator realizing this process is defined on the combined Hilbert space of the input qubit, the blank qubit and machine by U ′ √ √
We construct a probabilistic cloning and deleting machine which, taking several copies of an input quantum state, can output a linear superposition of multiple cloning and deleting states. Since the machine can perform cloning and deleting in a single unitary evolution, the probabilistic cloning and other cloning machines proposed in the previous literature can be thought of as special cases of our machine. A sufficient and necessary condition for successful cloning and deleting is presented, and it requires that the copies of an arbitrarily presumed number of the input states are linearly independent. This simply generalizes some results for cloning. We also derive an upper bound for the success probability of the cloning and deleting machine.