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Computer Engineering and Applications Journal

Deleting and cloning of quantum information

Yongming Li

Published Jan 1, 2007• 0 citations
OpenAlex

Abstract

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.

Related papers

Quantum Cloning and Deletion in Quantum Information Theory

Satyabrata Adhikari

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.

arXiv:0902.1622•Feb 10, 2009•2 citations

Probabilistic cloning and deleting of quantum states

Yuan Feng, Shengyu Zhang, Mingsheng Ying

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.

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Is It Possible To Clone Using An Arbitrary Blank State?

Roy, A, Ujjwal Sen, Sen, U

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 ′ √ √

•Jan 1, 2001•1 citation

Some analogies between quantum cloning and quantum deleting

Daowen Qiu

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.

Physical Review A•Apr 15, 2002•15 citations

Quantum deleting and cloning in a pseudo-unitary system

Yu‐Cheng Chen, Ming Gong, Peng Xue +2 more

In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to arbitrary two different and nonorthogonal pure qubit states. Furthermore, state discrimination, which is strongly related to quantum no-cloning theorem, is also discussed. Last but not least, we simulate the pseudo-unitary operators in conventional quantum mechanics with post-selection, and obtain the success probability of simulations. Pseudo-unitary operators are implemented with a limited efficiency due to the post-selections. Thus, the success probabilities of deleting and cloning in the simulation by conventional quantum mechanics are less than unity, which maintain the quantum no-deleting and no-cloning theorems.

arXiv:2103.15353•Frontiers of Physics•Apr 16, 2021•12 citations

Combinations of probabilistic and approximate quantum cloning and deleting

Daowen Qiu

We first construct a probabilistic and approximate quantum cloning machine (PACM) and then clarify the relation between the PACM and other cloning machines. After that, we estimate the global fidelity of the approximate cloning that improves the previous estimation for the deterministic cloning machine; and also derive a bound on the success probability of producing perfect multiple clones. Afterwards, we further establish a more generalized probabilistic and approximate cloning and deleting machine (PACDM) and discuss the connections of the PACDM to some of the existing quantum cloning and deleting machines. Finally the global fidelity and a bound on the success probability of the PACDM are obtained. Summarily, the quantum devices established in this paper improve and also greatly generalize some of the existing machines.

Physical Review A•May 13, 2002•19 citations
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.
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.
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.
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 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.
In conventional quantum mechanics, quantum no-deleting and no-cloning theorems indicate that two different and nonorthogonal states cannot be perfectly and deterministically deleted and cloned, respectively. Here, we investigate the quantum deleting and cloning in a pseudo-unitary system. We first present a pseudo-Hermitian Hamiltonian with real eigenvalues in a two-qubit system. By using the pseudo-unitary operators generated from this pseudo-Hermitian Hamiltonian, we show that it is possible to delete and clone a class of two different and nonorthogonal states, and it can be generalized to arbitrary two different and nonorthogonal pure qubit states. Furthermore, state discrimination, which is strongly related to quantum no-cloning theorem, is also discussed. Last but not least, we simulate the pseudo-unitary operators in conventional quantum mechanics with post-selection, and obtain the success probability of simulations. Pseudo-unitary operators are implemented with a limited efficiency due to the post-selections. Thus, the success probabilities of deleting and cloning in the simulation by conventional quantum mechanics are less than unity, which maintain the quantum no-deleting and no-cloning theorems.
We first construct a probabilistic and approximate quantum cloning machine (PACM) and then clarify the relation between the PACM and other cloning machines. After that, we estimate the global fidelity of the approximate cloning that improves the previous estimation for the deterministic cloning machine; and also derive a bound on the success probability of producing perfect multiple clones. Afterwards, we further establish a more generalized probabilistic and approximate cloning and deleting machine (PACDM) and discuss the connections of the PACDM to some of the existing quantum cloning and deleting machines. Finally the global fidelity and a bound on the success probability of the PACDM are obtained. Summarily, the quantum devices established in this paper improve and also greatly generalize some of the existing machines.