Elsevier

Physics Letters A

Volume 327, Issues 2–3, 28 June 2004, Pages 123-128
Physics Letters A

Strategies for state-dependent quantum deleting

https://doi.org/10.1016/j.physleta.2004.05.036Get rights and content

Abstract

A quantum state-dependent quantum deleting machine is constructed. We obtain a upper bound of the global fidelity on N-to-M quantum deleting from a set of K non-orthogonal states. Quantum networks are constructed for the above state-dependent quantum deleting machine when K=2. Our deleting protocol only involves a unitary interaction among the initial copies, with no ancilla. We also present some analogies between quantum cloning and deleting.

Introduction

Manipulation and extraction of quantum information are important tasks in building quantum computer. As is known, the copying and deleting of information in a classical computer are inevitable operations whereas similar operations cannot be realized perfectly in quantum computers. Linearity of quantum mechanics unveils that we cannot duplicate an unknown quantum state accurately [1]. This has been proven by Wootters and Zurek [1] and Dieks [2] which called the quantum no-cloning theorem. Though exact cloning is not possible, in the literature various cloning machines have been proposed [3], [4], [5], [6], [7], [8], [9], [10], [11] which operate either in a deterministic or probabilistic way. Corresponding to the quantum no-cloning theorem, Pati and Braunstein [12] demonstrated that the linearity of quantum mechanics also forbids one to delete one unknown state ideally against a copy [12], which is called the quantum no-deleting principle and complements the quantum no-cloning theorem in spirit. If quantum deleting could be done, then one would create a standard blank state onto which one could copy an unknown state approximately, by deterministic cloning or exactly, by probabilistic cloning process. When memory in a quantum computer is scare, quantum deleting may play an important role, and one could store new information in an already computed state by deleting the old information. At first glance it seems that quantum deleting is just the reverse of quantum cloning, actually it is not so. In Ref. [13], through the analysis of a approximate 2→1 deleting machine, the author shows the fidelity of the two modes are different for the deleting operation, whereas in the cloning operation the reduced density matrix of both the modes are same. As indicated emphatically in Refs. [12], [14], discussing probabilistic and approximate quantum deleting will not only contribute to processing quantum information, but further understand some connections between quantum cloning and quantum deleting. Recently some probabilistic deleting machines have been established [15], [16].

The purpose of this Letter is to investigate how well one can deleting quantum states. Here, we discuss the problem of approximate N-to-M quantum deleting from a set of K non-orthogonal states, and our deleting protocol only involves a unitary interaction among the initial copies, with no ancilla. In Section 2, we prove the impossibility of deleting a copy from two copies of K non-orthogonal states, then the analytic solution for N-to-M quantum deleting from a set of two non-orthogonal states is given, quantum networks are also constructed for this quantum deleting machine. In Section 3, we give an upper bound of the global fidelity on N-to-M quantum deleting from a set of K non-orthogonal states, and some analogies between quantum cloning and deleting are presented. A summary is given in Section 4.

Section snippets

State-dependent deleting machine

Firstly let us review the definition for N-to-M quantum deleting introduced by Pati and Braunstein [13]. In general the quantum deleting operation is defined for N unknown states |ΨN such that the linear operator acts on the combined Hilbert space and deletes NM copies and keeps M copies intact. It is defined by |Ψ〉⊗N|A〉→|Ψ〉⊗M|Σ〉⊗N−M|Aψ〉, where |Σ〉 is the blank state of a qubit, |A〉 is the initial and |AΨ〉 is the final state of the ancilla. Pati and Braunstein has proven the impossibility of

Upper bound of state-dependent deleting when the set contains N states

In this section, we will investigate the state-dependent quantum deleting when the state set contains more than two states. Similar to the above discussion, the ancilla is not involved. It is still difficult to solve this problem completely, we can only give some bounds for this problem. Now our purpose is to maximizing F under the condition of Eq. (2.4). In Ref. [10], the author have obtained the upper bound of F which is given by F⩽12+12n(n−1)k≠j=1ncos(aj,k−aj,k′) in the quantum deleting

Conclusions

In summary, we have constructed the state-dependent deleting machines which may be formally thought of the converse device of the corresponding cloning machines to a certain extent. We also analyze the connection and difference between quantum cloning and quantum deleting. Our results may have potential applications in information processing because it provides strategies for state-dependent quantum deleting in a quantum computer. It tells us how to control the deleting efficiencies from a set

Acknowledgements

We are grateful to the referee for valuable comments and helpful suggestions. This work is supported by Anhui Provincial Natural Science Foundation under Grant No. 03042401, the Key Program of the Education Department of Anhui Province under Grant No. 2004kj005zd, and the Talent Foundation of Anhui University.

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