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arXiv:1012.0115

Test of Common Sense in Quantum Copying Process

Mi-Ra Hwang, Eylee Jung, Kap Soo Jang, MuSeong Kim, DaeKil Park, Eui-Soon Yim, Hung-Soo Kim, Jin-Woo Son

Published Dec 1, 2010• 0 citations
arXivHTMLPDF

Abstract

It is believed that the more we have {\it a priori} information on input states, the better we can make the quality of clones in quantum cloning machines. This common sense idea was confirmed several years ago by analyzing a situation, where the input state is either one of two non-orthogonal states. If the {\it a priori} information is measured by the Shannon entropy, common sense predicts that the quality of the clone becomes poorer with increasing $N$, where $N$ is the number of possible input states. We show, however, that the {\it a priori} information measured by the Shannon entropy does not affect the quality of the clones. Instead the no-cloning theorem and `denseness' of the possible input states play important roles in determining the quality. Specifically, the factor `denseness' plays a more crucial role than the no-cloning theorem when $N \geq 3$.

DOI
10.48550/arxiv.1012.0115

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It is believed that the more we have {\it a priori} information on input states, the better we can make the quality of clones in quantum cloning machines. This common sense idea was confirmed several years ago by analyzing a situation, where the input state is either one of two non-orthogonal states. If the {\it a priori} information is measured by the Shannon entropy, common sense predicts that the quality of the clone becomes poorer with increasing NNN, where NNN is the number of possible input states. We show, however, that the {\it a priori} information measured by the Shannon entropy does not affect the quality of the clones. Instead the no-cloning theorem and denseness' of the possible input states play important roles in determining the quality. Specifically, the factor denseness' plays a more crucial role than the no-cloning theorem when N≥3N \geq 3N≥3.
We analyze the possibility of copying (that is, cloning) arbitrary states of a quantum-mechanical spin-1/2 system. We show that there exists a ``universal quantum-copying machine'' (i.e., transformation) which approximately copies quantum-mechanical states such that the quality of its output does not depend on the input. We also examine a machine which combines a unitary transformation and a selective measurement to produce good copies of states in the neighborhood of a particular state. We discuss the problem of measurement of the output states. \textcopyright{} 1996 The American Physical Society.
It is well known that (non-orthogonal) pure states cannot be cloned so one may ask: how much or what kind of additional (quantum) information is needed to supplement one copy of a quantum state in order to be able to produce two copies of that state by a physical operation? For classical information, no supplementary information is required. However for pure quantum (non-orthogonal) states, we show that the supplementary information must always be as large as it can possibly be i.e. the clone must be able to be generated from the additional information alone, independently of the first (given) copy.
We present quantum cloning machines that transform NNN identical qubits into M>NM>NM>N identical copies and we prove that the fidelity (quality) of these copies is optimal. The connection between cloning and measurement is discussed in detail. When the number of clones MMM tends towards infinity, the fidelity of each clone tends towards the optimal fidelity that can be obtained by a measurement on the input qubits. More generally, quantum cloning machines are universal devices to translate quantum information into classical information.
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.
An unknown quantum state cannot be copied and broadcast freely due to the no-cloning theorem. Approximate cloning schemes have been proposed to achieve the optimal cloning characterized by the maximal fidelity between the original and its copies. Here, from the perspective of quantum Fisher information (QFI), we investigate the distribution of QFI in asymmetric cloning machines which produce two nonidentical copies. As one might expect, improving the QFI of one copy results in decreasing the QFI of the other copy. It is perhaps also unsurprising that asymmetric phase-covariant cloning outperforms universal cloning in distributing QFI since a priori information of the input state has been utilized. However, interesting results appear when we compare the distributabilities of fidelity (which quantifies the full information of quantum states), and QFI (which only captures the information of relevant parameters) in asymmetric cloning machines. Unlike the results of fidelity, where the distributability of symmetric cloning is always optimal for any d-dimensional cloning, we find that any asymmetric cloning outperforms symmetric cloning on the distribution of QFI for d ≤ 18, whereas some but not all asymmetric cloning strategies could be worse than symmetric ones when d > 18.
In this project, quantum cloning machines are analyzed that take in N quantum systems in the same unknown pure state and output M quantum systems with M > N, such that the output best resembles the ideal, but impossible output of an M-fold tensor product of the pure input state. The proof of Keyl and Werner is reviewed in which a unique optimal solution is constructed, both in the case where the quality of the cloning is determined by the entire output, and in the case where the quality is determined by measurements on a single clone. Furthermore, it is shown that previous work on qubit cloning machines is a special case of the presented optimal solution.