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arXiv:quant-ph/9701034 · Physical Review A

Quantum copying: Fundamental inequalities

Mark S. Hillery, Vladimír Bužek

Published Aug 1, 1997• 89 citations
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Abstract

How well can one copy an arbitrary qubit? To answer this question we consider two arbitrary vectors in a two-dimensional state space and an abstract copying transformation which will copy these two vectors. If the vectors are orthogonal, then perfect copies can be made. If they are not, then errors will be introduced. The size of the error depends on the inner product of the two original vectors. We derive a lower bound for the amount of noise induced by quantum copying. We examine both copying transformations which produce one copy and transformations which produce many, and show that the quality of each copy decreases as the number of copies increases.

DOI
10.1103/physreva.56.1212

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How well can one copy an arbitrary qubit? To answer this question we consider two arbitrary vectors in a two-dimensional state space and an abstract copying transformation which will copy these two vectors. If the vectors are orthogonal, then perfect copies can be made. If they are not, then errors will be introduced. The size of the error depends on the inner product of the two original vectors. We derive a lower bound for the amount of noise induced by quantum copying. We examine both copying transformations which produce one copy and transformations which produce many, and show that the quality of each copy decreases as the number of copies increases.
We review our recent work on the universal (i.e. input state independent) optimal quantum copying (cloning) of qubits. We present unitary transformations which describe the optimal cloning of a qubit and we present the corresponding quantum logical network. We also present network for an optimal quantum copying ``machine'' (transformation) which produces N+1 identical copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copies, N. In addition, we present the machine which universaly and optimally clones states of quantum objects in arbitrary-dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers.
We analyze how information encoded in quantum systems can be optimally processed. In particular, we investigate copying (cloning) of quantum information (represented as states of 2-level quantum systems--qubits). We present unitary transformations which describe the optimal universal cloning of a qubit. Universality of the transformation guarantees that the fidelity of the cloning does not depend on the input state of the qubit, i.e. all states are cloned equally well. We present network for the optimal universal quantum cloning `machine' (transformation) which produces N + 1 copies from the original qubit. Here again the quality (fidelity) of the copies does not depend on the state of the original and is only a function of the number of copes, N. We also present the `machine' which universally and optimally clones states of quantum objects in arbitrary- dimensional Hilbert spaces. In particular, we discuss universal cloning of quantum registers. In addition to cloning of qubits we analyzed another universal operation-- the Universal NOT. We present the optimal transformation and the corresponding logical network which optimally complements an arbitrary input state of a qubit. We show that the fidelity of the performance of the Universal NOT operation increases as a function of the number of input qubits prepared in the same state.
We study the relative error of the state-dependent N=>L cloning. A copying transformation and dimension of state space are not specified. Only the unitarity of quantum mechanical transformations is used. The proposed approach is based on the notion of the angle between two states. Firstly, the notion of the angle between two states is discussed. The lower bound on the relative error of copying with multiple copies is examined. In addition, the lower bound on the absolute error is then studied. We compare the obtained bounds with the case of maximizing the global fidelity.
We propose new optimality criterion for the estimation of state-dependent cloning. We call this measure the relative error because the one compares the errors in the copies with contiguous size taking into account the similarity of states to be copied. A copying transformation and dimension of state space are not specified. Only the unitarity of quantum mechanical transformations is used. The presented approach is based on the notion of the angle between two states. Firstly, several useful statements simply expressed in terms of angles are proved. Among them there are the spherical triangle inequality and the inequality establishing the upper bound on the modulus of difference between probability distributions generated by two any states for an arbitrary measurement. The tightest lower bound on the relative error is then obtained. Hillery and Buzek originally examined an approximate state-dependent copying and obtained the lower bound on the absolute error. We consider relationship between the size of error and the corresponding probability distributions and obtain the tightest lower bound on the absolute error. Thus, the proposed approach supplements and reinforces the results obtained by Hillery and Buzek. Finally, the basic findings of investigation for the relative error are discussed.
We study the mapping that occurs when a single qubit in an arbitrary state interacts with another qubit in a given, fixed state resulting in some unitary transformation on the two qubit system that, in effect, makes two copies of the first qubit. The general problem of the quality of the resulting copies is discussed using a special representation, a generalization of the usual Schmidt decomposition, of an arbitrary two-dimensional subspace of a tensor product of two two-dimensional Hilbert spaces. We exhibit quantum circuits that can reproduce the results of any two-qubit copying machine of this type. A simple stochastic generalization (using a classical random signal) of the copying machine is also considered. These copying machines provide simple embodiments of previously proposed optimal eavesdropping schemes for the BB84 and B92 quantum cryptography protocols.
A quantum copying machine producing two (in general nonidentical) copies of an arbitrary input state of a two-dimensional Hilbert space (quantum bit) is studied using a quality measure based on distinguishability of states, rather than fidelity. The problem of producing optimal copies is investigated with the help of a Bloch sphere representation, and shown to have a well-defined solution, including cases in which the two copies have an unequal quality, or the quality depends upon the input state (is ``anisotropic'' in Bloch sphere language), or both. A simple quantum circuit yields the optimal copying machine. With a suitable choice of parameters it becomes an optimal eavesdropping machine for some versions of quantum cryptography, or reproduces the Bu\ifmmode \check{z}\else \v{z}\fi{}ek and Hillery result for isotropic copies.