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Physical Review A

Relative error of state-dependent cloning

Alexey E. Rastegin

Published Oct 10, 2002• 65 citations
Publisher

Abstract

We present a criterion for estimation of the quality of state-dependent $\stackrel{\ensuremath{\rightarrow}}{N}L$ cloning. We call this criterion the relative error because one compares the errors in copies with contiguous size, which is not independent of the closeness of the states to be copied. First, we prove several useful statements that maintain the interpretation for the proposed measure. The tightest lower bound on the relative error is then derived. We describe the optimal asymmetric state-dependent cloner that minimizes the relative error.

DOI
10.1103/physreva.66.042304

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We present a criterion for estimation of the quality of state-dependent N\ensuremath→L\stackrel{\ensuremath{\rightarrow}}{N}LN\ensuremath→​L cloning. We call this criterion the relative error because one compares the errors in copies with contiguous size, which is not independent of the closeness of the states to be copied. First, we prove several useful statements that maintain the interpretation for the proposed measure. The tightest lower bound on the relative error is then derived. We describe the optimal asymmetric state-dependent cloner that minimizes the relative error.
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 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.
Due to the no-cloning theorem, the unknown quantum state can only be cloned approximately or exactly with some probability. There are two types of cloners: universal and state-dependent cloner. The optimal universal cloner has been found and can be viewed as a special state-dependent quantum cloner that has no information about the states. In this paper, we investigate the state-dependent cloning when the state set contains more than two states. We get some bounds of the global fidelity for these processes. This method is not dependent on the number of the states contained in the state set. It is also independent of the numbers of copying.
Quantum state cloning is the task of approximately producing additional copies of an unknown quantum state from a finite number of input copies. The optimal cloning fidelity is known exactly for pure states, but no comparable characterization is known for general mixed states. We determine the optimal asymptotic copy complexity for ddd-dimensional states of rank at most rrr: producing MMM additional copies with worst-case fidelity at least 1−ε1-\varepsilon1−ε requires and is achievable with N=Θ(Mrd/ε)N=Θ(Mrd/\varepsilon)N=Θ(Mrd/ε) input copies. Remarkably, the lower bound already holds for a family of states with a fixed flat spectrum, while the matching upper bound is achieved by random purification followed by optimal pure-state cloning. For M=1M=1M=1, we further show that high-fidelity tomography can be coherently converted into cloning with comparable error, revealing an operational origin of the matching cloning and tomography complexities.
We consider the problem of approximate cloning of quantum states: given nnn copies of an unknown state ρ∈Cd×dρ\in \mathbb{C}^{d \times d}ρ∈Cd×d, prepare an (n+k)(n+k)(n+k)-copy state with high fidelity to ρ⊗(n+k)ρ^{\otimes (n+k)}ρ⊗(n+k). Werner's pure state cloner is the optimal channel for the pure state case, and shows that n=Θ(kd/ε)n = Θ(kd/\varepsilon)n=Θ(kd/ε) copies are necessary and sufficient to clone kkk additional copies of an unknown pure state to fidelity 1−ε1-\varepsilon1−ε. The random purification channel gives a straightforward extension of Werner's cloner to mixed state inputs: given nnn copies of a mixed state, randomly purify your input, apply Werner's channel in the larger Hilbert space, and then trace out the auxiliary registers. This gives a mixed state cloner using n=O(krd/ε)n = O(krd/\varepsilon)n=O(krd/ε) copies to clone rank-rrr states. Can one do any better? We show that the answer is no: one must use n=Ω(krd/ε)n = Ω(krd/\varepsilon)n=Ω(krd/ε) copies. We prove our lower bound by studying the special case of projector cloning, in which the input state ρρρ is promised to be of the form P/rP/rP/r, where PPP is a rank-rrr orthogonal projector. As a further application of our techniques, we consider the closely related problem of approximate transposition of quantum states, where one seeks to convert ρ⊗nρ^{\otimes n}ρ⊗n to a kkk-copy state with high fidelity to (ρT)⊗k(ρ^T)^{\otimes k}(ρT)⊗k. Here, we again show n=Θ(krd/ε)n = Θ(krd/\varepsilon)n=Θ(krd/ε) copies are necessary and sufficient for this task.
The relative error of cloning of quantum states with arbitrary prior probabilities is considered. It is assumed that the ancilla may contain some a priori information about the input state to be cloned. The lower bound on the relative error for general cloning scenario is derived. Both the case of two-state set and case of multi-state set are analyzed in details. The treated figure of merit is compared with other optimality criteria. The quantum circuit for optimal cloning of a pair of pure states is constructed.