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Lower bound on the relative error of mixed-state cloning and related operations

Alexey E. Rastegin

Published Aug 27, 2002• 23 citations
OpenAlex

Abstract

We extend the concept of the relative error to mixed-state cloning and related physical operations, in which the ancilla contains some information about the input state. The lower bound on the relative error is obtained.

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A lower bound on the relative error of mixed-state cloning and related operations

A E Rastegin

We extend the concept of the relative error to mixed-state cloning and related physical operations, in which the ancilla contains some information a priori about the input state. The lower bound on the relative error is obtained. It is shown that this result provides further support for a stronger no-cloning theorem.

arXiv:quant-ph/0208159•Journal of Optics B Quantum and Semiclassical Optics•Oct 16, 2003•35 citations

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Relative error of state-dependent cloning

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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.

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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 $d$-dimensional states of rank at most $r$: producing $M$ additional copies with worst-case fidelity at least $1-\varepsilon$ requires and is achievable with $N=Θ(Mrd/\varepsilon)$ 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 $M=1$, 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.

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Optimal cloning with respect to the relative error

Alexey E. Rastegin

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.

arXiv:0906.5494•Jun 30, 2009•0 citations
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.
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.
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.