arXiv:quant-ph/0301132 · Physical Review A

Global-fidelity limits of state-dependent cloning of mixed states

Alexey E. Rastegin

Published Sep 10, 2003• 8 citations

Abstract

By relevant modifications, the known global-fidelity limits of state-dependent cloning are extended to mixed quantum states. We assume that the ancilla contains some a priori information about the input state. As it is shown, the obtained results contribute to the stronger no-cloning theorem. An attainability of presented limits is discussed.

DOI: 10.1103/physreva.68.032303

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arXiv:quant-ph/0301132 · Physical Review A

Global-fidelity limits of state-dependent cloning of mixed states

Alexey E. Rastegin

Published Sep 10, 2003• 8 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physreva.68.032303

Related papers

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

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arXiv:quant-ph/0208159Journal of Optics B Quantum and Semiclassical OpticsOct 16, 200335 citations

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

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Aug 27, 200223 citations

Bounds for state-dependent quantum cloning

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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.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.

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State-dependent cloning machines that have so far been considered either deterministically copy a set of states approximately or probablistically copy them exactly. In considering the case of two equiprobable pure states, we derive the maximum global fidelity of N approximate clones given M initial exact copies, where $N>M$ . We also consider strategies that interpolate between approximate and exact cloning. A tight inequality is obtained that expresses a trade-off between the global fidelity and success probability. This inequality is found to tend, in the limit $N\stackrel{\ensuremath{\rightarrow}}{}\ensuremath{\infty}$, to a known inequality that expresses the trade-off between error and inconclusive result probabilities for state-discrimination measurements. Quantum-computational networks are also constructed for the kinds of cloning machine we describe. For this purpose, we introduce two gates: the distinguishability transfer and state separation gates. Their key properties are described and we show how they may be decomposed into basic operations.State-dependent cloning machines that have so far been considered either deterministically copy a set of states approximately or probablistically copy them exactly. In considering the case of two equiprobable pure states, we derive the maximum global fidelity of N approximate clones given M initial exact copies, where $N>M$. We also consider strategies that interpolate between approximate and exact cloning. A tight inequality is obtained that expresses a trade-off between the global fidelity and success probability. This inequality is found to tend, in the limit $N\stackrel{\ensuremath{\rightarrow}}{}\ensuremath{\infty}$, to a known inequality that expresses the trade-off between error and inconclusive result probabilities for state-discrimination measurements. Quantum-computational networks are also constructed for the kinds of cloning machine we describe. For this purpose, we introduce two gates: the distinguishability transfer and state separation gates. Their key properties are described and we show how they may be decomposed into basic operations.

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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.

We determine asymptotic global fidelities of cloning generic mixed states with simple (non-degenerate) full-rank spectra. In fixed dimension

d

, let

n

inputs and

m_n

outputs satisfy

m_n/n\toγ>1

. For a known simple full-rank spectrum

p_1>\cdots>p_d>0

, we prove that the optimal root fidelity between the entire output and the product target state is $$ F_\rm{orb}(γ,p) =\prod_{i<j} \frac{\sqrtγ+\sqrt{q_{ij}(γ-1+q_{ij})}} {γ+q_{ij}}, \qquad q_{ij}=\frac{p_j}{p_i}. $$ When the spectrum is unknown, one spectrum-independent cloner attains $$ F_\rm{univ}(γ,p) =\left(\frac{2\sqrtγ}{1+γ}\right)^{(d-1)/2} F_\rm{orb}(γ,p), $$ uniformly on compact subsets of the simple full-rank chamber and is minimax optimal. The two factors quantify spectral and eigenbasis uncertainty. Our channels combine classical processing of Schur--Weyl labels with a covariant Cartan-sector channel. Local asymptotic normality and a Gaussian fidelity bound against arbitrary channels yield the converses. For flat projector states

ρ=P/r

, with

P

a rank-

r

orthogonal projector, we separately prove the optimum

γ^{-r(d-r)/2}

. Furthermore, we estimate the exact asymptotic fidelity of purify--clone--trace (PCT), and show that our cloners strictly outperform PCT at every fixed gain for simple full-rank spectra and rank-

r

projector states with

1<r<d

N>M

. We also consider strategies that interpolate between approximate and exact cloning. A tight inequality is obtained that expresses a trade-off between the global fidelity and success probability. This inequality is found to tend, in the limit $N\stackrel{\ensuremath{\rightarrow}}{}\ensuremath{\infty}$, to a known inequality that expresses the trade-off between error and inconclusive result probabilities for state-discrimination measurements. Quantum-computational networks are also constructed for the kinds of cloning machine we describe. For this purpose, we introduce two gates: the distinguishability transfer and state separation gates. Their key properties are described and we show how they may be decomposed into basic operations.