Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:quant-ph/0108014

Some bounds for quantum copying

Alexey E. Rastegin

Published Aug 3, 2001• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.48550/arxiv.quant-ph/0108014

Related papers

Some bounds for quantum copying with multiple copies

Alexey E. Rastegin

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.

arXiv:quant-ph/0111085•Nov 15, 2001•0 citations

Relative error of state-dependent cloning

Alexey E. Rastegin

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.

Physical Review A•Oct 10, 2002•65 citations

Indeterminate Space-Time Quantum Mechanics, Geometrical Interpretation Of Quantum Information Compression, Hallgren's Efficient Quantum Algorithm For Solving Pell's Equation, Stronger No-Cloning Theorem, Quantum Template Matching, Perturbative And Nonpert

K.N.P. Kumar

Bužek and M. Hillery analyze the possibility of (e) copying (that is, cloning) arbitrary states of (e) a quantum-mechanical spin-1/2 system. Authors show that there exists (eb) universal quantum-copying machine‘‘ (i.e., transformation) which approximately copies quantum-mechanical states such that (e) the quality of its output does not (e) depend on the input. Authors also examine a machine which combines a unitary transformation and (e&eb) a selective measurement to produce good copies of (e) states in the neighborhood of a particular state. Authors discuss the problem of (e) measurement of the output states. © 1996 The American Physical Society Received 5 February 1996 DOI:http://dx.doi.org/10.1103/PhysRevA.54.1844 Quantum copying: Beyond the no-cloning theorem V. Bužek and M. Hillery Phys. Rev. A 54, 1844 – Published 1 September 1996 STOP Authors derive a necessary and sufficient condition for two pure states, each entangled in two remote systems, to be clonable by the sequential access to the two systems. The result shows (eb) to what extent... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-496948-501458.pdf

Advances in physics theories and applications•Jan 1, 2013•0 citations

Optimal copy complexity of quantum state cloning

Sangwoo Jeon, Vaughn Sohn, Changhun Oh

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.

quant-pharXiv:2608.24484•Aug 26, 2026•0 citations

Quantum copying: Fundamental inequalities

Mark S. Hillery, Vladimír Bužek

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.

arXiv:quant-ph/9701034•Physical Review A•Aug 1, 1997•89 citations

Strategies and networks for state-dependent quantum cloning

Anthony Chefles, Stephen Mark Barnett

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.

arXiv:quant-ph/9812035•Physical Review A•Jul 1, 1999•65 citations
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.
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.
Bužek and M. Hillery analyze the possibility of (e) copying (that is, cloning) arbitrary states of (e) a quantum-mechanical spin-1/2 system. Authors show that there exists (eb) universal quantum-copying machine‘‘ (i.e., transformation) which approximately copies quantum-mechanical states such that (e) the quality of its output does not (e) depend on the input. Authors also examine a machine which combines a unitary transformation and (e&eb) a selective measurement to produce good copies of (e) states in the neighborhood of a particular state. Authors discuss the problem of (e) measurement of the output states. © 1996 The American Physical Society Received 5 February 1996 DOI:http://dx.doi.org/10.1103/PhysRevA.54.1844 Quantum copying: Beyond the no-cloning theorem V. Bužek and M. Hillery Phys. Rev. A 54, 1844 – Published 1 September 1996 STOP Authors derive a necessary and sufficient condition for two pure states, each entangled in two remote systems, to be clonable by the sequential access to the two systems. The result shows (eb) to what extent... The full paper: http://www.iiste.org/PDFshare/APTA-PAGENO-496948-501458.pdf
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.
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.
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>MN>MN>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.