arXiv:quant-ph/9803058 · Physical Review Letters

Pauli Cloning of a Quantum Bit

Nicolas J. Cerf

Published May 8, 2000• 235 citations

Abstract

A family of asymmetric quantum cloning machines is introduced that produce two approximate copies of a single quantum bit, each copy emerging from a Pauli channel. A no-cloning inequality is derived, describing the balance between the quality of the copies. The Pauli cloning machine is also shown to put a limit on the quantum capacity of Pauli channels.

DOI: 10.1103/physrevlett.84.4497

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arXiv:quant-ph/9803058 · Physical Review Letters

Pauli Cloning of a Quantum Bit

Nicolas J. Cerf

Published May 8, 2000• 235 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevlett.84.4497

Related papers

Asymmetric quantum cloning machines

Nicolas J. Cerf

A family of asymmetric cloning machines for quantum bits and $N$ -dimensional quantum states is introduced. These machines produce two approximate copies of a single quantum state that emerge from two distinct channels. In particular, an asymmetric Pauli cloning machine is defined that makes two imperfect copies of a quantum bit, while the overall input-to-output operation for each copy is a Pauli channel. A no-cloning inequality is derived, characterizing the impossibility of copying imposed by quantum mechanics. If $p$ and $p'$ are the probabilities of the depolarizing channels associated with the two outputs, the domain in $(\\sqrt{p},\\sqrt{p'})$ -space located inside a particular ellipse representing close-to-perfect cloning is forbidden. This ellipse tends to a circle when copying an $N$ -dimensional state with $N\\to\\infty$ , which has a simple semi-classical interpretation. The symmetric Pauli cloning machines are then used to provide an upper bound on the quantum capacity of the Pauli channel of probabilities $p_x$ , $p_y$ and $p_z$ . The capacity is proven to be vanishing if $(\\sqrt{p_x},\\sqrt{p_y},\\sqrt{p_z})$ lies outside an ellipsoid whose pole coincides with the depolarizing channel that underlies the universal cloning machine. Finally, the tradeoff between the quality of the two copies is shown to result from a complementarity akin to Heisenberg uncertainty principle.A family of asymmetric cloning machines for quantum bits and $N$-dimensional quantum states is introduced. These machines produce two approximate copies of a single quantum state that emerge from two distinct channels. In particular, an asymmetric Pauli cloning machine is defined that makes two imperfect copies of a quantum bit, while the overall input-to-output operation for each copy is a Pauli channel. A no-cloning inequality is derived, characterizing the impossibility of copying imposed by quantum mechanics. If $p$ and $p'$ are the probabilities of the depolarizing channels associated with the two outputs, the domain in $(\\sqrt{p},\\sqrt{p'})$-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. This ellipse tends to a circle when copying an $N$-dimensional state with $N\\to\\infty$, which has a simple semi-classical interpretation. The symmetric Pauli cloning machines are then used to provide an upper bound on the quantum capacity of the Pauli channel of probabilities $p_x$, $p_y$ and $p_z$. The capacity is proven to be vanishing if $(\\sqrt{p_x},\\sqrt{p_y},\\sqrt{p_z})$ lies outside an ellipsoid whose pole coincides with the depolarizing channel that underlies the universal cloning machine. Finally, the tradeoff between the quality of the two copies is shown to result from a complementarity akin to Heisenberg uncertainty principle.

Acta Physica Slovaca Reviews and TutorialsJan 1, 19985 citations

Asymmetric quantum cloning in any dimension

Nicolas J. Cerf

A family of asymmetric cloning machines for N-dimensional quantum states is introduced. These machines produce two imperfect copies of a single state that emerge from non-identical Heisenberg channels. The trade-off between the quality of the copies imposed by quantum mechanics is shown to result from a complementarity akin to the Heisenberg uncertainty principle. More specifically, the probability distributions of the error operators affecting the two copies are the square modulus of two functions related by a Fourier transform. A no-cloning inequality is derived for the special case of isotropic cloners, quantifying the impossibility of perfect cloning: if π a and π b are the depolarizing fractions associated with the two copies, the domain in (π1/2 a , π1/2 b )-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. More generally, an entropic no-cloning uncertainty relation is also discussed. Finally, the class of asymmetric cloning machines for quantum bits is investigated in detail, and a connection with the capacity of the Pauli channel is displayed.A family of asymmetric cloning machines for N-dimensional quantum states is introduced. These machines produce two imperfect copies of a single state that emerge from non-identical Heisenberg channels. The trade-off between the quality of the copies imposed by quantum mechanics is shown to result from a complementarity akin to the Heisenberg uncertainty principle. More specifically, the probability distributions of the error operators affecting the two copies are the square modulus of two functions related by a Fourier transform. A no-cloning inequality is derived for the special case of isotropic cloners, quantifying the impossibility of perfect cloning: if π a and π b are the depolarizing fractions associated with the two copies, the domain in (π1/2 a , π1/2 b )-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. More generally, an entropic no-cloning uncertainty relation is also discussed. Finally, the class of asymmetric cloning machines for quantum bits is investigated in detail, and a connection with the capacity of the Pauli channel is displayed.

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Abstrac t. A family of asymmetric cloning machines for N-dimensional quantum states is introduced. These machines produce two imperfect copies of a single state that emerge from non-identical Heisenberg channels. The trade-oA between the quality of the copies imposed by quantum mechanics is shown to result from a complementarity akin to the Heisenberg uncertainty principle. More speci® cally, the probability distributions of the error operators aA ecting the two copies are the square modulus of two functions related by a Fourier transform. A no-cloning inequality is derived for the special case of isotropic cloners, quantifying the impossibility of perfect cloning: if oa and ob are the depolarizing fractions associated with the two copies, the domain in …o 1=2 a ;o 1=2 b †-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. More generally, an entropic no-cloning uncertainty relation is also discussed. Finally, the class of asymmetric cloning machines for quantum bits is investigated in detail, and a connection with the capacity of the Pauli channel is displayed.Abstrac t. A family of asymmetric cloning machines for N-dimensional quantum states is introduced. These machines produce two imperfect copies of a single state that emerge from non-identical Heisenberg channels. The trade-oA between the quality of the copies imposed by quantum mechanics is shown to result from a complementarity akin to the Heisenberg uncertainty principle. More speci® cally, the probability distributions of the error operators aA ecting the two copies are the square modulus of two functions related by a Fourier transform. A no-cloning inequality is derived for the special case of isotropic cloners, quantifying the impossibility of perfect cloning: if oa and ob are the depolarizing fractions associated with the two copies, the domain in …o 1=2 a ;o 1=2 b †-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. More generally, an entropic no-cloning uncertainty relation is also discussed. Finally, the class of asymmetric cloning machines for quantum bits is investigated in detail, and a connection with the capacity of the Pauli channel is displayed.

Jan 1, 20000 citations

Pauli Cloners for Pauli Channels

S. F. Kerstan, M. Gallezot, T. Decker +2 more

We present a quantum circuit architecture for the one-to-two cloning of $N$ -qubit registers. It implements the broad class of Pauli cloners by extending the Niu--Griffiths architecture to multi-qubit systems. In the single-qubit case, we provide explicit constructions for asymmetric universal, phase covariant and biased cloners. We explore the fundamental relationship between Pauli errors, mutually unbiased bases and Pauli cloning. Furthermore, we demonstrate how Pauli cloners can be tailored to specific noise models in the context of quantum communication, especially quantum key distribution.We present a quantum circuit architecture for the one-to-two cloning of $N$-qubit registers. It implements the broad class of Pauli cloners by extending the Niu--Griffiths architecture to multi-qubit systems. In the single-qubit case, we provide explicit constructions for asymmetric universal, phase covariant and biased cloners. We explore the fundamental relationship between Pauli errors, mutually unbiased bases and Pauli cloning. Furthermore, we demonstrate how Pauli cloners can be tailored to specific noise models in the context of quantum communication, especially quantum key distribution.

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Quantum copying: Beyond the no-cloning theorem

Vladimír Bužek, Mark S. Hillery

We analyze the possibility of copying (that is, cloning) arbitrary states of a quantum-mechanical spin-1/2 system. We show that there exists a ``universal quantum-copying machine'' (i.e., transformation) which approximately copies quantum-mechanical states such that the quality of its output does not depend on the input. We also examine a machine which combines a unitary transformation and a selective measurement to produce good copies of states in the neighborhood of a particular state. We discuss the problem of measurement of the output states. \textcopyright{} 1996 The American Physical Society.We analyze the possibility of copying (that is, cloning) arbitrary states of a quantum-mechanical spin-1/2 system. We show that there exists a ``universal quantum-copying machine'' (i.e., transformation) which approximately copies quantum-mechanical states such that the quality of its output does not depend on the input. We also examine a machine which combines a unitary transformation and a selective measurement to produce good copies of states in the neighborhood of a particular state. We discuss the problem of measurement of the output states. \textcopyright{} 1996 The American Physical Society.

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Quantum cloning of identical mixed qubits

Heng Fan, B.-Y. Liu, Kangjie Shi

Quantum cloning of two identical mixed qubits $\rho \otimes \rho$ is studied. We propose the quantum cloning transformations not only for the triplet (symmetric) states but also for the singlet (antisymmetric) state. We can copy these two identical mixed qubits to $M$ ( $M\ge 2$ ) copies. This quantum cloning machine is optimal in the sense that the shrinking factor between the input and the output single qubit achieves the upper bound. The result shows that we can copy two identical mixed qubits with the same quality as that of two identical pure states.Quantum cloning of two identical mixed qubits $\rho \otimes \rho$ is studied. We propose the quantum cloning transformations not only for the triplet (symmetric) states but also for the singlet (antisymmetric) state. We can copy these two identical mixed qubits to $M$ ($M\ge 2$) copies. This quantum cloning machine is optimal in the sense that the shrinking factor between the input and the output single qubit achieves the upper bound. The result shows that we can copy two identical mixed qubits with the same quality as that of two identical pure states.

arXiv:quant-ph/0601017Quantum Information and ComputationJul 1, 20079 citations

A family of asymmetric cloning machines for quantum bits and

N

-dimensional quantum states is introduced. These machines produce two approximate copies of a single quantum state that emerge from two distinct channels. In particular, an asymmetric Pauli cloning machine is defined that makes two imperfect copies of a quantum bit, while the overall input-to-output operation for each copy is a Pauli channel. A no-cloning inequality is derived, characterizing the impossibility of copying imposed by quantum mechanics. If

p

and

p'

are the probabilities of the depolarizing channels associated with the two outputs, the domain in

(\\sqrt{p},\\sqrt{p'})

-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. This ellipse tends to a circle when copying an

N

-dimensional state with

N\\to\\infty

, which has a simple semi-classical interpretation. The symmetric Pauli cloning machines are then used to provide an upper bound on the quantum capacity of the Pauli channel of probabilities

p_x

p_y

and

p_z

. The capacity is proven to be vanishing if

(\\sqrt{p_x},\\sqrt{p_y},\\sqrt{p_z})

lies outside an ellipsoid whose pole coincides with the depolarizing channel that underlies the universal cloning machine. Finally, the tradeoff between the quality of the two copies is shown to result from a complementarity akin to Heisenberg uncertainty principle.

We present a quantum circuit architecture for the one-to-two cloning of

N

-qubit registers. It implements the broad class of Pauli cloners by extending the Niu--Griffiths architecture to multi-qubit systems. In the single-qubit case, we provide explicit constructions for asymmetric universal, phase covariant and biased cloners. We explore the fundamental relationship between Pauli errors, mutually unbiased bases and Pauli cloning. Furthermore, we demonstrate how Pauli cloners can be tailored to specific noise models in the context of quantum communication, especially quantum key distribution.

Quantum cloning of two identical mixed qubits

\rho \otimes \rho

is studied. We propose the quantum cloning transformations not only for the triplet (symmetric) states but also for the singlet (antisymmetric) state. We can copy these two identical mixed qubits to

M

(

M\ge 2

) copies. This quantum cloning machine is optimal in the sense that the shrinking factor between the input and the output single qubit achieves the upper bound. The result shows that we can copy two identical mixed qubits with the same quality as that of two identical pure states.