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arXiv:quant-ph/9805024 · Journal of Modern Optics

Asymmetric quantum cloning in any dimension

Nicolas J. Cerf

Published Feb 1, 2000• 194 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1080/09500340008244036

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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.
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.
A family of asymmetric cloning machines for quantum bits and NNN-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 ppp and p′p'p′ are the probabilities of the depolarizing channels associated with the two outputs, the domain in (sqrtp,sqrtp′)(\\sqrt{p},\\sqrt{p'})(sqrtp,sqrtp′)-space located inside a particular ellipse representing close-to-perfect cloning is forbidden. This ellipse tends to a circle when copying an NNN-dimensional state with NtoinftyN\\to\\inftyNtoinfty, 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 pxp_xpx​, pyp_ypy​ and pzp_zpz​. The capacity is proven to be vanishing if (sqrtpx,sqrtpy,sqrtpz)(\\sqrt{p_x},\\sqrt{p_y},\\sqrt{p_z})(sqrtpx​,sqrtpy​,sqrtpz​) 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 study the problem of universal quantum cloning – taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to perfectly reproduce the state, how well the copies can be cloned can be quantified using the fidelity. We examine how individual fidelities can be traded against each other, and how different fidelity measures can be incorporated. The broadly applicable formalism into which we transform the cloning problem is described as a series of quadratic constraints which are amenable to mathematical and computational scrutiny. As such, we reproduce all known results on optimal universal cloning, and push the recent results on asymmetric cloning much further, giving new trade-off relations between fidelities for broad classes of optimal cloning machines. We also provide substantial evidence that motivates why other parameter ranges (number of input copies) have not, and will not, yield to similar analysis.
We study the problem of universal quantum cloning -- taking several identical copies of a pure but unknown quantum state and producing further copies. While it is well known that it is impossible to perfectly reproduce the state, how well the copies can be cloned can be quantified using the fidelity. We examine how individual fidelities can be traded against each other, and how different fidelity measures can be incorporated. The broadly applicable formalism into which we transform the cloning problem is described as a series of quadratic constraints which are amenable to mathematical and computational scrutiny. As such, we reproduce all known results on optimal universal cloning, and push the recent results on asymmetric cloning much further, giving new trade-off relations between fidelities for broad classes of optimal cloning machines. We also provide substantial evidence that motivates why other parameter ranges (number of input copies) have not, and will not, yield to similar analysis.