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arXiv:1311.2772 · Physical Review A

Group-representation approach to 1→N universal quantum cloning machines

Michał Studziński, Piotr Ćwikliński, Michał Horodecki, Marek Mozrzymas

Published May 22, 2014• 14 citations
arXivHTMLPDF

Abstract

In this work, we revisit the problem of finding an admissible region of fidelities obtained after the application of an arbitrary $1\ensuremath{\rightarrow}N$ universal quantum cloner which has been recently solved in A. Kay et al. [Quant. Inf. Comput 13, 880 (2013)] from the side of cloning machines. Using group-theory formalism, we show that the allowed region for fidelities can be alternatively expressed in terms of overlaps of pure states with recently found irreducible representations of the commutant $U\ensuremath{\bigotimes}U\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}U\ensuremath{\bigotimes}{U}^{*}$, which gives the characterization of the allowed region where states being cloned are a figure of merit. Additionally, it is sufficient to take pure states with real coefficients only, which makes calculations simpler. To obtain the allowed region, we make a convex hull of possible ranges of fidelities related to a given irrep. Subsequently, two cases, $1\ensuremath{\rightarrow}2$ and $1\ensuremath{\rightarrow}3$ cloners, are studied for different dimensions of states as illustrative examples.

DOI
10.1103/physreva.89.052322

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In this work, we revisit the problem of finding an admissible region of fidelities obtained after the application of an arbitrary 1\ensuremath→N1\ensuremath{\rightarrow}N1\ensuremath→N universal quantum cloner which has been recently solved in A. Kay et al. [Quant. Inf. Comput 13, 880 (2013)] from the side of cloning machines. Using group-theory formalism, we show that the allowed region for fidelities can be alternatively expressed in terms of overlaps of pure states with recently found irreducible representations of the commutant U\ensuremath⨂U\ensuremath⨂\ensuremath⋯\ensuremath⨂U\ensuremath⨂U∗U\ensuremath{\bigotimes}U\ensuremath{\bigotimes}\ensuremath{\cdots}\ensuremath{\bigotimes}U\ensuremath{\bigotimes}{U}^{*}U\ensuremath⨂U\ensuremath⨂\ensuremath⋯\ensuremath⨂U\ensuremath⨂U∗, which gives the characterization of the allowed region where states being cloned are a figure of merit. Additionally, it is sufficient to take pure states with real coefficients only, which makes calculations simpler. To obtain the allowed region, we make a convex hull of possible ranges of fidelities related to a given irrep. Subsequently, two cases, 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 and 1\ensuremath→31\ensuremath{\rightarrow}31\ensuremath→3 cloners, are studied for different dimensions of states as illustrative examples.
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 investigate the universal asymmetric cloning of states in a Hilbert space of arbitrary dimension. We derive the class of optimal and fully asymmetric 1→31\rightarrow 31→3 cloners, which produce three copies, each having a different fidelity. A simple parametric expression for the maximum achievable cloning fidelity triplets is then provided. As a side-product, we also prove the optimality of the 1→21\rightarrow 21→2 asymmetric cloning machines that have been proposed in the literature.
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.
Beyond the no-cloning theorem, the universal symmetric quantum cloning machine was first addressed by Bu\ifmmode \check{z}\else \v{z}\fi{}ek and Hillery. Here, we realized the $\stackrel{\ensuremath{\rightarrow}}{1}$2 qubit Bu\ifmmode \check{z}\else \v{z}\fi{}ek-Hillery cloning machine with linear optical devices. This method relies on the representation of several qubits by a single photon. We showed that the fidelities between the two output qubits and the original qubit are both 56\frac{5}{6}65​ (which proved to be the optimal fidelity of $\stackrel{\ensuremath{\rightarrow}}{1}$2 qubit universal cloner) for arbitrary input pure states.
We investigate the problem of cloning a set of states that is invariant under the action of an irreducible group representation. We then characterize the cloners that are extremal in the convex set of group covariant cloning machines, among which one can restrict the search for optimal cloners. For a set of states that is invariant under the discrete Weyl-Heisenberg group, we show that all extremal cloners can be unitarily realized using the so-called double-Bell states, whence providing a general proof of the popular ansatz used in the literature for finding optimal cloners in a variety of settings. Our result can also be generalized to continuous-variable optimal cloning in infinite dimensions, where the covariance group is the customary Weyl-Heisenberg group of displacements.