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Physical Review A

Optimal real state cloning in d dimensions

Wen‐Hai Zhang, Tao Wu, Liu Ye, Dai Jie-lin

Published Apr 20, 2007• 53 citations
Publisher

Abstract

We present an explicit transformation of optimal asymmetric $1\ensuremath{\rightarrow}2$ real state cloning in two dimensions. Interestingly, the distribution of the two fidelities of the cloning is the same as its counterpart in optimal asymmetric phase-covariant cloning. We also generalize the cloner to the $d$-dimensional case and derive the explicit transformation of the optimal symmetric $1\ensuremath{\rightarrow}2$ real state cloning in the $d$-dimensional system. The maximal fidelity of two clones obtained coincides with the theoretical value [P. Vavez and N. J. Cerf, Phys. Rev. A 68, 032313 (2003)].

DOI
10.1103/physreva.75.044303

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We present an explicit transformation of optimal asymmetric 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 real state cloning in two dimensions. Interestingly, the distribution of the two fidelities of the cloning is the same as its counterpart in optimal asymmetric phase-covariant cloning. We also generalize the cloner to the ddd-dimensional case and derive the explicit transformation of the optimal symmetric 1\ensuremath→21\ensuremath{\rightarrow}21\ensuremath→2 real state cloning in the ddd-dimensional system. The maximal fidelity of two clones obtained coincides with the theoretical value [P. Vavez and N. J. Cerf, Phys. Rev. A 68, 032313 (2003)].
We generalize three symmetric cloners to the asymmetric cases and present three explicit cloning transformations as well as their corresponding fidelity distributions in d dimensions. The three asymmetric cloners, including the optimal asymmetric phase-covariant cloning (APCC) and the suboptimal asymmetric economical phase-covariant cloning (AEPCC) working without ancilla and the optimal asymmetric real state cloning (ARSC), together with the optimal asymmetric universal quantum cloning (AUQC) construct a generic cloning, where the quantum information of initial systems of different pure input states in d dimensions with their information not completely known can be optimally distributed to different final systems. By comparison of the fidelity distributions of the four asymmetric cloners, some interesting results can be obtained.
We consider cloning transformations of d-dimensional states of the form eiϕ0∣0>+eiϕ1∣1>+...+eiϕd−1∣d−1>e^{i\phi_0}|0> + e^{i\phi_1}|1> +...+ e^{i\phi_{d-1}}|d-1>eiϕ0​∣0>+eiϕ1​∣1>+...+eiϕd−1​∣d−1> that are covariant with respect to rotations of the phases ϕi\phi_iϕi​'s. The optimal cloning maps are easily obtained using a well-defined general characterization of state-dependent 1→21 \rightarrow 21→2 cloning transformations in arbitrary dimensions. Our results apply to symmetric as well as asymmetric cloners, so that the balance between the fidelity of the two clones can be analyzed.
In this work, we study asymmetric (1→2)(1\to2)(1→2) quantum cloning for pure states belonging to orbits generated by the orthogonal group. For every dimension (d≥3)(d\geq3)(d≥3), we determine the full region of achievable pairs of average single-copy fidelities. Using orthogonal covariance and the Brauer algebra, we reduce the optimization over all CPTP maps to a finite-dimensional spectral problem and construct explicit optimal cloning channels. We then apply the general results to four-mode fermionic states in the even-parity sector. Using triality, we identify the fixed-concurrence fermionic families with the orthogonal orbits considered in the first part of the paper. In particular, for pure Gaussian states we show that the symmetric locally optimal cloner is unique. We also compare local and global cloning and find that the channel optimal for the joint two-copy fidelity is different from the one optimal for the single-copy fidelities. Finally, we show that the locally optimal cloner cannot be implemented using only operations which preserve fermionic Gaussian states.