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arXiv:quant-ph/9804011 · Physical Review A

Quantum cloning in d dimensions

Paolo Zanardi A

Published Nov 1, 1998• 40 citations
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Abstract

The quantum state space S over a $d$-dimensional Hilbert space is represented as a convex subset of a $(D\ensuremath{-}1)$-dimensional sphere ${S}_{D\ensuremath{-}1}\ensuremath{\subset}{\mathbf{R}}^{D},$ where ${D=d}^{2}\ensuremath{-}1.$ Quantum transformations ($\mathrm{CP}$ maps) are then associated with the affine transformations of ${\mathbf{R}}^{D}$ and $N\ensuremath{\mapsto}M$ cloners induce polynomial mappings. In this geometrical setting it is shown that an optimal cloner can be chosen to be covariant and induces a map between reduced density matrices given by a simple contraction of the associated $D$-dimensional Bloch vectors.

DOI
10.1103/physreva.58.3484

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The quantum state space S over a ddd-dimensional Hilbert space is represented as a convex subset of a (D\ensuremath−1)(D\ensuremath{-}1)(D\ensuremath−1)-dimensional sphere SD\ensuremath−1\ensuremath⊂RD,{S}_{D\ensuremath{-}1}\ensuremath{\subset}{\mathbf{R}}^{D},SD\ensuremath−1​\ensuremath⊂RD, where D=d2\ensuremath−1.{D=d}^{2}\ensuremath{-}1.D=d2\ensuremath−1. Quantum transformations (CP\mathrm{CP}CP maps) are then associated with the affine transformations of RD{\mathbf{R}}^{D}RD and N\ensuremath↦MN\ensuremath{\mapsto}MN\ensuremath↦M cloners induce polynomial mappings. In this geometrical setting it is shown that an optimal cloner can be chosen to be covariant and induces a map between reduced density matrices given by a simple contraction of the associated DDD-dimensional Bloch vectors.
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.
Binary coherent state alphabets can be represented in a two-dimensional Hilbert space. We capitalize this formal connection between the otherwise distinct domains of qubits and continuous variable states to map binary phase-shift keyed coherent states onto the Bloch sphere and to derive their quantum-optimal clones. We analyze the Wigner function and the cumulants of the clones, and we conclude that optimal cloning of binary coherent states requires a nonlinearity above second order. We propose several practical and near-optimal cloning schemes and compare their cloning fidelity to the optimal cloner.
The possibility of cloning a ddd-dimensional quantum system without an ancilla is explored, extending on the economical phase-covariant cloning machine for qubits found in Phys. Rev. A 60, 2764 (1999). We prove the impossibility of constructing an economical version of the optimal universal $1\ensuremath{\rightarrow}2$ cloning machine in any dimension. We also show, using an ansatz on the generic form of cloning machines, that the ddd-dimensional $1\ensuremath{\rightarrow}2$ phase-covariant cloner, which optimally clones all balanced superpositions with arbitrary phases, can be realized economically only in dimension d=2d=2d=2. The used ansatz is supported by numerical evidence up to d=7d=7d=7. An economical phase-covariant cloner can nevertheless be constructed for d>2d>2d>2, albeit with a slightly lower fidelity than that of the optimal cloner requiring an ancilla. Finally, using again an ansatz on cloning machines, we show that an economical version of the $1\ensuremath{\rightarrow}2$ Fourier-covariant cloner, which optimally clones the computational basis and its Fourier transform, is also possible only in dimension d=2d=2d=2.
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 ddd-dimensional case and derive the explicit transformation of the optimal symmetric $1\ensuremath{\rightarrow}2$ 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 consider nonuniversal cloning maps, namely, cloning transformations that are covariant under a proper subgroup G of the universal unitary group U(d),U(d),U(d), where d is the dimension of the Hilbert space H\mathcal{H}H of the system to be cloned. We give a general method for optimizing cloning for any cost function. Examples of applications are given for phase-covariant cloning (cloning of equatorial qubits and qutrits) and for the Weyl-Heisenberg group (cloning of ``continuous variables'').
Following the work of Niu and Griffiths, in \emph{Phys.Rev.A 58, 4377(1998)}, we shall investigate the problem, how to design the optimal quantum cloning machines (QCMs) for qubit system, with the help of Bloch-sphere representation. In stead of the quality factor there, the Fiuráusek's optimal condition, where the optimal cloning machine should maximize a convex mixture of the average fidelity, is used as the optimality criterion in present protocol. Almost all of the known optimal QCMs in previous works, the cloning for states with fixed polar angle, the phase-covariant cloning, the universal QCMs, the cloning for two arbitrary pure states, and the mirror phase-covariant cloning, should be discussed in a systematic way. The known results, the optimal fidelities for various input ensembles according to different optimality criteria, are recovered here. Our present scheme also offers a general way of constructing the unitary transformation to realize the optimal cloning.