Economical quantum cloning in any dimension

Abstract

The possibility of cloning a $d$-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 $d$-dimensional $1\ensuremath{\rightarrow}2$ phase-covariant cloner, which optimally clones all balanced superpositions with arbitrary phases, can be realized economically only in dimension $d=2$. The used ansatz is supported by numerical evidence up to $d=7$. An economical phase-covariant cloner can nevertheless be constructed for $d>2$, 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 $d=2$.