Quantum cloning in<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mi>d</mml:mi></mml:math>dimensions

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.