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Remote-state preparation in higher dimension and the parallelizable manifold<mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant="italic">S</mml:mi></mml:mrow><mml:mrow><mml:mi mathvariant="italic">n</mml:mi><mml:mi>โ</mml:mi><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math>
This paper proves that the remote-state preparation (RSP) scheme in real Hilbert space can only be implemented when the dimension of the space is 2, 4, or 8. This fact is shown to be related to the parallelizability of the $(n\ensuremath{-}1)$-dimensional sphere ${S}^{n\ensuremath{-}1}.$ When the dimension is 4 and 8 โฆ