THE QUANTUM INTELLIGENCER

A paper now on the arXiv settles an open question in the theory of quantum copying: to clone an unknown mixed state, no scheme can improve upon the straightforward method of random purification followed by Werner's channel. The authors prove that n = Ω(krd/ε) copies are required to prepare k additional copies of a rank-r state in dimension d to fidelity 1−ε, thus matching the upper bound already established for the task. The demonstration proceeds by considering the special case of projector cloning, in which the input is promised to be of the form P/r, where P is a rank-r orthogonal projector. The same techniques yield a like conclusion for the closely related problem of approximate transposition, where n = Θ(krd/ε) copies are shown to be both necessary and sufficient to convert the n-fold tensor power of ρ into a k-copy state with high fidelity to the transpose. The result is one of tight resource accounting, fixing the cost of a common operation with finality; it does not by itself alter the practical art, but it draws a clean boundary around what is possible. So the catalogue of theoretical certainties grows, as it must, one bound at a time. To measure what this paper accomplishes, one must hold the earlier benchmark in view. Werner's pure-state cloner, long the standard instrument, requires on the order of kd/ε copies to produce k additional copies of an unknown pure state at fidelity 1−ε; that bound was known to be tight. The mixed-state case, where the input may be rank-r and the auxiliaries of purification complicate the count, had resisted a matching lower bound until now. The authors' projector-cloning argument supplies that bound, and the transposition corollary closes the account with a Θ(krd/ε) figure on both sides. —Dr. Octavia Blythe Dispatch from The Prepared E0