### On monotone metrics' of the classical channel space:non-asymptotic theoryOn monotone metrics' of the classical channel space:non-asymptotic theory

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 Author Matsumoto, Keiji Source arXiv.org Content type Text File Format PDF Date of Submission 2010-01-20 Language English
 Subject Domain (in DDC) Computer science, information & general works ♦ Natural sciences & mathematics ♦ Physics Subject Keyword Quantum Physics ♦ physics:quant-ph Abstract The aim of the manuscript is to characterize monotone metric' in the space of Markov map. Here, metric' means the square of the norm defined on the tangent space, and not necessarily induced from an inner product (this property hereafter will be called inner-product-assumption), different from usual metric used in differential geometry. As for metrics in So far, there have been plenty of literatures on the metric in the space of probability distributions and quantum states. Among them, Cencov proved the monotone metric in probability distribution space is unique up to constant multiple, and identical to Fisher information metric. Petz characterized all the monotone metrics in the quantum state space using operator mean. As for channels, however, only a little had been known. In this paper, we impose monotonicity by concatenation of channels before and after the given channel families, and invariance by tensoring identity channels. (Notably, we do not use the inner-product-assumption.) To obtain this result, resource conversion' technique, which is widely used in quantum information, is used. We consider distillation from and formation to a family of channels. Under these axioms, we identify the largest and the smallest metrics'. Interestingly, they are not induced from any inner product, i.e., not a metric. Indeed, one can prove that any `metric' satisfying our axioms can not be a metric. This result has some impact on the axiomatic study of the monotone metric in the space of classical and quantum states, since both conventional theory relies on the inner-product-assumption. Also, we compute the lower and the upper bound for some concrete examples. Educational Use Research Learning Resource Type Article