An information-theoretic treatment of quantum dichotomies

Francesco Buscemi; David Sutter; Marco Tomamichel

doi:10.22331/q-2019-12-09-209

An information-theoretic treatment of quantum dichotomies

Francesco Buscemi¹, David Sutter², and Marco Tomamichel³

¹Graduate School of Informatics, Nagoya University, Nagoya, Japan
²Institute for Theoretical Physics, ETH Zurich, Switzerland
³Centre for Quantum Software and Information and School of Computer Science, University of Technology Sydney, Sydney

Published:	2019-12-09, volume 3, page 209
Eprint:	arXiv:1907.08539v4
Doi:	https://doi.org/10.22331/q-2019-12-09-209
Citation:	Quantum 3, 209 (2019).

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Abstract

Given two pairs of quantum states, we want to decide if there exists a quantum channel that transforms one pair into the other. The theory of quantum statistical comparison and quantum relative majorization provides necessary and sufficient conditions for such a transformation to exist, but such conditions are typically difficult to check in practice. Here, by building upon work by Keiji Matsumoto, we relax the problem by allowing for small errors in one of the transformations. In this way, a simple sufficient condition can be formulated in terms of one-shot relative entropies of the two pairs. In the asymptotic setting where we consider sequences of state pairs, under some mild convergence conditions, this implies that the quantum relative entropy is the only relevant quantity deciding when a pairwise state transformation is possible. More precisely, if the relative entropy of the initial state pair is strictly larger compared to the relative entropy of the target state pair, then a transformation with exponentially vanishing error is possible. On the other hand, if the relative entropy of the target state is strictly larger, then any such transformation will have an error converging exponentially to one. As an immediate consequence, we show that the rate at which pairs of states can be transformed into each other is given by the ratio of their relative entropies. We discuss applications to the resource theories of athermality and coherence, where our results imply an exponential strong converse for general state interconversion.

► BibTeX data

@article{Buscemi2019information,
  doi = {10.22331/q-2019-12-09-209},
  url = {https://doi.org/10.22331/q-2019-12-09-209},
  title = {An information-theoretic treatment of quantum dichotomies},
  author = {Buscemi, Francesco and Sutter, David and Tomamichel, Marco},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {3},
  pages = {209},
  month = dec,
  year = {2019}
}

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