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
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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.
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[1] Xin Wang and Mark M. Wilde, “Resource theory of asymmetric distinguishability”, arXiv:1905.11629.
[2] Bartosz Regula, Varun Narasimhachar, Francesco Buscemi, and Mile Gu, “Coherence manipulation with dephasing-covariant operations”, arXiv:1907.08606.
[3] Bartosz Regula, Kaifeng Bu, Ryuji Takagi, and Zi-Wen Liu, “Characterizing one-shot distillation in general resource theories”, arXiv:1909.11677.
[4] Philippe Faist, Takahiro Sagawa, Kohtaro Kato, Hiroshi Nagaoka, and Fernando G. S. L. Brandão, “Macroscopic Thermodynamic Reversibility in Quantum Many-Body Systems”, Physical Review Letters 123 25, 250601 (2019).
[5] Soorya Rethinasamy and Mark M. Wilde, “Relative Entropy and Catalytic Relative Majorization”, arXiv:1912.04254.
[6] Michele Dall’Arno, Francesco Buscemi, and Valerio Scarani, “Extension of the Alberti-Ulhmann criterion beyond qubit dichotomies”, arXiv:1910.04294.
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Source: https://quantum-journal.org/papers/q-2019-12-09-209/
