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arXiv:2106.16110 · Science China Physics Mechanics and Astronomy

Maximum relative entropy of coherence for quantum channels

Zhi-Xiang Jin, Long-Mei Yang, Shao-Ming Fei, Xianqing Li-Jost, Zhi-Xi Wang, Gui-Lu Long, Cong-Feng Qiao

Published Jun 29, 2021• 21 citations
arXivHTMLPDF
DOI
10.1007/s11433-021-1709-9

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•Oct 22, 2015•4 citations
Abstract For the well-known relative entropy of coherence (REC), a quantum state attains maximum coherence only in optimal bases, which are mutually unbiased bases to the eigenvectors of the given state. In this paper, we explore the conditions of obtaining the maximum of REC under noisy channels and explain the geometric significance of optimal basis form the viewpoint of the Bloch sphere. We also show that the eigenvectors of the state will remain constant throughout the whole process of evolution, as long as the effects of channels induce a synchronous change for the components of the initial state Bloch vector. Furthermore, as a special example, we investigate the dynamics of the maximum REC for any finite dimensional single quantum system under a depolarizing (DP) channel, and obtain an analytical form of maximum coherence for a d -dimensional maximally coherent state. Meanwhile, a concise analytical expression of maximum coherence for two classes of an N -qubit system under the DP channel is also provided.
The operational characterization of quantum coherence is the cornerstone in the development of the resource theory of coherence. We introduce a new coherence quantifier based on maximum relative entropy. We prove that the maximum relative entropy of coherence is directly related to the maximum overlap with maximally coherent states under a particular class of operations, which provides an operational interpretation of the maximum relative entropy of coherence. Moreover, we show that, for any coherent state, there are examples of subchannel discrimination problems such that this coherent state allows for a higher probability of successfully discriminating subchannels than that of all incoherent states. This advantage of coherent states in subchannel discrimination can be exactly characterized by the maximum relative entropy of coherence. By introducing a suitable smooth maximum relative entropy of coherence, we prove that the smooth maximum relative entropy of coherence provides a lower bound of one-shot coherence cost, and the maximum relative entropy of coherence is equivalent to the relative entropy of coherence in the asymptotic limit. Similar to the maximum relative entropy of coherence, the minimum relative entropy of coherence has also been investigated. We show that the minimum relative entropy of coherence provides an upper bound of one-shot coherence distillation, and in the asymptotic limit the minimum relative entropy of coherence is equivalent to the relative entropy of coherence.
In this paper, we consider the cohering power of unitary operation, which is defined by the maximum coherence caused by an unitary operation acting on incoherent states. We obtain that the optimizations can be reduced to a simple discrete set that depends only on the fixed reference basis. Then, we investigate that the de-cohering power of quantum channel, which is defined by the maximum decay of coherence caused by quantum channel acting on the maximally coherent states. Based on the relative entropy, we discuss the cohering power of 1-qubit and 2-qubit unitary operations and the de-cohering power of quantum channel on 1-qubit system. Finally, we study the relation between the coherence and the discord via incoherent operation.