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arXiv:1607.01591

Ordering states of Tsallis relative $α$-entropies of coherence

Fu-Gang Zhang, Lian-He Shao, Yu Luo, Yongming Li

Published Jul 6, 2016• 1 citation
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Abstract

In this paper, we study the ordering states with Tsallis relative $α$-entropies of coherence and $l_{1}$ norm of coherence for single-qubit states. We show that any Tsallis relative $α$-entropies of coherence and $l_{1}$ norm of coherence give the same ordering for single-qubit pure states. However, they don't generate the same ordering for some high dimensional pure states, even though these states are pure. We also consider three special Tsallis relative $α$-entropies of coherence, such as $C_{1}$ ,$C_{2}$ and $C_{\frac{1}{2}}$, and show any one of these three measures and $C_{l_{1}}$ will not generate the same ordering for single-qubit mixed states. Furthermore, we find that any two of these three special measures generate different ordering for single-qubit mixed states.

DOI
10.48550/arxiv.1607.01591

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Abstract In this paper, we study coherence-induced state ordering with Tsallis relative entropy of coherence, relative entropy of coherence and l 1 norm of coherence, and give the sufficient conditions of the same state order induced by above coherence measures. First, we show that the above measures give the same ordering for single-qubit states in some special cases. Second, we consider some special states in a d -dimensional quantum system. We show that the above measures generate the same ordering for these special states. Finally, we discuss dynamics of coherence-induced state ordering under Markovian channels. We find amplitude damping channel changes the coherence-induced ordering even though for single-qubit states with fixed mixedness.
Quantifying coherence is an essential endeavor for both quantum mechanical foundations and quantum technologies. We present a bona fide measure of quantum coherence by utilizing the Tsallis relative operator (α,β)-entropy. We first prove that the proposed coherence measure fulfills all the criteria of a well defined coherence measure, including the strong monotonicity in the resource theories of quantum coherence. We then study the ordering of the Tsallis relative operator (α,β)-entropy of coherence, Tsallis relative α-entropies of coherence, Rényi α-entropy of coherence and l1 norm of coherence for both pure and mixed qubit states. This provides a new method for defining new coherence measure and entanglement measure, and also provides a new idea for further study of quantum coherence.
Abstract This paper studies the order relations among various quantum coherence measures in a single-qubit system. Ten representative coherence measures are selected from three categories: distance-based, fidelity-based, and relative entropy-based measures. The analytical expressions for the coherence values are derived in the Bloch sphere representation of the single-qubit state, and the order relations within the same category and across categories are systematically analyzed. The study shows that coherence measures within the same category generally maintain consistent order relations in the single-qubit system. However, there is typically no fixed order relation between fidelity-based coherence measures (represented by geometric coherence) and relative entropy-based coherence measures (represented by α -affinity coherence measures). Yet, in pure single-qubit states and certain restricted sets of mixed states, these two types of coherence measures exhibit the same order relation. The results indicate that the order relations between coherence measures depend not only on their construction but also on the quantum state space structure and the constraints on parameters.
We study the dynamics of coherence-induced state ordering under incoherent channels, particularly four specific Markovian channels: amplitude damping channel, phase damping channel, depolarizing channel and bit flit channel for single-qubit states. We show that the amplitude damping channel, phase damping channel, and depolarizing channel do not change the coherence-induced state ordering by l 1 norm of coherence, relative entropy of coherence, geometric measure of coherence, and Tsallis relative α -entropies, while the bit flit channel does change for some special cases.
We study the dynamics of coherence-induced state ordering under incoherent channels, particularly four specific Markovian channels: amplitude damping channel, phase damping channel, depolarizing channel and bit flit channel for single-qubit states. We show that the amplitude damping channel, phase damping channel, and depolarizing channel do not change the coherence-induced state ordering by l1 norm of coherence, relative entropy of coherence, geometric measure of coherence, and Tsallis relative α-entropies, while the bit flit channel does change for some special cases.