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arXiv:2205.07265 · Physical Review A

Unification of coherence and quantum correlations in tripartite systems

Dong-Dong Dong, Geng-Biao Wei, Xue‐Ke Song, Dong Wang, Ye Liu

Published Oct 13, 2022• 21 citations
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Abstract

In quantum resource theories (QRTs), evidence of intrinsic connections between coherence and quantum correlations, including Bell nonlocality, entanglement, quantum steering, and so on, exists. However, building these relationships is a vital yet challenging task in multipartite quantum systems. Here, we focus on a unified framework of interpreting the interconversions among coherence and different quantum correlations in tripartite systems. In particular, an exact relation between the generalized geometric measure and the genuinely multipartite concurrence is derived for tripartite entanglement states. Then we obtain the trade-off relation between the first-order coherence and the genuine tripartite entanglement by the genuinely multipartite concurrence and concurrence fill. Furthermore, the trade-off relation between the maximum steering inequality violation and concurrence fill for arbitrary three-qubit states is found. In addition, we investigate the close relation between the maximum steering inequality violation and the first-order coherence. The results show that coherence and quantum correlations are intrinsically related and can be converted to one another in the framework of QRTs.

DOI
10.1103/physreva.106.042415

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In quantum resource theories (QRTs), evidence of intrinsic connections between coherence and quantum correlations, including Bell nonlocality, entanglement, quantum steering, and so on, exists. However, building these relationships is a vital yet challenging task in multipartite quantum systems. Here, we focus on a unified framework of interpreting the interconversions among coherence and different quantum correlations in tripartite systems. In particular, an exact relation between the generalized geometric measure and the genuinely multipartite concurrence is derived for tripartite entanglement states. Then we obtain the trade-off relation between the first-order coherence and the genuine tripartite entanglement by the genuinely multipartite concurrence and concurrence fill. Furthermore, the trade-off relation between the maximum steering inequality violation and concurrence fill for arbitrary three-qubit states is found. In addition, we investigate the close relation between the maximum steering inequality violation and the first-order coherence. The results show that coherence and quantum correlations are intrinsically related and can be converted to one another in the framework of QRTs.
Coherence and nonlocal correlations are two fundamental features of quantum mechanics and the relations between them have received much attention recently. While the relations of coherence to Bell nonlocality and to entanglement have been established, an interesting question left open is how to quantitatively characterize the relation of coherence to steering, which is one kind of nonlocal correlation lying between Bell nonlocality and entanglement. In this paper, we address this question by considering the first-order coherence and the three-setting linear steering inequality for a two-qubit system. Interestingly, we find that the first-order coherence for both subsystems has a complementary relation with the maximum violation of the three-setting linear steering inequality. A weighted sum of their squares keeps invariant under unitary operations, although each of them changes in general. This suggests that the maximum violation of the three-setting linear steering inequality can be taken as a measure of hidden coherence unveiled by unitary operations.
Abstract Quantum resources play crucial roles for displaying superiority in many quantum communication and computation tasks. To reveal the intrinsic relations hidden in these quantum resources, many efforts have been made in recent years. In this work, the correlations of the tripartite W‐type states based on bipartite quantum resources are investigated. The inter‐relations among the degree of coherence, concurrence, Bell nonlocality, and purity are presented. Considering Bell nonlocal and Bell local (satisfied the Clauser–Horne–Shimony–Holt inequality) states for the two‐qubit subsystems derived from the tripartite W‐type states, exact lower and upper boundaries of the degree of coherence versus concurrence are obtained. Interestingly, exact relation among the degree of coherence, concurrence, and purity is obtained. Moreover, coherence is also closely related to entanglement in two specific scenarios: the tripartite W‐type state under decoherence and a practical system for a renormalized spin‐1/2 Heisenberg model.
We put forward complementary relations of entanglement, coherence, steering inequality violation, and Bell nonlocality for arbitrary three-qubit states. We show that two families of genuinely entangled three-qubit pure states with a single parameter exist, and they exhibit maximum coherence and steering inequality violation for a fixed amount of negativity, respectively. It is found that the negativity is exactly equal to the geometric mean of bipartite concurrences for the three-qubit pure states, although the negativity is always less than or equal to the latter for three-qubit mixed states. Moreover, the complementary relation between negativity and first-order coherence for tripartite entanglement states is established. Furthermore, we investigate the close relation between the negativity and the maximum steering inequality violation. In addition, the complementary relation between negativity and the maximum Bell-inequality violation for arbitrary three-qubit states is obtained. The results provide reliable evidence of fundamental connections among entanglement, coherence, steering inequality violation, and Bell nonlocality.
Steerability is a characteristic of quantum correlations lying between entanglement and Bell nonlocality. Understanding how these steering correlations can be shared between different parties has profound applications in ensuring the security of quantum communication protocols. Here we show that at most two bipartite reduced states of a three-qubit state can violate the three-setting CJWR linear steering inequality, contrary to the two-setting linear steering inequality. This result explains that quantum steering correlations have limited shareability properties apart from the conventional 'nonshareable' monogamy constraint. In contrast to the two-measurement-setting scenario, the three-setting scenario turns out to be more useful for developing a deeper understanding of the shareability of tripartite steering correlations. Apart from the distribution of steering correlations, several relations between reduced bipartite steering, different measures of bipartite entanglement of reduced states, and genuine tripartite entanglement are presented here. The results enable the detection of different kinds of tripartite entanglement.
Entanglement and coherence are two essential quantum resources for quantum information processing. A natural question arises of whether there are direct link between them. And by thinking about this question, we propose a new measure for quantum state that contains concurrence and is called intrinsic concurrence. Interestingly, we discover that the intrinsic concurrence is always complementary to coherence. Note that the intrinsic concurrence is related to the concurrence of a special pure state ensemble. In order to explain the trade-off relation more intuitively, we apply it in some composite systems composed by a single-qubit state coupling four typical noise channels with the aim at illustrating their mutual transformation relationship between their coherence and intrinsic concurrence. This unified trade-off relation will provide more flexibility in exploiting one resource to perform quantum tasks and also provide credible theoretical basis for the interconversion of the two important quantum resources.
We investigate the relation between tripartite entanglement of the three-qubit system and the maximum steering inequality violation of the reduced two-qubit states. Firstly, it is found that a single parameter family of entangled three-qubit pure states have the maximum steering inequality violation among all of the three-qubit pure states for a fixed amount of tripartite entanglement. The tripartite entanglement is quantified by the genuinely multipartite concurrence, generalized geometric measure, and tangle. Subsequently, the complementary relation between tripartite entanglement and the maximum steering inequality violation for an arbitrary three-qubit pure state can be established. Particularly, the result also holds for three-qubit mixed states if the entanglement measure is tangle. The complementary relations indicate that the maximum steering inequality violation of the reduced two-qubit system is at the expense of tripartite entanglement.