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arXiv:1610.07052 · Journal of Physics A Mathematical and Theoretical

Measuring coherence with entanglement concurrence

Xianfei Qi, Ting Gao, Fengli Yan

Published Jun 1, 2017• 147 citations
arXivHTMLPDF

Abstract

Abstract Quantum coherence is a fundamental manifestation of the quantum superposition principle. Recently, Baumgratz et al (2014 Phys. Rev. Lett . 113 140401) presented a rigorous framework to quantify coherence from the view of theory of physical resource. Here we propose a new valid quantum coherence measure which is a convex roof measure, for a quantum system of arbitrary dimension, essentially using the generalized Gell-Mann matrices. Rigorous proof shows that the proposed coherence measure, coherence concurrence, fulfills all the requirements dictated by the resource theory of quantum coherence measures. Moreover, strong links between the resource frameworks of coherence concurrence and entanglement concurrence is derived, which shows that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. Our work provides a clear quantitative and operational connection between coherence and entanglement based on two kinds of concurrence. This new coherence measure, coherence concurrence, may also be beneficial to the study of quantum coherence.

DOI
10.1088/1751-8121/aa7638

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Abstract Quantum coherence is a fundamental manifestation of the quantum superposition principle. Recently, Baumgratz et al (2014 Phys. Rev. Lett . 113 140401) presented a rigorous framework to quantify coherence from the view of theory of physical resource. Here we propose a new valid quantum coherence measure which is a convex roof measure, for a quantum system of arbitrary dimension, essentially using the generalized Gell-Mann matrices. Rigorous proof shows that the proposed coherence measure, coherence concurrence, fulfills all the requirements dictated by the resource theory of quantum coherence measures. Moreover, strong links between the resource frameworks of coherence concurrence and entanglement concurrence is derived, which shows that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. Our work provides a clear quantitative and operational connection between coherence and entanglement based on two kinds of concurrence. This new coherence measure, coherence concurrence, may also be beneficial to the study of quantum coherence.
Quantum coherence is an essential ingredient in quantum information processing and plays a central role in emergent fields such as nanoscale thermodynamics and quantum biology. However, our understanding and quantitative characterization of coherence as an operational resource are still very limited. Here we show that any degree of coherence with respect to some reference basis can be converted to entanglement via incoherent operations. This finding allows us to define a novel general class of measures of coherence for a quantum system of arbitrary dimension, in terms of the maximum bipartite entanglement that can be generated via incoherent operations applied to the system and an incoherent ancilla. The resulting measures are proven to be valid coherence monotones satisfying all the requirements dictated by the resource theory of quantum coherence. We demonstrate the usefulness of our approach by proving that the fidelity-based geometric measure of coherence is a full convex coherence monotone, and deriving a closed formula for it on arbitrary single-qubit states. Our work provides a clear quantitative and operational connection between coherence and entanglement, two landmark manifestations of quantum theory and both key enablers for quantum technologies.
Quantum coherence is regarded as a fundamental ingredient in quantum information processing tasks. Quantifying coherence of states is a crucial problem, which has attracted a growing interest for scientists. In this paper, we put forward a quantitative measure of coherence based on the angle between quantum states. Rigorous proof shows that the proposed coherence measure fulfills all the axioms dictated by the resource theory of quantum coherence measure. Furthermore, a strong relation between coherence measure based on the angle and entanglement measure based on fidelity is derived.
Based on the theory of quantum mechanics, intrinsic randomness in measurement distinguishes quantum effects from classical ones. From the perspective of states, this quantum feature can be summarized as coherence or superposition in a specific (classical) computational basis. Recently, by regarding coherence as a physical resource, Baumgratz et al. [Phys. Rev. Lett. 113, 140401 (2014)] presented a comprehensive framework for coherence measures. Here, we propose a quantum coherence measure essentially using the intrinsic randomness of measurement. The proposed coherence measure provides an answer to the open question in completing the resource theory of coherence. Meanwhile, we show that the coherence distillation process can be treated as quantum extraction, which can be regarded as an equivalent process of classical random number extraction. From this viewpoint, the proposed coherence measure also clarifies the operational aspect of quantum coherence. Finally, our results indicate a strong similarity between two types of quantumness---coherence and entanglement.
Recently, how to measure quantum coherence is a hot topic in the quantum resource theory. In 2014, Baumgratz et al. proposed a rigorous framework to quantify coherence. In this framework, a particular basis of the Hilbert space is required to define the coherence measure, and hence, the proposed coherence measure of the same quantum state does not remain the same under the basis transformations. In this paper, a measure of coherence based on the normalized modulus of the coherent vector is proposed in coherent vector representation, where the property of the coherent vector under basis transformation is considered. The measure of coherence proposed here is independent of the choice of the basis. The incoherent states and incoherent operators based on this measure can also be well defined as the classical maximal mixed states and the unitary operators respectively. And three important properties can be obtained, namely, non-negativity, convexity and invariance on the incoherent operators.
We establish a general operational one-to-one mapping between coherence measures and entanglement measures: Any entanglement measure of bipartite pure states is the minimum of a suitable coherence measure over product bases. Any coherence measure of pure states, with extension to mixed states by convex roof, is the maximum entanglement generated by incoherent operations acting on the system and an incoherent ancilla. Remarkably, the generalized cnot gate is the universal optimal incoherent operation. In this way, all convex-roof coherence measures, including the coherence of formation, are endowed with (additional) operational interpretations. By virtue of this connection, many results on entanglement can be translated to the coherence setting, and vice versa. As applications, we provide tight observable lower bounds for generalized entanglement concurrence and coherence concurrence, which enable experimentalists to quantify entanglement and coherence of the maximal dimension in real experiments.
To describe the nonclassicality of quantum channels, it is necessary to quantify their properties. In the framework of resource theory of quantum channel, we present three new entanglement measures of channels for a quantum system of arbitrary finite dimension, based on the Choi relative entropy of channels, concurrence, and kkk-ME concurrence, respectively. Rigorous proofs show that these proposed measures fulfill all the requirements dictated by the resource theory of quantum channels, including non-negativity, weak monotonicity, strong monotone, convexity, additivity, and subadditivity. By using the measure, we describe the probability that the initial hypothesis is true in the hypothesis testing of channels. Compared with other known measures of the quantum channels, these measures have an obvious advantage in computability. According to the power of generating entanglement, we give a classification of quantum channels. Moreover, we provide several examples that characterize the properties of quantum channels and the conversions between quantum coherence states and entangled states and between fully separable states and multipartite entangled states.